OPTIMIZATION-BASED MESHING TECHNIQUES FOR MESH QUALITY IMPROVEMENT AND DEFORMATION

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1 The Pennsylvania State University The Graduate School Department of Computer Science and Engineering OPTIMIZATION-BASED MESHING TECHNIQUES FOR MESH QUALITY IMPROVEMENT AND DEFORMATION A Dissertation in Computer Science and Engineering by Jibum Kim c 2012 Jibum Kim Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2012

2 The dissertation of Jibum Kim was reviewed and approved by the following: Suzanne M. Shontz Adjunct Assistant Professor of Computer Science and Engineering Dissertation Advisor Chair of Committee Padma Raghavan Distinguished Professor of Computer Science and Engineering Professor of Information Sciences and Technology Director, Institute for CyberScience Jesse Barlow Professor of Computer Science and Engineering Professor of Statistics Qiang Du Verne M. Willaman Professor of Mathematics Professor of Materials Science and Engineering Raj Acharya Head and Professor of the Computer Science and Engineering Signatures are on file in the Graduate School.

3 iii Abstract High quality meshes are important for the accuracy, stability, and efficiency of numerical techniques in computational simulations involving partial differential equations (PDEs), nonlocal peridynamics, mesh deformation, or shape matching. Several applications involving these mesh-based computational techniques include ocean dynamics (PDEs), surface cracks (nonlocal peridynamics), hydrocephalus (mesh deformation), and object recognition (shape matching). The first part of the dissertation explores the best combinations of mesh quality metrics, preconditioners, and sparse linear solvers for solving various elliptic PDEs, multiobjective mesh optimization methods, and the effect of mesh anisotropy, mesh refinement, and kernel functions on the conditioning of nonlocal peridynamics models. Engineers use various mesh quality improvement methods for solving PDEs to improve the efficiency and accuracy as well as various types of preconditioners and sparse linear solvers for various PDE problems. However, little research has been performed with respect to choosing the most efficient combinations of these three factors for solving various elliptic PDE problems. First, we investigate the effect of choosing various combinations of mesh quality metrics, preconditioners, and sparse linear solvers on the numerical solution of elliptic PDEs. Many PDE-based engineering and scientific applications have multiple requirements for the finite element mesh discretizing the geometric domain; however, most traditional mesh optimization algorithms improve only one aspect of the mesh, Second, we propose a multiobjective mesh optimization framework for simultaneous mesh quality improvement and mesh untangling for PDE-based applications for optimizing two or more aspects of the mesh. Recently, a new paradigm called nonlocal peridynamics, which employs integral equations, was proposed to model discontinuous domains. Third, we investigate the effect of mesh anisotropy, mesh refinement, and kernel functions on the conditioning of the global stiffness matrix for a nonlocal peridynamic model. The second part of the dissertation studies mesh deformation algorithms for robust anisotropic mesh deformation and for shape matching. First, we propose a robust mesh deformation algorithm using the anisotropy of the boundary deformation and multiobjective mesh optimization. When mesh deformation occurs, it is challenging to preserve element shape and noninverted mesh elements. In order to achieve this on the deformed domain, we use the direction of the boundary deformation to estimate the interior vertex positions and employ multiobjective mesh optimization for simultaneously preserving element shape and untangling the mesh.

4 Second, we propose an improved shape matching algorithm for deformable objects modeled by triangular meshes. We use dynamic programming to find the optimal mapping from the source image to the target image. iv

5 i Table of Contents List of Tables iv List of Figures Acknowledgments vii xv Chapter 1. Introduction Chapter 2. A Numerical Investigation on the Interplay Amongst Geometry, Meshes, and Linear Algebra in the Finite Element Solution of Elliptic PDEs Introduction Finite Element Solution of Elliptic PDEs Mesh Quality Metrics Mesh Optimization Iterative Linear Solvers Preconditioners Numerical Experiments Experimental Setup Description of Experiments Numerical Experiments Preliminary experiment for determination of restart value of the GMRES solver Numerical Results for Poisson s Equation Numerical Results for General Second-order Elliptic PDEs Numerical Results for the Linear Elasticity Problem Conclusions and Future Work Chapter 3. A Multiobjective Mesh Optimization Framework for Mesh Quality Improvement and Mesh Untangling Introduction Single Objective Mesh Optimization Mesh Quality Metrics Single Objective Functions

6 3.3 Multiobjective Mesh Optimization Methods Multiobjective Optimization Problems Nonlinear Optimization Problems Comparison among the Exponential Sum, Objective Product, and Equal Sum Methods Numerical Experiments Numerical Results for Optimizing Shape and Size Numerical Results for Optimizing Shape and Interpolation Error Numerical Results for Optimizing Shape and Untangling Numerical Results for Optimizing Shape, Size, and Untangling Application of Our Multiobjective Mesh Optimization Methods to Real- World Applications Mesh Warping Problem on 2D Hydrocephalus Domains Mesh Warping Problem on 3D Domains Effect of Simultaneously Optimizing Both Shape and Size Metrics on the Efficiency for Solving Possion s Equation Conclusions and Future Work Chapter 4. The Effect of Anisotropy, Mesh Refinement, and Kernel Functions on the Conditioning of the Stiffness Matrix for Nonlocal Peridynamic Models Introductions A Bond-based Nonlocal Peridynamic Model Quadrature Rules and Basis Functions Quadrature rules for a triangle Basis functions Connections among the horizon, mesh refinement, anisotropy of the mesh element and the condition number of the global stiffness matrix Condition number of the global stiffness matrix for a nonlocal peridynamic model Condition number of the global stiffness matrix for general secondorder elliptic PDEs Numerical results on 2D rectangular meshes Piecewise constant basis function with an integrable kernel function Piecewise linear basis function with an integrable kernel function Piecewise linear basis function with the nonintegrable kernel function Conclusions Introduction Background FEMWARP FEMWARP and anisotropic boundary deformation Hybrid Mesh Deformation Algorithm Step 1: Anisotropic FEMWARP using anisotropic PDE coefficients Step 2: Multiobjective mesh optimization with shape and untangling Numerical Experiments ii

7 Moving cylinder domain for anisotropic boundary deformation aligned in x-axis Moving bar domain for anisotropic boundary deformation aligned in y-axis Moving gate domain for y axis aligned anisotropic boundary deformation Cylinder in a channel domain subject to a diagonal deformation Bending bar domain for nonlinear 2D deformation D moving sphere domain for z axis aligned anisotropic boundary deformation Conclusions Chapter 5. An Improved Shape Matching Algorithm for Deformable Objects using a Global Image Feature Introduction Shape matching process Determination of the boundary vertices approximating the source image boundary Generation of the triangular mesh on the source image using the constrained Delaunay triangulation method Solution of the shape matching problem Experiments Conclusions and Future Work Chapter 6. Conclusions and Future Work References iii

8 iv List of Tables 2.1 Notation used in the definition of the 2D mesh quality metrics in Table The mesh quality metric definitions Listing of numerical experiments and examples of PDE problems. The letters (a) through (c) are representative examples of the three types of PDE problems under consideration Properties of meshes on geometric domains. Initial angle distributions for the meshes are given in Table The sixteen combinations of preconditioners and solvers. For example, 10 refers to using the ILU(0) preconditioner with the GMRES solver Mesh smoothing time (sec) for various mesh quality metrics Angle (θ) distribution for various mesh quality metrics. The reported values indicate a percentage of angles in the mesh. The initial mesh is the mesh with 50% of the interior vertices perturbed Linear solver time (secs) and number of iterations required to converge for Poisson s equation (problem (A) in Table 2.3) as a function of mesh quality metric for the 16 preconditioner-solver combinations (see Table 2.5) on the wrench and hinge domains. A * denotes failure. For each quality metric, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively Linear solver time (secs) and number of iterations required to converge for Poisson s equation (problem (A) in Table 2.3) as a function of vertex perturbation for the 16 preconditioner-solver combinations (see Table 2.5) on the two geometric domains. A * denotes failure. For each percentage of vertices perturbed, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively Linear solver time (secs) and number of iterations required to converge for general second-order elliptic PDEs (problem (B) in Table 2.3) as a function of mesh quality metric for the 16 preconditioner-solver combinations (see Table 2.5) on the two geometric domains. A * denotes failure. For each quality metric, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively

9 2.11 Linear solver time (secs) and number of iterations required to converge for the linear elasticity problem (problem (C) in Table 2.3) as a function of mesh quality metric for the 16 preconditioner-solver combinations (see Table 2.5) on the two geometric domains. A * denotes failure. For each quality metric, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively Linear solver time (secs) and number of iterations required to converge for the linear elasticity problem (problem (C) in Table 2.3) as a function of vertex perturbation for the 16 preconditioner-solver combinations (see Table 2.5) on the two geometric domains. A * denotes failure. For each percentage of vertices perturbed, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively Listing of numerical experiments, goals, and a description of goals Smoothing time (secs) for various mesh optimization methods in terms of the number of iterations of smoothing on the 3D disk mesh Mesh smoothing time (secs) for various mesh optimization methods in terms of the number of iterations of smoothing on the 3D disk domain Number of inverted elements as a function of the number of iterations of mesh optimization. The initial mesh has 543 inverted elements Number of inverted elements as a function of the number of iterations of mesh optimization. The initial mesh has 111 inverted elements Number of inverted elements as a function of the number of iterations of mesh optimization. The initial 3D disk mesh has 111 inverted elements Number of inverted elements as a function of the number of iterations of mesh optimization. The initial mesh has 13 inverted elements Maximum eigenvalue, minimum eigenvalue, condition number of P 1 A, and the number of iterations required to converge to solve Poisson s equation on the 2D hydrocephalus domain. Here, P 1 A is the preconditioned stiffness matrix. A * denotes invalid PDE solution due to inverted elements on the mesh Maximum eigenvalue, minimum eigenvalue, condition number of P 1 A, and the number of iterations required to converge to solve Poisson s equation on the 3D bore domain. Here, P 1 A is the preconditioned stiffness matrix. A * denotes invalid PDE solution due to inverted elements on the mesh Maximum eigenvalue, minimum eigenvalue, and condition number of P 1 A, and number of iterations to converge to solve Poisson s equation. Here, P 1 A is the preconditioned stiffness matrix Cond(A) of the global stiffness matrix for piecewise constant basis functions with an integrable basis kernel function (p=1). The number of elements is the number of elements in Ω. Here, the ratio is the ratio of the condition number of the current level of mesh refinement compared with that of the previous level of mesh refinement for a fixed δ. Therefore, these ratios are not defined for the 200 elements and are denoted by * for these cases v

10 4.2 Cond(A) of a global stiffness matrix for piecewise linear basis functions with an integrable basis kernel function (p=1). The total number of elements changes with respect to the horizon, δ. The number of elements indicates a number of elements in Ω. Here, the ratio is defined as the ratio of the condition number of the current level of mesh refinement compared with that of the previous level of mesh refinement for a fixed δ. Therefore, these ratios are not defined for the 200 elements and are denoted as * for these cases Cond(A) of a stiffness matrix for piecewise linear basis functions with a nonintegrable basis kernel function (p=1). The number of elements indicates the number of elements in Ω. Here, the ratio is defined as the ratio of the condition number of the current level of mesh refinement compared with that of the previous level of mesh refinement for a fixed δ. Therefore, these ratios are not defined for the 200 element case and are denoted as * for such cases Recognition rate results on the Brown dataset vi

11 vii List of Figures 1.1 Output meshes (a) before applying mesh qualiy improvement and (b) after applying mesh quality improvement One example of a tangled mesh which includes inverted elements Fragmentation of the domain [55]. Discontinuous domains occur due to fragmentation Bending bar deformation Some of the binary images in the Brown dataset [108] This flowchart represents the two parts of the dissertation and the dependency among the chapters Contour plots of the quality metric of a triangle as a function of a free vertex when the two other vertices held fixed at (0,0) and (0,1) Coarse initial meshes on the wrench and hinge geometric domains indicative of the actual meshes to be smoothed Effect of the mesh size on the number of iterations (a) and solver time (b) to convergence as a function of GMRES restart value, m, for Poisson s equation (problem (A) in Table 2.3) on the wrench domain Percentage increase (PI) as a function of the solver time for different combinations of preconditioners and solvers for Poisson s equation (problem (A) in Table 2.3). Preconditioner-solver combinations which fail to generate a preconditioner or do not converge correspond to the missing bars in these figures. Note that PI values for the hinge domain are significantly greater than those for the wrench domain The order of convergence for the solver time based on the number of iterations for various combinations of preconditioners and solvers for Poisson s equation (problem (A) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear leastsquares method Similar to Figure 2.5, this figure displays the order of convergence based on the solver time for Poisson s equation (problem (A) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method

12 2.7 PI as a function of the solver time for various combinations of preconditioners and solvers after vertex perturbation for general second-order elliptic PDEs (problem (B) in Table 2.3) The order of convergence for the solver time based on the number of iterations for the combinations of preconditioners and solvers for general second-order elliptic PDEs (problem (B) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method Similar to Figure 2.8, this figure displays the order of convergence based on the solver time for general second-order elliptic PDEs (problem (B) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method PI as a function of the solver time after vertex perturbation for the combinations of preconditioners and solvers for the linear elasticity problem (problem (C) in Table 2.3). The missing bar (combination 10) for the hinge domain corresponds to a preconditioner-solver combination which does not converge The order of convergence for the solver time based on the number of iterations for the different combinations of preconditioners and solvers for the linear elasticity problem (problem (C) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method Similar to Figure 2.11, this figure displays the order of convergence based on the solver time for the linear elasticity problem (problem (C) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method Contour plots of the quality metric of a triangle as a function of a free vertex when the two vertices are fixed at (0,0) and (1,0). In (e), β = Comparison among the exponential sum, objective product, and equal sum multiobjective mesh optimization methods Coarse initial meshes on the disk and barrier geometric domains indicative of the actual meshes to be smoothed (a) Average element quality in terms of the shape metric (IMR) on the 3D disk mesh; (b) average element quality in terms of the size metric (volume) on the same mesh (a) Worst element quality in terms of the shape metric (IMR) on the 3D disk mesh; (b) worst element quality in terms of the size metric (volume) on the same mesh (a) Average element quality as measured by the shape metric (IMR) on the 3D disk mesh; (b) average element quality as measured by the interpolation error (SS) on the same mesh (a) Worst element quality in terms of the shape metric (IMR) on the 3D disk mesh; (b) worst element quality in terms of the interpolation error (SS) metric on the same mesh viii

13 3.8 2D barrier mesh: (a) the initial mesh with inverted elements (the elements circled in red); (b) the final mesh with inverted elements (the elements circled in red) when employing the SI metric with the single objective method. The number of inverted elements increases after mesh optimization; (c) the final mesh when employing the untangling metric with the single objective method; (d) the final mesh for the exponential sum multiobjective method; (e) the final mesh for the objective product multiobjective method; (f) the final mesh for the equal sum multiobjective method. The final mesh by employing the exponential sum multiobjective method yields the best element quality while eliminating all inverted elements (a) Average element quality in terms of the shape metric (IMR) on the 3D disk mesh; (b) worst element quality in terms of the size metric (volume) on the same mesh (a) Average element quality in terms of the shape metric (SI) on the 3D disk mesh; (b) Average element quality in terms of the size (volume) metric on the same mesh Segmented images and meshes of initial and deformed domains Meshes of initial and deformed domains Connection between horizon (δ), force ( f ) between v and v, inside the neighborhood, H, over the domain Ω. The point v does not interact with any points beyond the distance δ One example of the sparsity pattern of a matrix A for the nonlocal peridynamic model. Here nz is the number of nonzeros in A Boundary conditions for (a) classical PDE models and (b) nonlocal peridynamic models. For classical PDE problems, the boundary condition is only specified on the boundary points. For nonlocal peridynamic models, all mesh elements belonging to BΩ belong to the boundary condition Three different cases for computing T when the intersection of T with δ is computed. Here, we only compute the intersection areas inside the horizon The approximation of the area of intersection of the triangular element and the horizon for case 2 in Fig (a) Initial isotropic mesh with h=0.1 and δ =0.2; (b) refined mesh with h=0.05 and δ =0.2; (c) Two level-refined mesh with h=0.025 and δ = Anisotropic mesh (aspect ratio=4) with δ= The definition of the aspect ratio of a triangle. The aspect ratio is defined as W/w [75] Condition number of A for fixed h and varying anisotropy of the elements and δ Condition number of A as a function of δ 2 for fixed h and varying anisotropy of the elements and δ Condition number of A for fixed anisotropy and varying h and δ Condition number of A as a function of δ 2 for fixed anisotropy and varying h and δ Condition number of A for fixed h and varying anisotropy of the elements and δ Condition number of A for fixed h and varying anisotropy of the elements and δ. 128 ix

14 4.15 Condition number of A for fixed anisotropy and varying h and δ Condition number of A for fixed anisotropy and varying h and δ Condition number of A when h=0.05 for fixed h and varying anisotropy and δ Condition number of A for fixed anisotropy (aspect ratio=1 and 4) and varying h and δ Condition number of A for fixed h and anisotropy and various power of a kernel function, p (a) Initial mesh on the rectangular domain. (b) Deformed mesh using FEMWARP. This mesh has 20 inverted elements Generic element in structured triangular mesh with uniform elements (a) Deformed mesh using FEMWARP. This mesh has 20 inverted elements. (b) Deformed mesh using anisotropic FEMWARP with coefficients α=0 and β =1 in (4.11). This deformed mesh using anisotropic FEMWARP does not have any inverted elements Connection between the PDE coefficients (α and β) and the direction of motion. If the deformation occurs (a) aligned with the x-axis or (b) aligned with the y-axis, we only consider neighbors aligned with the same axis. Here, the cross-out indicates that we do not consider those neighbors. If deformation occurs that is not aligned with either the x or y axes (c), we use the angle of direction of the deformation to choose the appropriate PDE coefficients. Here, a and b are α and β in (4.19), respectively Moving cylinder domain for anisotropic boundary deformation aligned with the x-axis: (a) Initial mesh for a moving cylinder in a channel. (b) Deformed mesh using FEMWARP for a moving cylinder. This mesh has 38 inverted elements. (c) Deformed mesh using anisotropic FEMWARP for a moving cylinder. This mesh has zero inverted elements. (d) Optimized mesh on the deformed domain with no inverted elements using anisotropic FEMWARP followed by multiobjective mesh optimization with TMP shape and untangling Mesh quality measured by the TMP shape metric on the deformed cylinder domain. A smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 80.4% less than the quantity of the one using UBN [95]. Note that FEMWARP includes 38 inverted elements, while the deformed meshes resulting from both our hybrid algorithm and UBN do not have any inverted elements Number of inverted elements with respect to the translation of inner boundary after applying (a) FEMWARP (α=1 and β=1) and anisotropic FEMWARP (α=1 and β=0). The inner boundary shifts right. (b) Time to untangle inverted elements using multiobjective mesh optimization. This figure shows the time to untangle inverted meshes using multiobjective mesh optimization x

15 4.27 Moving bar domain for anisotropic boundary deformation aligned in y-axis:(a) Initial mesh and (b) zoomed-in on the bar domain. (c) Deformed mesh with FEMWARP and (d) zoomed in mesh with FEMWARP. Deformed mesh with FEMWARP has 118 inverted elements. (e) Deformed mesh using anisotropic FEMWARP and (f) zoomed-in mesh with anisotropic FEMWARP. The deformed mesh with anisotropic FEMWARP has 17 inverted elements. (g) Optimized mesh and (h) zoomed-in mesh with anisotropic FEMWARP followed by multiobjective mesh optimization. This optimized mesh does not have any inverted elements Mesh quality measured by the TMP shape metric on the deformed bar domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 93.5% less than the quality of the mesh obtained from the use of FEMWARP [95]. Note that FEMWARP includes 118 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements. For this domain, UBN fails to find the deformed mesh due to numerical tolerance issues Number of inverted elements with respect to the translation of inner boundary after applying (a) FEMWARP (α=1 and β=1) and anisotropic FEMWARP (α=0 and β=1). The inner bar moves down. (b) We apply multiobjective mesh optimization to simultaneously smooth and untangle inverted elements. This figure shows timing to untangle inverted elements using multiobjective mesh optimization. Our multiobjective mesh optimization is able to improve element qualities while eliminating inverted elements but anisotropic FEMWARP takes up to 63.7% less time to eliminate inverted elements compared with FEMWARP since anisotropic FEMWARP has fewer iterations to untangle inverted elements than FEMWARP Moving gate domain for anisotropic boundary deformation aligned in y-axis: (a) Initial mesh and (b) zoomed-in on the gate domain. (c) The deformed mesh with FEMWARP and (d) zoomed-in mesh with FEMWARP. Deformed mesh with FEMWARP has 103 inverted elements. (e) The deformed mesh using anisotropic FEMWARP and (f) zoomed-in mesh with anisotropic FEMWARP. The deformed mesh with anisotropic FEMWARP has 14 inverted elements. (g) The optimized mesh and (h) zoomed-in optimized mesh on the deformed domain with no inverted elements using anisotropic FEMWARP followed by multiobjective mesh optimization with TMP shape and untangling Mesh quality measured by the TMP shape metric on the deformed gate domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 9.85% less than the one using FEMWARP [95]. Note that FEMWARP includes 103 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements. For this domain, UBN fails to find the deformed mesh due to numerical tolerance issues. 168 xi

16 4.32 Number of inverted elements with respect to the translation of outer boundary after applying (a) FEMWARP (α=1 and β=1) and anisotropic FEMWARP (α=0 and β=1). The corner in the middle moves down. (b) We apply multiobjective mesh optimization to simultaneously smooth and untangle inverted elements. This figure shows the time to untangle inverted elements using multiobjective mesh optimization. Our multiobjective mesh optimization is able to improve element qualities while eliminating inverted elements but anisotropic FEMWARP takes up to 90.6% less time to eliminate inverted elements compared with FEMWARP Moving cylinder domain for anisotropic diagonal boundary deformation: (a) Initial mesh for a moving cylinder in a channel. (b) The deformed mesh using FEMWARP for a diagonal deformation. It has 101 inverted elements. (c) The deformed mesh using anisotropic FEMWARP for a diagonal deformation. It has 15 inverted elements. (d) The optimized mesh on the deformed domain with no inverted elements using anisotropic FEMWARP followed by multiobjective mesh optimization with TMP shape and untangling Mesh quality measured by the TMP shape metric on the deformed cylinder domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 79.2% less than the one using FEMWARP [95]. Note that FEMWARP includes 103 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements Bending bar domain for anisotropic boundary deformation:(a) Initial mesh and (b) zoomed-in mesh on the bar domain. (c) Deformed mesh with FEMWARP and (d) zoomed-in mesh with FEMWARP. Deformed mesh with FEMWARP has 159 inverted elements. (e) Deformed mesh using anisotropic FEMWARP and (f) zoomed-in mesh with anisotropic FEMWARP. Deformed mesh with anisotropic FEMWARP has 48 inverted elements. (g) Optimized mesh and (h) zoomed-in mesh with anisotropic FEMWARP followed by multiobjective mesh optimization. This optimized mesh does not have any inverted elements Mesh quality measured by the TMP shape metric on the deformed bar domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 92.5% less than the one using FEMWARP [95]. Note that FEMWARP includes 159 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements. For this domain, UBN fails to find the deformed mesh due to numerical tolerance issues Moving sphere in a 3D cube domain: (a) Surface mesh on the initial domain (b) Volume mesh on the initial domain (c) Optimized volume mesh using our hybrid algorithm on the deformed domain. This optimized mesh does not have any inverted elements xii

17 4.38 Mesh quality measured by the TMP shape metric on the deformed 3D sphere domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 86.2% less than the one using FEMWARP [95]. Note that FEMWARP includes 120 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements. For this domain, UBN fails to find the deformed mesh due to numerical tolerance issues The number of inverted elements with respect to the translation of outer boundary after applying (a) FEMWARP (α=1, β =1, and γ=1) and anisotropic FEMWARP (α=0, β =0, γ=1). (b) We apply multiobjective mesh optimization to simultaneously smooth and untangle inverted elements. This figure shows the time to untangle inverted elements using multiobjective mesh optimization. Our multiobjective mesh optimization is able to improve element qualities while eliminating inverted elements but anisotropic FEMWARP takes up to 71.1% less time to eliminate inverted elements compared with FEMWARP Overview of the shape matching process. The function f maps triangles in the triangular mesh on the source image to a triangular mesh on the target image. (a) Equally-spaced boundary vertices are generated. (b) The triangular mesh is created. (c) The detected image is illustrated on the target image (a) A sample image shape from the Brown dataset [108]. (b) The dot in the middle represents the center of mass for the image (this is denoted as C in the target image), and the arrow represents the maximum distance from the center of mass in the image to the boundary vertices (i.e., d max,s or d max,t ) Poor shape matching result of the algorithm in [100] (left) and improved matching result using the center of mass in the target image (right). For these figures, the detected mesh is illustrated on the target image. For this experiment, θ = d max,t / Sample images in the Brown dataset [108]. Three sample images are shown per category Good shape matching results for the Brown dataset on three source (query) images and comparison with [100]. For each source image, the 10 best matching results are shown with the smallest (left) to the largest (right) energy. The top figures in each group represent the matching results obtained from our algorithm, whereas the bottom figures in each group represent matching results using the algorithm in [100]. For these experimental sets, only two matching results of [100] (i.e., the bottom right images) fail to match Poor shape matching results for the Brown dataset on three source (query) images and comparison with [100]. For each source image, the 10 best matching results are shown with the smallest (left) to the largest (right) energy. The top figures in each group represent the matching results obtained from our algorithm, and the bottom figures in each group represent matching results using the algorithm in [100]. For this experimental data set, both our algorithm and [100] show poor matching results. However, our algorithm shows better matching results than does the method in [100] xiii

18 5.7 Example matching results including detected meshes on the target image using images from the Brown dataset [108]. For this experiment, Felzenszwalb s algorithm shows poor matching results because triangles are placed at poor positions xiv

19 xv Acknowledgments I would like to express my sincere appreciation to my advisor Dr. Suzanne Shontz. When I first started my Ph.D research with Dr. Shontz, I was a novice in meshing. I was really fortunate to work with my advisor, and her guidance and encouragement helped me to finish my Ph.D. This dissertation could not have completed without her help. I appreciate the advice and guidance of my committee members, Drs. Padma Raghavan, Jesse Barlow, and Qiang Du, which helped strengthen my dissertation. I would like to thank my collaborators: Shankar Prasad Sastry, Nicholas Voshell, Thap Panitanarak, Li Tian, Brian J. Miller, Lori Diachin, and Seungkyu Lee. My Ph.D work benefitted on greatly from discussions and their feedback. I would like to thank to Shankar Prasad Sastry and Nicholas Voshell for giving helpful advice on my second chapter. The third chapter benefitted from discussion with Thap Panitanark. The collaboration with Li Tian significantly helped complete the fourth chapter. Finally, I have to thank my summer internship mentor, Brian Miller, and the Director of the Center for Applied Scientific Computing, Lori Diachin, at Lawrence Livermore National Laboratory, for their help on my summer internship project. I had a wonderful summer at Lawrence Livermore National Laboratory, and their guidance and discussions significantly help me to finish the fifth chapter. I would also like to thank to my officemates, Manaschai Kunaseth and Perry Huang, during my summer internship at Lawrence Livermore National Laboratory. I really enjoyed my summer internship with them, and they also helped me to be exposed to and to understand current parallel computing issues. I would like to thank to Seungkyu Lee for helpful discussions on the last chapter. My gratitude also goes to our Scalable Scientific Computing Lab members: Jeonghyung Park, Michael Frasca, Anirban Chatterjee, Shad Kirmani, and Manu Shantharam. I would like to thank the Mesquite team members: Patrick Knupp, Jason Kraftcheck, and Lori Diachin. Three out of five topics in this dissertation use the Mesquite software for our numerical experiments. They also provided some of the test meshes used in this dissertation. I would like to thank my family for their patience and love during my Ph.D study. I deeply appreciate my brothers, Jihyun Kim and Jiwan Kim, for discussing my graduate study and life. Also, their encouragement led me to start and finish Ph.D study in the United States. I also thank to my parents for their seamless support for my Ph.D study. Finally, I would like to acknowledge my sponsors for their support during my Ph.D. My research was funded in part by NSF CAREER Award OCI , a grant from the the Center

20 for Applied Scientific Computing at the Lawrence Livermore National Laboratory, and through instrumentation funded by the NSF through grant OCI xvi

21 1 Chapter 1 Introduction Many scientific applications require a discretization of the domain, which is called a mesh. It is well-known that mesh element shapes and mesh qualities affect both the efficiency and accuracy for solving many scientific applications, especially partial differential equation (PDE)-based applications [4, 5, 41]. Figure 1.1 shows a simple example of meshes (a) before and (b) after mesh quality improvement. A mesh is said to be tangled if it has inverted elements which have negative orientations. Figure 1.2 show an example of a tangled mesh which includes inverted elements. Tangled meshes are often generated during mesh generation, mesh optimization, and mesh deformation. Meshes with noninverted elements are important for PDE-based applications since tangled meshes with inverted elements result in invalid PDE solutions [1]. Recently, a new finite element method to handle tangled meshes was proposed [2]. However, this method can result in a system with a large condition number when the stiffness matrix is close to singular [2]. Also, there is no guarantee that this method is able to handle highly tangled meshes. In any case, it is desirable to eliminate inverted elements or to minimize the number of inverted elements before employing this new finite element method.

22 2 In this dissertation, we propose optimization-based unstructured meshing algorithms to improve the mesh qualities and to perform mesh deformations for various problems such as PDEs, nonlocal peridynamics problems, mesh deformation problems, and shape matching problems. In the first part of the dissertation, we focus on optimization-based meshing techniques for mesh quality improvements and mesh untangling. The accuracy and the efficiency of PDE solution highly depends on the quality of the mesh, preconditioner, and sparse linear solver. In Chapter 2, we study the effect the choice of mesh quality metric, preconditioner, and sparse linear solver have on the numerical solution of elliptic PDEs. We smooth meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We also investigate the effect of vertex perturbation and the effect of increasing the problem size on the number of iterations required to converge and on the solver time. We consider various elliptic PDE problems such as Poisson s equation, general second order elliptic PDEs, and linear elasticity problems. Many PDE-based engineering and scientific applications have multiple requirements for the finite element mesh discretizing the geometric domain [41, 44, 45]. For example, such requirements may include having non-inverted mesh elements, elements that are well-shaped, elements with uniform element size, and/or elements which yield small PDE interpolation error. Despite there being multiple mesh requirements for various PDE applications, most traditional mesh optimization algorithms optimize only a single objective function and hence improve only one aspect of the mesh. In Chapter 3, we propose a multiobjective mesh optimization framework for use in simultaneous improvement of multiple aspects of the mesh with the goal of improving

23 3 (a) Mesh before applying mesh quality improvement (b) Mesh after applying mesh quality improvement Fig. 1.1 Output meshes (a) before applying mesh qualiy improvement and (b) after applying mesh quality improvement Fig. 1.2 One example of a tangled mesh which includes inverted elements.

24 4 the accuracy, efficiency, stability, and conditioning of the associated finite element solver. Our framework combines two or more competing objective functions into a single objective function to be solved using one of various multiobjective optimization methods. Methods within our framework are able to optimize various aspects of the mesh such as the element shape, element size, associated PDE interpolation error, and number of inverted elements, but the improvement is not limited to these categories. Many applications in solid mechanics such as damage, surface cracks, and fracture include discontinuities [56, 63, 64]. Figure 1.3 shows one example of a cracked domain which includes discontinuities [55]. However, classical PDEs have some limitations in modeling and computation of these discontinuous domains since we are not able to compute derivatives in these domains. In order to improve these limitations of classical PDEs, a peridynamics nonlocal model using integral equations was recently developed. In Chapter 4, we use the continuous Galerkin finite element methods for discretizing a linear peridynamics system and study the effect the mesh anisotropy, mesh refinement, and kernel functions have on the conditioning of the stiffness matrix for a bond-based nonlocal peridynamics model on 2D geometric domains. Fig. 1.3 Fragmentation of the domain [55]. Discontinuous domains occur due to fragmentation.

25 5 In the second part of the dissertation, we investigate meshing techniques for mesh deformation problems and shape matching problems. There are numerous applications where discretized geometric domains vary with respect to time such as the solution of Arbitrary- Lagrangian-Eulerian (ALE) flow simulations [76], deformation of the human face in computer graphics [78], deformation of sequences of medical images [81, 82], and deformations of biomedical applications [79, 80]. Figure 1.4 shows an example of a bending bar deformation. In Chapter 5, we propose a hybrid mesh deformation algorithm which uses the direction of the boundary deformation. Our goal is to produce meshes on deformed domains which maintains element shapes and possesses no inverted elements. The hybrid mesh deformation algorithm consists of two steps, i.e., anisotropic finite element-based mesh warping (FEMWARP) followed by multiobjective mesh optimization. The idea of multiobjective mesh optimization comes from Chapter 3, but we extend this idea to mesh deformation problems. The first step estimates the interior vertex positions on the deformed mesh using the boundary deformation to choose appropriate PDE coefficients in the anisotropic FEMWARP method. As a second step, we find a locally optimal mesh with no inverted elements on the deformed domain by employing a multiobejctive mesh optimization with one term controlling element shape and a second term designed to untangle inverted elements. Shape matching is an important problem in many computer vision applications such as object tracking and image-based searches [98, 99]. The goal of shape matching is to match the source image to the target image, i.e., the deformed image. Figure 1.5 shows some binary images which are frequently used for testing shape matching algorithms in the Brown dataset. This image dataset includes various transformations and occlusions of shapes. In Chapter 6, we propose a shape matching algorithm for deformable objects using both local and global shape

26 6 (a) Mesh on the initial domain (b) Mesh on the deformed domain Fig. 1.4 Bending bar deformation. information. We focus on shape matching problems for binary images. For our approach, we use triangular meshes to represent deformable objects and use dynamic programming to find the optimal mapping from the source image to the target image which minimizes a new energy function. Our energy function includes a new cost term that takes into account the center of mass of an image. We use the well-known Brown dataset shown in Fig. 1.5 to test our algorithm. Fig. 1.5 Some of the binary images in the Brown dataset [108]. Our goal is to improve the accuracy and efficiency for the solution of various applications by employing our optimization-based meshing techniques. We will show that our optimizedbased meshing techniques significantly improve the accuracy and efficiency for the solution of various problems: PDEs, nonlocal peridynamics, mesh deformation, and shape matching.

27 7 In the following paragraph, we summarize the software we use for our numerical experiments. In Chapter 2, we use Mesquite [16], a mesh quality improvement toolkit, and PETSc [17], a linear solver toolkit, to perform a numerical study investigating the performance of several mesh quality metrics, preconditioners, and sparse linear solvers on the solution of various elliptic PDEs. We use Mesquite and PETSc in their native state with the default parameters. Only these two toolkits are employed so that differences in solver implementations, data structures, and other such factors would not influence the results. In Chapter 3, we use Mesquite for our numerical experiments, but we implement our multiobjective mesh optimization framework. We also use PETSc to solve a sparse linear system, which is discretized from Poisson s equation for real-world applications. We use the default options of PETSc to compute the maximum eigenvalue, minimum eigenvalue, condition number of the stiffness matrix, and to solve the resulting linear system. The Jacobi preconditioner (P) with the minimal residual (MINRES) solver is used to solve the linear system. In Chapter 4, no existing software is used, but instead we use C++ to implement a nonlocal peridynamics model. In Chapter 5, we use a finite-element based mesh warping algorithm (FEMWARP) [88] and Mesquite for our numerical experiments. We modify the FEMWARP code, which was provided by Dr. Suzanne Shontz, to implement the first step of our hybrid mesh deformation algorithm, i.e., anisotropic FEMWARP. We use Mesquite for implementing the second step of our hybrid mesh deformation algorithm, i.e., multiobjective mesh optimization. In Chapter 6, we do not use any existing software, but instead C++ is used to implement our shape matching algorithm. The outline of the dissertation is given on the next page. Figure 1.6 shows the flowchart which describes the overall theme and subthemes of the dissertation as well as the connections between the chapters.

28 8 Dissertation Outline Fig. 1.6 This flowchart represents the two parts of the dissertation and the dependency among the chapters.

29 9 Chapter 2 A Numerical Investigation on the Interplay Amongst Geometry, Meshes, and Linear Algebra in the Finite Element Solution of Elliptic PDEs 2.1 Introduction Discretization methods, such as the finite element (FE) method, are commonly used in the numerical solution of partial differential equations (PDEs). The accuracy of the computed PDE solution depends on the degree of the approximation scheme, the number of elements in the mesh [4], and the quality of the mesh [5, 6]. More specifically, it is known that as the element dihedral angles become too large, the discretization error in the finite element solution increases [4]. In addition, the stability and convergence of the finite element method is affected by poor quality elements [7]. In particular, it is known that as the angles become too small, the condition number of the finite element matrix increases [7].

30 10 Analytical studies have been performed at the intersection of meshing and linear solvers. For example, mathematical connections between mesh geometry, interpolation errors, and stiffness matrix conditioning for triangular and tetrahedral finite element meshes have been studied [41]. Quality metrics which determine the relevant fitness of elements for the purposes of interpolation or for creating a global stiffness matrix with a low condition number have been developed [41]. Further mathematical research has been performed at the intersection of finite element meshes and linear solvers. A mesh and solver co-adaptation strategy for anisotropic problems has been developed [8]. In addition, relationships between the spectral condition number of the stiffness matrix and mesh geometry for second-order elliptic problems for general finite element spaces defined on simplicial meshes have been determined [9]. Several computational studies have been performed which examined the connections between finite element meshes and linear solvers in various contexts. For example, the effect of unstructured meshes on the preconditioned conjugate gradient solver performance for the solution of the Laplace and Poisson equations has been examined [10, 11]. In [12], the relative performance of multigrid methods for unstructured meshes was studied on fluid flow and radiation diffusion problems. Trade-offs associated with the cost of mesh improvement in terms of solution efficiency has been examined for problems in computational fluid dynamics [13, 14]. Composite linear solvers which provide better average performance and reliability than single linear solvers for large-scale, nonlinear PDEs have been designed [15]. We examine the connections between geometry, mesh smoothing, and linear solver convergence for elliptic PDEs via an experimental approach. In particular, we seek answers to the following questions pertaining to the solution of an elliptic PDE on a given geometric domain.

31 11 What is the most efficient combination of mesh quality metric, preconditioner, and solver for solving each PDE problem? Which combinations are most and least sensitive to vertex perturbation? What is the effect of increasing the problem size on the number of iterations required to converge and on the solver time? Our goal is to determine the best combination of quality metric, preconditioner, and linear solver which results in a small condition number of the preconditioned matrix and fast solver convergence for a given PDE, geometric domain, and initial mesh. To answer the above questions, we use Mesquite [16], a mesh quality improvement toolkit, and PETSc [17], a linear solver toolkit, to perform a numerical study investigating the performance of several mesh quality metrics, preconditioners, and sparse linear solvers on the solution of various elliptic PDEs (i.e., Poisson s equation, general second-order elliptic PDEs, and linear elasticity) of interest. Mesh quality metrics used in this study are as follows: inverse mean ratio (IMR) [18], radius ratio (RR) [41], a conditioning-based scale-invariant metric (SI) [41], and an interpolation-based size-and-shape metric (SS) [41]. Furthermore, we investigate the performance of the following preconditioners: Jacobi [19], symmetric successive over relaxation (SSOR) [19], incomplete LU (ILU) [19] and algebraic multigrid [20]; and linear solvers: conjugate gradient (CG) [21], minimum residual solver (MINRES) [22], generalized minimal residual solver (GMRES) [23], and bi-conjugate gradient stabilized solver (Bi-CGSTAB) [24]. The quality metric/preconditioner/linear solver combinations are compared on the basis of efficiency, robustness, and complexity in solving several elliptic PDEs on realistic unstructured tetrahedral finite element meshes. We use Mesquite and PETSc in their native state with the default parameters. Only these two toolkits are employed so that differences in solver implementations, data structures, and other such factors would not influence the results.

32 12 This chapter builds on our preliminary results, which were published in [25, 53]. In particular, this chapter extends our previous work in two ways. First, we also consider the linear elasticity problem, since this problem has a different sparsity pattern than our previously-studied elliptic PDE problems. Second, we also investigate the effect of vertex perturbation and the effect of increasing the problem size on the number of iterations required to converge and on the solver time. The contents of this chapter have been published in [73]. 2.2 Finite Element Solution of Elliptic PDEs We consider the solution of three elliptic PDE problems: Poisson s equation, general second-order elliptic PDE problems, and linear elasticity. On a given geometric domain, we consider only homogeneous Dirichlet boundary conditions, since we observed in [25] that modifying the boundary conditions does not alter the efficiency ranking of the solver time for combinations of mesh quality metrics, preconditioners, and linear solvers. To solve the elliptic PDE problems, triangular meshes are used to discretize the domain, Ω. The standard Galerkin finite element (FE) method [26] is used to solve the given PDE problem resulting in the linear system Aξ = b. (2.1) The approximate solution, ξ, of the analytical solution, u, can be computed by solving (2.1). In general, (2.1) is a sparse linear system, and iterative methods such as CG and GMRES are often used to solve the system.

33 Poisson s equation. Poisson s equation is used to model many mechanical and electromagnetic problems. Poisson s equation is given by 13 2 u x 2 2 u = f on Ω, (2.2) y2 where f is a given function. For Poisson s equation, the matrix A is given by A=K+N, where K is the stiffness matrix and N is a matrix containing boundary information. General second-order elliptic PDE problem. The general second-order elliptic PDE problem on Ω is defined as α 2 u x 2 β 2 u + au = f on Ω, (2.3) y2 where α and β are PDE coefficients and a and f are given functions. If a = 0, (2.3) reduces to Poisson s equation. We consider the case when a is nonzero. The coefficients α and β form a coefficient matrix, C, given by C = α 0. 0 β For example, C is the identity matrix, I, for (2). The major difference between (2.3) and Poisson s equation in (2.2) is the existence of a mass matrix, M, in the decomposition for matrix A. For this problem, the matrix A is given by A=K+M+N, where K is the stiffness matrix, M is the mass matrix, and N is a matrix containing boundary information. One application of the general second-order elliptic PDE problem is the scattering problem.

34 14 Linear elasticity. Linear elasticity is a common problem in structural mechanics and is used to compute the displacement vector, u, given a body force, f, and (or) a traction load. We do not consider a traction load. The linear elasticity problem with the Lamé parameters λ and µ is defined as τ = λ τ = f ( α u x + β u y ) I + 2µε(u), (2.4) where α and β are PDE coefficients and τ and ε(u) are the stress and strain tensors, respectively. The strain tensor is given by ε i j (u) = 1 2 ( ui + u ) j,i, j = 1,2. x j x i For homogeneous materials, the Lamé parameters are given by λ=eν/((1 + ν)(1 2ν)) and µ=e/(2(1 + ν)), where E and ν are the elastic modulus and Poisson s ratio, respectively [27]. Elliptic PDE coefficients and optimal element shape. The efficiency of elliptic PDE solvers is highly connected with the condition number, κ, of the matrix A. A smaller κ(a) results in a faster convergence time for solving (1). The optimal element shape that reduces κ(a) depends on the coefficients of the elliptic PDEs [9, 28, 41]. It has been shown that for C = γi over Ω, where γ is a constant, the optimal shape of a triangular element is an equilateral triangle [9, 28, 41]. If C is a constant, but is not given γi over Ω, the optimal shape is different from an equilateral triangle, and anisotropic triangular elements can be used to obtain a smaller κ(a) [9, 41]. If C is non-constant over Ω, the optimal shape of a triangular element varies over Ω [28]. We consider elliptic PDEs with continuous, isotropic coefficients which are constant over Ω (i.e.,

35 C = γi). Therefore, for the PDEs considered in the thesis, the ideal triangular element shape is an equilateral triangle Mesh Quality Metrics Table 2.1 provides the notation used to define the following mesh quality metrics: inverse mean ratio (IMR) [18], radius ratio (RR) [41], a conditioning-based scale-invariant metric (SI) [41], and an interpolation-based size-and-shape metric (SS) [41]. Table 2.2 defines IMR, RR, SI, and SS. These four quality metrics were chosen based on their geometric features, which result in varying contour plots as shown in Fig 3.1. The plots show the contour lines of the quality of a triangle as a function of a free vertex when the two other vertices held fixed at (0,0) and (0,1). IMR is one of the most well-known mesh quality metrics for mesh quality improvement and is evaluated using the position vectors in the element. RR is a quality metric which is computed using the radius of the triangular element s circumscribing and inscribing circles. SI is a quality metric based on stiffness matrix conditioning, whereas SS is a quality metric based on interpolation error bounds. Note that a conditioning-based size-and-shape quality metric is not defined for 2D cases (triangles) [41]. 2.4 Mesh Optimization We denote the elements of a mesh and the number of mesh elements by E and E, respectively. The overall quality of the mesh, Q, is a function of the individual element qualities, q i, where q i is the quality of the i th element in the mesh. The mesh quality depends on the choice of q i (Section 2.3) and the objective function used to combine the individual element qualities.

36 16 Notation in Definition a, b, and c Position vectors for vertices in a triangular element C = [b ( a; c a] Jacobian of a triangular element 1 1 ) W = Incidence matrix for an equilateral triangle 2 Area Area of a triangular element r circ Radius of a triangular element s circumscribing circle r in Radius of a triangular element s inscribing circle s 1, s 2, and s 3 Edge lengths of a triangular element Table 2.1 Notation used in the definition of the 2D mesh quality metrics in Table 2.2. Quality Metric Inverse mean ratio (IMR) Radius ratio (RR) Formula CW 1 2 F 2 det(cw 1 ) [18] rcirc /r in [41] Conditioning-based scale-invariant (SI) Area/(s s2 2 + s2 3 ) [41] Interpolation-based size-and-shape (SS) Area/(s 1 s 2 s 3 ) [41] Table 2.2 The mesh quality metric definitions.

37 17 (a) IMR (b) RR (c) SI (d) SS Fig. 2.1 Contour plots of the quality metric of a triangle as a function of a free vertex when the two other vertices held fixed at (0,0) and (0,1).

38 18 The value of the IMR quality metric lies between 1 and, whereas the RR quality metric s value lies between 0.5 and. For IMR and RR, a lower value indicates a higher quality element. For the IMR and RR quality metrics, we define the overall mesh quality, Q, as the sum of squares of the individual element qualities: Q = E i=1 q 2 i. (2.5) The SI quality metric s value lies between 0 and 1/(4 3), whereas the SS quality metric s value lies between 0 and. For SI and SS, a higher value indicates a higher quality element. For the SI and SS quality metrics, we define the overall mesh quality, Q, as the sum of the squares of the reciprocal of the individual element qualities: Q = E i=1 1. (2.6) q 2 i We use the reciprocal of the quality metric instead of the additive inverse to optimize the mesh because the optimization process for the latter results in meshes with elements of both very good and very poor quality, which is not desirable. Using the reciprocal (as was done in [29]) results in smoothed meshes with a more even distribution of element qualities. We use the Fletcher-Reeves nonlinear conjugate gradient method [30] in Mesquite [16] to minimize Q (as defined by either (3.4) or (3.5)), since this method is the default NLCG method in Mesquite. The local mesh smoothing technique is used for the mesh optimization procedure. Mesquite employs a line search version of the nonlinear conjugate gradient method. The implementation ensures that the triangular elements do not tangle as a result of vertex movement.

39 19 Our preliminary experiments indicate that the total time, which is the sum of the smoothing time and the solver time, is minimal when the reduction in the objective function is 95% of the possible reduction (when the mesh is completely smoothed). The smoothing time significantly dominates the solver time when the mesh is completely smoothed. Thus, we perform inaccurate mesh smoothing, i.e., we smoothe the mesh only to the extent required to yield the optimal total time for numerically solving the PDE on the mesh (as in [31]). 2.5 Iterative Linear Solvers Four iterative Krylov subspace methods are employed to solve the preconditioned linear system. The conjugate gradient (CG) solver [21] is a well-known iterative method for solving systems with symmetric positive definite matrices. It produces a sequence of orthogonal vectors on successive iterations. Let P be the preconditioner (to be described in Section 2.6). The convergence rate of the preconditioned CG method depends upon the condition number, κ, of P 1 A. In the 2-norm, κ is defined as κ 2 (P 1 A) = λ max (P 1 A)/λ min (P 1 A), where λ max and λ min are the maximum and minimum eigenvalues of P 1 A, respectively. The fastest convergence occurs when eigenvalues are clustered around the same non-null value, and hence, κ is near 1 [3]. Some theoretical convergence results for the CG solver are given in [14, 32]. In particular, if the eigenvalues of A are uniformly distributed, the number of iterations required to converge, n, to reduce the error in the energy norm by a factor of ε satisfies the

40 20 following inequality [14]: n κ 2 log + 1. ε The minimal residual (MINRES) algorithm [22] solves linear systems with symmetric indefinite matrices. It generates a sequence of orthogonal vectors and attempts to minimize the residual in each iteration. Similar to CG, a small condition number for P 1 A and clustering of eigenvalues around the same non-null value results in fast convergence. For nonsymmetric matrices, the generalized minimal residual (GMRES) method [23] is one of the most widely used iterative solvers. Similar to CG and MINRES, GMRES computes orthogonal vectors on each iteration; however, the entire sequence needs to be stored. Therefore, the version of GMRES which restarts GMRES every m steps, i.e., GMRES(m), is used in practice. It is known that a large value of m is effective in decreasing the number of iterations required to converge; however, the optimal value of m depends upon the problem [33]. The biconjugate gradient stabilized (Bi-CGSTAB) method [24] is a biorthogonalization technique, which generates two sets of biorthogonal vectors instead of producing long orthogonal vectors. Bi-CGSTAB is known to have comparable or even faster convergence than other biorthogonalization methods such as the conjugate gradient squared method. However, Bi- CGSTAB sometimes shows an irregular convergence rate similar to other biorthogonalization methods [19]. Bi-CGSTAB and GMRES are the most widely-used iterative methods for solving systems based on nonsymmetric matrices.

41 Preconditioners The objective of introducing a preconditioner, P, into the solution of a linear system is to make the system easier to solve, whereby reducing the convergence time of the iterative solver. The reader is referred to [34] (and the references therein) for further information on iterative solvers and preconditioners. In this thesis, four preconditioners are employed. The first is the Jacobi preconditioner, which is simply the diagonal of A. The second is the symmetric successive over relaxation (SSOR) preconditioner. SSOR is similar to Jacobi but decomposes A into L (the strictly lower triangular part), D (the diagonal), and U (the strictly upper triangular part), i.e., A=L+D+U. The SSOR preconditioner is given by P = (D ωl)d 1 (D ωu), where ω represents the relaxation coefficient. The default ω value in PETSc is 1. The incomplete LU (ILU) preconditioner with level zero fill-in (ILU(0)) is the third preconditioner. ILU is commonly used in the solution of elliptic PDE problems. The basic idea of the ILU preconditioner is to determine lower ( L) and upper triangular (Ũ) matrices such that the matrix LŨ-A satisfies certain constraints [34]. The ILU preconditioner works well for many problems but fails when it encounters negative or zero pivots. The fourth is the algebraic multigrid (AMG) preconditioner. Different from geometric multigrid methods, the AMG preconditioner does not need any mesh information to generate the preconditioner and hence is known as black-box technique. AMG only requires the matrix A to generate the preconditioner. Therefore, the AMG preconditioner can be used to solve linear systems which arise from unstructured meshes. The main idea of the AMG preconditioner is to

42 22 eliminate the smooth error using restriction and interpolation which is not removed by relaxation on the fine grid [20]. We use the default options for HYPRE BoomerAMG in PETSc. The default option in PETSc employs 25 levels of V-cycles. For further infomation on HYPRE BoomerAMG, the reader is referred to [17]. 2.7 Numerical Experiments Experimental Setup We consider the following questions which we investigate on the three elliptic PDE problems shown in Table 2.3. What is the most efficient combination of mesh quality metric, preconditioner, and solver for solving each PDE problem? Which combinations are most and least sensitive to vertex perturbation? What is the effect of increasing the problem size on the number of iterations required to converge? Table 2.3 summarizes the experiments and corresponding PDE problems to be solved. For all three PDE problems in Table 2.3, we consider only a simple homogeneous Dirichlet boundary condition with u=0 on the boundary because we already observed that modifying boundary condition does not affect the efficiency ranking [25]. For the elasticity problem in Table 2.3, E and ν are set to 1 and 0.3, respectively. The body force, f, is set to [1 0] T. The machine employed for this study is equipped with an Intel Nehalem processor (2.66 GHz) and 24GB of RAM [35]. We consider elliptic PDEs with isotropic coefficients, which is constant (i.e., C = αi) over 2D geometric domains. The optimal element shapes for solving elliptic PDEs with anisotropic coefficients are different from the optimal shapes for elliptic PDEs with isotropic coefficients [9, 28, 41]. Therefore, our experimental results cannot be generalized to the solution of elliptic

43 23 PDEs with anisotropic coefficients. The quality metrics used to determine the mesh element quality are different for 2D and 3D elements. In addition, the sparsity patterns of the matrices in the linear systems obtained from unstructured meshes are different for 2D and 3D meshes. Therefore, our experimental results cannot be generalized to 3D meshes. Exp. Exp. Examples of No. Name. PDE Problems Determination of restart valuesr of the GMRES solver (A) u = 1 on Ω, u = 0 on Ω Poisson s equation (A) u = 1 on Ω, u = 0 on Ω General second-order elliptic PDE problem (B) u u = 1 on Ω, u = 0 on Ω Elasticity problem (C) τ = f on Ω, u = 0 on Ω, where τ = λ ( u)i + 2µε(u) Table 2.3 Listing of numerical experiments and examples of PDE problems. The letters (a) through (c) are representative examples of the three types of PDE problems under consideration. Geometric Domains. The two 2D geometric domains, wrench and hinge, considered in our experiments are shown in Figure 3.3. Triangle [36] is used to generate initial meshes on the two domains. Half the interior vertices in each mesh are perturbed to create test meshes that are further from optimal. Properties of the test meshes and the corresponding finite element matrices are shown in Table 2.4. Finite Element Solution. The FE method described in Section 2.2 is used to discretize the domain, Ω, and to generate a linear system of the form Aξ =b. PETSc [17] is used to generate the preconditioners, P, and to solve the linear system, P 1 Aξ =P 1 b. We employ the solvers and preconditioners described in Sections 2.5 and 2.6, respectively, to solve the linear system. Table

44 24 (a) Wrench mesh (b) Hinge mesh Fig. 2.2 Coarse initial meshes on the wrench and hinge geometric domains indicative of the actual meshes to be smoothed. mesh # vertices # elements mesh # vertices # elements Wrench (10K) 10,142 19,796 Hinge (10K) 9,989 18,359 Wrench (50K) 50,161 99,197 Hinge (50K) 49,986 96,483 Wrench (100K) 100, ,055 Hinge (100K) 99, ,131 Wrench (200K) 199, ,764 Hinge (200K) 200, ,743 Wrench (500K) 499, ,496 Hinge (500K) 498, ,162 Table 2.4 Properties of meshes on geometric domains. Initial angle distributions for the meshes are given in Table 2.7.

45 2.5 enumerates the 16 preconditioner-solver combinations used in our experiments. The default parameters for each preconditioner and solver were employed. 25 Preconditioner Solver CG GMRES MINRES BI-CGSTAB Jacobi SSOR ILU(0) AMG Table 2.5 The sixteen combinations of preconditioners and solvers. For example, 10 refers to using the ILU(0) preconditioner with the GMRES solver. The default stopping criteria in PETSc were employed. For example, the absolute tolerance, abstol, and the relative tolerance, rtol, are set to 1e-50 and 1e-05, respectively. The maximum number of iterations for solving the preconditioned linear system is set to 10,000. When the preconditioned linear system is solved, ξ 0 is set to the default value of 0. The preconditioned linear system converges on the i th iteration if the following inequality is satisfied: r i < max(rtol r 0,abstol), (2.7) where r i is the residual at the i th iteration and r 0 is the initial residual. Accuracy of the Solution. The exact solutions of the PDE problems in Table 2.3 on the geometric domains in Figure 3.3 are unknown. Therefore, we conduct a mesh-independence study using meshes with 10K, 50K, 100K, 200K, and 500K vertices and verify that the angle distribution and resulting PDE solution is independent of the mesh size. Our experimental results show that the finite element method converges to the same PDE solution if the mesh consists of more

46 26 than 10K vertices. We also observe that the finite element solution is not affected by choice of quality metric, preconditioner, and linear solver for these mesh sizes. We will investigate the accuracy of the finite element solution for boundary value PDEs as future work. Timing. In our experiments, the total time is defined as the sum of the smoothing time and the solver time. The smoothing time is the time to achieve an accurately smoothed mesh as described in Section 2.4. The time required to smoothe the test meshes using various quality metrics is shown in Table 2.6. The solver time is the time the solver takes to satisfy (2.7) and includes the time to generate P and to solve P 1 Aξ =P 1 b. For our experiments, we use Mesquite version and PETSc version Mesquite and PETSc are widely used for solving linear systems [14, 15, 37] and mesh quality improvement [25, 31, 38], respectively. Both PETSc and Mesquite are numerical libraries containing data structures and routines. Algorithms in each software package are coded using similar data structures and routines. We report both the timing (in seconds) and the number of iterations required to converge. Note the latter is not affected by the software version or the hardware used to solve the problem. However, the timing can be affected by these factors. We expect that the relative difference amongst different combinations of quality metric, preconditioner, and linear solvers will remain the same. Mesh Mesh quality metrics IMR RR SI SS Wrench (500K) Hinge (500K) Table 2.6 Mesh smoothing time (sec) for various mesh quality metrics

47 Description of Experiments Experiment 1: Best combination of mesh quality metric, preconditioner and solver. In this experiment, we seek to determine the most efficient combination of mesh quality metric, preconditioner, and solver for each type of PDE problem in Table 2.3. We will examine the most efficient combinations for both the solver time and the total time. Experiment 2: Effect of perturbation. We discuss the effect of varying the percentage of vertex perturbation on the efficiency of the preconditioner-solver combinations. In [14], it is reported that the CG solver is more robust to perturbation than is the GMRES solver for the Jacobi preconditioner. We determine the robustness for various combinations of preconditioners and solvers to such perturbations. We randomly perturb a certain number of vertices in each mesh while ensuring that no perturbation results in a very poorly-shaped element (e.g., an inverted element). We perturb the interior vertices such that they move less than half the distance at which element inversion would occur. In our experiments, we perturb 5%, 25%, and 50% of the elements and investigate the robustness of the preconditioner-solver combinations to the vertex perturbations. We define the relative increase, i.e., the percentage increase (PI), in the solver time due to the vertex perturbation as follows: PI = T max T min T min, where T max and T min are the maximum and minimum convergence time among 0% (i.e., fullysmoothed unperturbed mesh), 5%, 10%, 25%, and 50% perturbed elements, respectively. Note

48 28 that we observe monotonic convergence behavior with all four mesh quality metrics. Experiment 3: Increasing the problem size. In this experiment, we examine the convergence rates of various preconditioner-solver combinations with an increasing number of vertices in a mesh for a given domain. We execute our numerical experiments for meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain. Let N be the number of vertices in the mesh. We compute the order of convergence, denoted O(N s ), using the linear least-squares method for both the number of iterations to converge and the solver time. Thus, the total work required to solve the system is O(N s+1 ) for each combination of preconditioner and solver Numerical Experiments We now discuss the results for the three numerical experiments discussed in Section for each PDE problem in Table Preliminary experiment for determination of restart value of the GMRES solver The default restart value, m, in PETSc is 30, and we examine whether that value is optimal to solve each of the PDE problems. We experimentally determine the optimal m value, which yields the fastest solver time, for each application by varying it as follows: 10, 30, 50, 100, and 200. Figure 2.3(a) shows the number of iterations required to converge for different m values as the problem size increases for the wrench domain. For this experiment, the meshes are smoothed with the IMR quality metric, and the SSOR preconditioner with the GMRES solver is employed to solve Poisson s equation (problem (A) in Table 2.3). The results obtained are

49 29 representative of the results obtained with other geometric domains, quality metrics, and preconditioners. We observe that as we increase the values of m, the total number of iterations to converge reduces. This is consistent with the theoretical analysis presented in [33]. Figure 2.3(b) shows the solver time for different m values as we increase the problem size. We observe that the solver time is least when the restart value, m, is 30. As we reduce or increase the restart value, the time taken to converge to a solution increases. Large restart values (e.g., m = 100 and 200) result in slower solver times because the cost of orthogonalization increases with an increase in m. Small restart values (e.g., m = 10) also slow the solver since it needs a greater number of iterations to converge. For the other types of PDE problems in Table 2.3, we observe similar results. The GM- RES solver with m = 30 yields the least solver time, although m = 200 results in the fewest number of iterations to converge. Therefore, we choose the GMRES solver with m = 30 for all elliptic PDE problems in Table Numerical Results for Poisson s Equation Exp. 1: Determination of the best combinations of mesh quality metric, preconditioner, and solver for Poisson s equation. Table 2.8 shows the solver time and the number of iterations required for convergence as a function of mesh quality metrics for different combinations of preconditioners and solvers. The most efficient combinations (e.g., the IMR mesh quality metric with the AMG preconditioner and the CG solver) are 97% faster than the least efficient combinations (e.g., the IMR quality metric with the Jacobi preconditioner and the GMRES solver). We observe that the AMG preconditioner with any choice of quality metric and solver is faster to converge than other combinations of quality metric, preconditioner and solver. When

50 30 (a) Number of iterations (b) Solver time Fig. 2.3 Effect of the mesh size on the number of iterations (a) and solver time (b) to convergence as a function of GMRES restart value, m, for Poisson s equation (problem (A) in Table 2.3) on the wrench domain.

51 31 we solve linear equations derived from a mesh, high frequency terms are not eliminated only by relaxation on the fine grid. Thus, the AMG preconditioner used in the experiment employs 25 level of V-cycles and efficiently eliminates high frequency terms (which correspond to large eigenvalues in the matrix A) using coarse grid correction techniques. The most time-consuming step during the generation of the AMG preconditioner is an aggregation step which finds the vertex-neighborhood information from the matrix A. However, the aggregation step is fast for Poisson s equation as it contains a single degree of freedom. Note that the number of iterations required to converge for combinations with the AMG preconditioner represent the number of outer iterations only (not including V-cycles). We also observe that the Bi-CGSTAB solver (combinations 4, 8, 12, and 16 in Table 2.5) is most sensitive to the choice of different mesh quality metric, whereas, the CG solver is least affected by the choice of the quality metric. These results are related with the irregular convergence behavior discussed in [25]. The number of iterations required to converge and the solver time taken by the BiCG-STAB solver with a given preconditioner varies by more than 20% in many cases. For other combinations, however, the variation is small and is restricted to less than 10% in most cases. On the choice of mesh quality metrics, the SS metric is the least efficient mesh quality metrics in most cases. The method using the SS metric often fails to converge because meshes smoothed by the SS metric fail to yield a positive definite preconditioner. Note that both the CG and the MINRES solvers require the preconditioner to be a positive definite matrix. In most cases, the maximum eigenvalue of the matrix A is larger and the minimum eigenvalue of the matrix A is smaller than the eigenvalues of the corresponding matrix A for the other quality metrics. These eigenvalues are connected to the shape (angle) of the elements in the mesh.

52 32 Table 2.7 shows the angle (θ) distributions for elements in the initial mesh (i.e., with 50% of the interior vertices perturbed) and different quality metrics on the wrench (500K) and hinge (500k) domains. We observe that all four quality metrics try to generate nearly equilateral elements, but the SS metric yields meshes containing elements with more small and large angles than do the other quality metrics. This is because the SS metric penalizes small angles less than do other quality metrics [41]. For elliptic PDEs with isotropic coefficients, it is well-known that equilateral triangles are desirable for efficiency, whereas small angles have a bad effect on the condition number of matrix A and on the efficiency [41]. This explains why the SS metric is less efficient for mesh smoothing than other quality metrics. We also observe that the elements with small angles or large angles (i.e., poorly-shaped) occur most often on the boundary of the mesh. Interestingly, among the four quality metrics in Table 2.7, the RR metric has the fewest poorly-shaped elements (even fewer than the IMR). In terms of the total time, the IMR metric outperforms the other quality metrics, because numerical computation of the IMR metric for mesh optimization is highly optimized in Mesquite. Numerical computation of the other mesh quality metrics, i.e., RR, SI, and SS, are not as optimized and hence are less efficient to compute. Similar to the solver time, the SS metric is the least efficient quality metric in terms of the total time. The total time for the most efficient combination (i.e., AMG with CG) for the IMR metric is 90% less than it is for SS. The least efficient combination (i.e., Jacobi with GMRES) for the IMR metric is 45% faster than that of the least efficient quality metric, i.e., SI. Exp. 2: Effect of perturbation for Poisson s equation. Table 2.9 shows the linear solver time and the number of iterations required to converge as a function of the amount of vertex

53 33 (a) Wrench (500K) Quality Angle (θ) distribution Metric θ < < θ < < θ < < θ < 100 θ > 100 Initial IMR RR SI SS (b) Hinge (500K) Quality Angle (θ) distribution Metric θ < < θ < < θ < < θ < 100 θ > 100 Initial mesh IMR RR SI SS Table 2.7 Angle (θ) distribution for various mesh quality metrics. The reported values indicate a percentage of angles in the mesh. The initial mesh is the mesh with 50% of the interior vertices perturbed. perturbation on the wrench and hinge meshes, respectively. Figures 2.4(a) and 2.4(b) show the PI for different combinations of preconditioners and solvers when the number of perturbed vertices is increased. For these experiments, the meshes are smoothed with the IMR quality metric. The results obtained are representative of the results obtained with other quality metrics. More preconditioner-solver combinations are able to solve the PDEs on the wrench domain than on the hinge domain. The PDE on the hinge domain is harder to solve because there are many holes in the domain and the holes create thin areas near the boundary. The vertices are highly constrained in these areas and perturbation produces poorly-shaped elements in many cases. Table 2.9 shows that the CG and MINRES solvers fail to converge in many cases, whereas the GMRES and Bi-CGSTAB solvers do not. The CG and the MINRES solver often fail to converge because they fail to generate a positive definite preconditioner for two reasons. First,

54 34 (a) Wrench (500K) Quality Combinations of preconditioners and solvers Metric IMR RR SI SS * * 76.7 * * 3657 * 616 * Quality Combinations of preconditioners and solvers Metric IMR RR SI SS * * (b) Hinge (500K) Quality Combinations of preconditioners and solvers Metric IMR * * * 2404 * RR * * * 2360 * SI * * * 2380 * SS * * 63.5 * 87.8 * 43.9 * 2531 * 501 * 517 * 226 Quality Combinations of preconditioners and solvers Metric IMR RR SI SS Table 2.8 Linear solver time (secs) and number of iterations required to converge for Poisson s equation (problem (A) in Table 2.3) as a function of mesh quality metric for the 16 preconditioner-solver combinations (see Table 2.5) on the wrench and hinge domains. A * denotes failure. For each quality metric, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively.

55 35 generation of the preconditioners fails when there are poorly-shaped elements in the mesh and when the minimum eigenvalue of the matrix A is too small relative to its maximum eigenvalue. Poorly-shaped elements represent the elements which have large (e.g., angle > 100 ) or small angles (e.g., angle < 20 ). This was also observed in the above experiment for the SS metric. Secondly, preconditioners for the CG and MINRES solvers fail when the element lengths are very small. In these cases, the minimum eigenvalue of the matrix A is very small and corresponds to a large condition number. The minimum eigenvalue of the matrix A depends on the edge lengths in the mesh. For these reasons, when the element lengths are very small or when the meshes include lots of poorly-shaped elements, we see that GMRES and Bi-CGSTAB are more robust than CG and MINRES are to vertex perturbation. We also observe the following rank-ordering of preconditioners for robustness to vertex perturbation: ILU(0) > SSOR AMG > Jacobi. The ranking is in order of most robust to least robust. Note that the solver time for the AMG preconditioner is less than 10 seconds, whereas it is greater than 40 seconds for the other preconditioners. Hence, the PI values are sensitive to small changes of the solver time. Exp. 3: Increasing the problem size for Poisson s equation. We observe that the maximum eigenvalues of P 1 A stay constant, but the minimum eigenvalue of P 1 A rapidly decreases as N increases. This is consistent with the results observed in [41]. In [41], it was discussed that the minimum eigenvalue in matrix A is a function of the edge lengths in the mesh. It was also discussed in Section 2.5 that clustering of eigenvalues around the same non-null value results in a faster convergence rate. Thus, as the minimum eigenvalues of P 1 A decrease, the eigenvalue spectrum becomes larger, and the number of iterations required to converge increases.

56 36 (a) Wrench (500K) Quality Combinations of preconditioners and solvers Metric % % * * * * 53.4 * 2981 * 738 * 647 * % * * 93.3 * * 62.4 * 2930 * 724 * 670 * % * * 95.6 * * 58.9 * 3869 * 809 * 713 * 294 Quality Combinations of preconditioners and solvers Metric % % % % (b) Hinge (500K) Quality Combinations of preconditioners and solvers Metric % * * * 2404 * % * * * * * * * * 5432 * 1500 * % * * * 54.0 * * * * * 4122 * 2412 * % * * * * * * * * * * * 1676 * 740 Quality Combinations of preconditioners and solvers Metric % % * 84.2 * 42.6 * 7.3 * 20.0 * 505 * 234 * 15 * 27 25% * 90.0 * 44.4 * * 493 * 230 * % * 94.4 * 45.2 * 7.5 * 22.1 * 535 * 237 * 15 * 34 Table 2.9 Linear solver time (secs) and number of iterations required to converge for Poisson s equation (problem (A) in Table 2.3) as a function of vertex perturbation for the 16 preconditionersolver combinations (see Table 2.5) on the two geometric domains. A * denotes failure. For each percentage of vertices perturbed, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively.

57 37 (a) Wrench (500K) (b) Hinge (500K) Fig. 2.4 Percentage increase (PI) as a function of the solver time for different combinations of preconditioners and solvers for Poisson s equation (problem (A) in Table 2.3). Preconditionersolver combinations which fail to generate a preconditioner or do not converge correspond to the missing bars in these figures. Note that PI values for the hinge domain are significantly greater than those for the wrench domain.

58 38 Figure 2.5(a) shows the value of s in O(N s ) for the number of iterations required to converge for different combinations of preconditioners and solvers on the wrench domain. In this experiment, we use meshes smoothed by employing the IMR quality metric. The results are similar for other quality metrics. The AMG preconditioner with any solver (i.e., combinations in Table 2.5)yields values of s less than 0.1. The other combinations of preconditioners and solvers yield s values around 0.5. The Jacobi-GMRES combination has the largest value with s approximately 0.8. Figure 2.5(b) shows the same results on the hinge domain. Figures 2.6(a) and 2.6(b) show the order of convergence of the solver time for the wrench and the hinge domain, respectively. The best combination of quality metric, preconditioner, and solver is not affected by increasing the problem size. The AMG preconditioner with any solver beats the other combinations. In terms of the quality metric, the RR metric is the most efficient, whereas the SS metric is the least efficient in most cases Numerical Results for General Second-order Elliptic PDEs Exp. 1: Determination of the best combinations of mesh quality metric, preconditioner, and solver for general second-oder elliptic PDEs. Table 2.10 shows the solver time as a function of quality metric for various combinations of preconditioners and solvers for problem (B) in Table 2.3. We observe that the overall solver time for problem (B) is lower than it is for the other problems. The results show the most efficient combination for the solution of problem (B) is similar to the best combinations to solve problem (A). The most efficient combination is the AMG preconditioner with any choice of quality metric and solver. Similar to Poisson s equation, general second-order elliptic PDEs have one

59 39 (a) Wrench (b) Hinge Fig. 2.5 The order of convergence for the solver time based on the number of iterations for various combinations of preconditioners and solvers for Poisson s equation (problem (A) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method.

60 40 (a) Wrench (b) Hinge Fig. 2.6 Similar to Figure 2.5, this figure displays the order of convergence based on the solver time for Poisson s equation (problem (A) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method.

61 41 degree of freedom during the aggregation step for generation of the AMG preconditioner. Therefore, it results in fast solver time. The solver time for the most efficient combination is 95% less than the least efficient combination (e.g., the SI quality metric with the Jacobi preconditioner and the GMRES solver). For the choice of mesh quality metric, RR is more efficient than other quality metrics in most cases, and the SS quality metric is the least efficient quality metric. This result is also related to the angle distribution of mesh elements as discussed before. Similar to Poisson s equation (Problem (A)), the Bi-CGSTAB solver is most sensitive to the choice of different mesh quality metric, whereas, the CG solver is least affected by the choice of quality metric. We also observe that the use of any preconditioner significantly mitigates poorly-shaped elements for the SS metric when solving problem (B). This implies that, for this problem, some poorly-shaped mesh elements can be overcome, and the solver time does not increase when the best preconditioner is not chosen. In terms of the total time, the overall trend is similar to that seen for Poisson s equation (problem (A)). In most cases, we observe the following rank ordering of quality metrics with respect to the total time: IMR > RR > SI > SS. The ranking is in order of fastest to slowest. The most efficient combination is the IMR quality metric and the AMG preconditioner with any choice of the solver. Note that for problem (B), the smoothing time dominates the solver time more when compared with problem (A) because the overall solver time for solving problem (B) takes less time compared with that for problem (A). We also conducted experiments in which we modify the PDE coefficients in (2.3) such that the coefficient matrix, C = γi, where γ is a constant. We verified that the efficiency rankings are not affected by these modifications because they do not affect the optimal triangular shape,

62 42 and hence κ(a) is not affected. Also, the sparsity pattern of A is not changed by these modifications. We also modify the functions, f in (2.3) and observe consistent efficiency ranking results because modifying f only affects b of the linear system in (1). This is consistent with the theoretical analysis, which is explained in Section 2. In terms of another PDE parameter, a in (2.3), we consider a to be a constant and modify a values. We investigate the existence of a value on the efficiency rankings. If a is too large (i.e., a > 100) or too small (i.e., a < -100) compared with f (e.g., f = 1), the problem is dominated by the linear term and affects the efficiency rankings. Therefore, we assume -100 < a < 100 and study the presence of the mass matrix on the efficiency. Further discussion on the effect of existence of parameter a on efficiency rankings is presented in our previous paper [25]. Exp. 2: Effect of perturbation for general second-oder elliptic PDEs. Figures 2.7(a) and 2.7(b) show the changes in the solver time to compute the solution for the perturbed wrench (500K) and hinge meshes (500K), respectively. For this experiment, the meshes are smoothed with the IMR quality metric. The results obtained are representative of the results obtained with other quality metrics. The Jacobi preconditioner is most sensitive to the vertex perturbation. Note that the solver time for the AMG preconditioner is less than 10 seconds, whereas it is greater than 40 seconds for the other preconditioners. Hence the values are more sensitive to small changes. Similar to Poisson s equation, the SSOR preconditioner with the CG or MINRES solver (combinations 5 and 7 in Table 2.5) shows the most robust performance with respect to perturbation for both the wrench and hinge domains. A few poor quality elements can increase the values of the maximum eigenvalues and the condition number of the linear system. We observe that both CG and MINRES are able to circumvent the numerical difficulties associated with

63 43 (a) Wrench (500K) Quality Combinations of preconditioners and solvers Metric IMR RR SI SS * 3657 * 616 * Quality Combinations of preconditioners and solvers Metric IMR RR SI SS (b) Hinge (500K) Quality Combinations of preconditioners and solvers Metric IMR RR SI SS Quality Combinations of preconditioners and solvers Metric IMR RR SI SS Table 2.10 Linear solver time (secs) and number of iterations required to converge for general second-order elliptic PDEs (problem (B) in Table 2.3) as a function of mesh quality metric for the 16 preconditioner-solver combinations (see Table 2.5) on the two geometric domains. A * denotes failure. For each quality metric, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively.

64 44 large eigenvalues [41]. The GMRES solver (GMRES(30)) is most sensitive to the poor elements because it restarts in every 30 iterations without executing the additional iterations needed to converge faster. The least robust combinations are the Jacobi preconditioner with any choice of the solver. Exp. 3: Increasing the problem size for general second-oder elliptic PDEs. Figure 2.8(a) and 2.8(b) show the order of convergence for the number of iterations required to converge as we increase N. The overall trend is similar to that seen for problem (A). The Jacobi preconditioner with the GMRES solver has the largest s value which is The s value of the CG solver with any preconditioner is approximately 0.5. We observe that the AMG preconditioner is not sensitive to an increase in N. Four solvers when combined with the AMG preconditioner have s values less than 0.1. Figures 2.9(a) and 2.9(a) show the same results for the solver times. The results are similar to the results for problem (A). The best combinations (the RR quality metric and the AMG preconditioner with any choice of the solver) is not affected by increasing the problem size Numerical Results for the Linear Elasticity Problem Exp. 1: Determination of the best combinations of mesh quality metric, preconditioner, and solver for linear elasticity. Table 2.11 shows the solver time and number of iterations required for convergence as a function of various combinations of preconditioners and solvers for problem (C) in Table 2.3. We observe that the efficiency rankings are different from those obtained from Poisson s equation (problem (A)) and general elliptic PDEs (problem (B)).

65 45 (a) Wrench (500K) (b) Hinge (500K) Fig. 2.7 PI as a function of the solver time for various combinations of preconditioners and solvers after vertex perturbation for general second-order elliptic PDEs (problem (B) in Table 2.3).

66 46 (a) Wrench (b) Hinge Fig. 2.8 The order of convergence for the solver time based on the number of iterations for the combinations of preconditioners and solvers for general second-order elliptic PDEs (problem (B) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method.

67 47 (a) Wrench (b) Hinge Fig. 2.9 Similar to Figure 2.8, this figure displays the order of convergence based on the solver time for general second-order elliptic PDEs (problem (B) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method.

68 48 The ILU(0)-preconditioned solvers are more efficient than the AMG-preconditioned solvers. The reason for the difference is the difference in sparsity pattern of the A matrix for each application. For Poisson s equation, the matrix A has a sparsity pattern that corresponds to the mesh connectivity. Thus, the AMG preconditioner works very well when an aggregation step is performed. For the linear elasticity equations, A has twice the number of rows and columns as the number of vertices in the mesh. Thus, the AMG coarsening algorithm generates aggregates of physically-incompatible degrees of freedom [39]. This results in an increased solver time for the AMG preconditioner to solve the linear elasticity problem. Also, the BiCG-STAB solver shows an approximately 20% variation in the number of iterations required for preconditioner-solver combinations as a function of quality metric. The variation of other solvers is in the range of 10%. We observe the power of preconditioners in some cases. Although the meshes smoothed by the SS metric have more poorly-shaped elements than those obtained by smoothing using other quality metrics, the solver time for the SS metric is 50% less than that for the IMR metric. Among the 16 combinations of preconditioners and solvers, the most efficient combinations are the IMR or the SI quality metric with the ILU(0) preconditioner and the CG solver with respect to the solver time. On the wrench domain, the most efficient combination is 92% faster than the least efficient combination (i.e., the RR quality metric with the Jacobi preconditioner and the GMRES solver) with respect to the solver time. In terms of the total time, the IMR quality metric with the ILU(0) preconditioner and the MINRES solver outperforms other combinations. The least efficient combination is the SS quality metric with the Jacobi preconditioner and the GMRES solver. The total time for the most efficient combination (i.e., ILU(0) with MINRES) with the IMR metric is 71% less than it is

69 49 for this combination with SS. The least efficient combination (i.e., Jacobi with GMRES) for the IMR metric is 16% faster than that of the least efficient quality metric, i.e., SI. In this case, the total time is more significantly affected by the solver time. We conducted experiments in which we modify the PDE coefficients in (2.4) such that the coefficient matrix, C = γi, where γ is a constant. Similar to the second-order general elliptic PDE problems, experimental results show that the sparsity pattern of A, the optimal triangular shape, and hence κ(a) are not affected by these modifications as discussed in Section 2. Exp. 2: Effect of perturbation for linear elasticity. Table 2.12 shows the effect of the vertex perturbation on the solution of the linear elasticity problem. For these experiments, the meshes are smoothed with the IMR quality metric. The results shown here are typical. We observe that the effect of perturbation is different from problem (A) and (B). As was explained before, this is because the sparsity pattern of the linear system in this application is different from that of the previous applications. Figures 2.10(a) and 2.10(b) show the PI for different preconditioner-solver combinations when the number of perturbed vertices is increased. The AMG preconditioner with the GMRES solver (combination 14 in Table 2.5) is least sensitive to vertex perturbation. Similar to problem (B), the SSOR preconditioner with the CG and MINRES solvers (combinations 5 and 7 in Table 2.5) are also not very sensitive to perturbation. We observe that the SSOR preconditioner with the GMRES solver or Bi-CGSTAB solver and the ILU(0) preconditioner with the GMRES solver (combinations 6, 8, and 10 in Table 2.5) are most sensitive to perturbation. These two solvers are not efficient to circumvent the numerical difficulties associated with poor eigenvalues. The SSOR preconditioner with the Bi-CGSTAB solver takes 52% more time to converge than does the most robust combination (combination 14

70 50 (a) Wrench (500K) Quality Combinations of preconditioners and solvers Metric IMR RR SI SS Quality Combinations of preconditioners and solvers Metric IMR RR SI SS (b) Hinge (500K) Quality Combinations of preconditioners and solvers Metric IMR RR SI SS Quality Combinations of preconditioners and solvers Metric IMR RR SI SS Table 2.11 Linear solver time (secs) and number of iterations required to converge for the linear elasticity problem (problem (C) in Table 2.3) as a function of mesh quality metric for the 16 preconditioner-solver combinations (see Table 2.5) on the two geometric domains. A * denotes failure. For each quality metric, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively.

71 51 (a) Wrench (500K) (b) Hinge (500K) Fig PI as a function of the solver time after vertex perturbation for the combinations of preconditioners and solvers for the linear elasticity problem (problem (C) in Table 2.3). The missing bar (combination 10) for the hinge domain corresponds to a preconditioner-solver combination which does not converge.

72 52 (a) Wrench (500K) Quality Combinations of preconditioners and solvers Metric % % % % Quality Combinations of preconditioners and solvers Metric % % % % (b) Hinge (500K) Quality Combinations of preconditioners and solvers Metric % % % % Quality Combinations of preconditioners and solvers Metric % % % % 144 * * Table 2.12 Linear solver time (secs) and number of iterations required to converge for the linear elasticity problem (problem (C) in Table 2.3) as a function of vertex perturbation for the 16 preconditioner-solver combinations (see Table 2.5) on the two geometric domains. A * denotes failure. For each percentage of vertices perturbed, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively.

73 53 (a) Wrench (b) Hinge Fig The order of convergence for the solver time based on the number of iterations for the different combinations of preconditioners and solvers for the linear elasticity problem (problem (C) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear least-squares method.

74 54 (a) Wrench (b) Hinge Fig Similar to Figure 2.11, this figure displays the order of convergence based on the solver time for the linear elasticity problem (problem (C) in Table 2.3). Meshes with 10K, 50K, 100K, 200K, and 500K vertices on each domain are employed in computing s using the linear leastsquares method.

75 55 in Table 2.5). On both domains, the AMG preconditioner is less affected by vertex perturbation than the other preconditioners because the coarse grid correction during the V-cycles removes large eigenvalues effectively. Exp. 3: Increasing the problem size for linear elasticity. Figures 2.11(a) and 2.11(b) show the s values for the number of iterations required to converge for the wrench and hinge domains, respectively. For this application, the experiments are performed on meshes smoothed using the IMR quality metric. For the simple wrench domain, the s values for the AMG preconditioner are less than for the other preconditioners. For the more complex hinge domain, however, the order of convergence of the number of iterations for all the combinations are all close to 0.5. Note that for the Jacobi preconditioner with the GMRES solver, s is approximately 0.8. Figures 2.12(a) and 2.12(b) show the order of convergence of the solver time for the wrench and the hinge domain, respectively. In most cases, the best combinations are the RR quality metric with the ILU(0) preconditioner and the MINRES (or CG) solver. The least efficient combinations are the SS quality metric with the Jacobi preconditioner and the GMRES solver. 2.8 Conclusions and Future Work We studied the most efficient combinations of quality metric, preconditioner, and sparse linear solver for the numerical solution of various elliptic PDEs on 2D geometric domains. This is the first time these three important factors, which affect the efficiency of the solution for various elliptic PDEs, have been studied. According to our experimental results, by choosing

76 56 the most efficient combination, solver time and total time can be reduced by 90% and 97%, respectively, when compared to those with the most inefficient combination. For all elliptic PDEs considered here, we observe that the radius ratio (RR) metric is the most efficient metric for minimizing the solver time, as the mesh smoothed by the RR metric contains the fewest poorly-shaped elements. Poorly-shaped elements increase both the maximum eigenvalue and the condition number of the linear system. The interpolation-based size-andshape (SS) metric is the least efficient mesh quality metric in terms of its effect on the solver time because meshes smoothed with the SS metric have more poorly-shaped elements than the meshes smoothed by other quality metrics. We also observe that the choice of the preconditioner is the most important factor that affects the solver time for all elliptic PDEs. The choice of a good preconditioner is more important than the choice of a good quality metric or good linear solver even if the initial mesh quality is extremely poor. For solving Poisson s equation and general second-order elliptic PDEs, the most efficient combination for minimizing the solver time is the RR quality metric with the algebraic multigrid (AMG) preconditioner and any choice of linear solver. For these problems, which have a single degree of freedom, the AMG preconditioner efficiently eliminates large eigenvalues, and its aggregation step is fast. This results in a solver time which is 97% faster than the least efficient combination, i.e., the Jacobi preconditioner and the GMRES solver with any choice of quality metric. For linear elasticity problems, the RR metric with the ILU(0) preconditioner and the MINRES (or CG) solver is the most efficient combination, which are up to 92% faster than the least efficient combination, which is the Jacobi preconditioner and the GMRES solver with any choice of the quality metric. Due to its higher degree of freedom and different sparsity pattern,

77 57 where displacements for x and y are computed in a coupled manner, the AMG preconditioner is not as efficient as it is with the other two problems. The most efficient combinations are also studied with respect to the total time, which includes both the smoothing and the solver time. For all elliptic PDEs, choosing an efficient quality metric is more important than the other factors in order to minimize the total time (as opposed to just the solver time). The inverse mean ratio (IMR) metric computation is highly optimized in Mesquite, and thus combinations with the IMR metric appear in the best combination. For Poisson s equation and general second-order elliptic PDEs, the IMR metric with the AMG preconditioner and the CG solver is 90% faster than the same combination with the SS metric. For linear elasticity problems, the IMR metric with the ILU(0) preconditioner and MINRES (or CG) solver outperforms other combinations. We also investigated the robustness of the combinations with respect to vertex perturbation. These results are useful for choosing the most robust preconditioner-solver combination when the initial mesh quality is poor and mesh smoothing is not performed. In most cases, the SSOR preconditioner with the CG or MINRES solver is more robust than other combinations because they are able to circumvent numerical difficulties associated with large eigenvalues as discussed above. The AMG preconditioner is less affected by vertex perturbation than the other preconditioners because the coarse grid correction during the V-cycles removes large eigenvalues effectively. Finally, we examined the effect of increasing the problem size on the number of iterations required to converge and on the solver time. For all three PDEs, the AMG preconditioner with any quality metric or solver exhibits up to 94% faster convergence than the other combinations of preconditioners and solvers as the problem size is increased. The order of convergence for

78 58 Poisson s equation and general second-order elliptic PDEs, combinations with the AMG preconditioner are asymptotically much faster than the other combinations. Our experimental results can be generalized to the solution of other elliptic PDE problems with constant isotropic coefficients on various 2D geometric domains and homogeneous boundary conditions. However, they cannot be generalized to elliptic PDE problems with anisotropic coefficients or the 3D geometric domains for the following two reasons. First, elliptic PDEs with anisotropic coefficients require different ideal element shapes. Second, the quality metrics for 3D geometric domains are different from those for 2D elements. In this chapter, we investigated the best combinations of mesh quality metric, preconditioner, and sparse linear solver for solving various elliptic PDE problems. We focused on optimizing only one aspect of the mesh, such as the element shape or interpolation error, when we considered the effect the choice of various mesh quality metrics had on the solution of elliptic PDEs. In the next chapter, we propose a multiobjective mesh optimization framework which is able to simultaneously optimize multiple aspects of the mesh.

79 59 Chapter 3 A Multiobjective Mesh Optimization Framework for Mesh Quality Improvement and Mesh Untangling 3.1 Introduction Many partial differential equation (PDE)-based engineering and scientific applications have multiple requirements for the finite element mesh discretizing the geometric domain. For example, such requirements may include having non-inverted mesh elements, elements that are well-shaped, elements with uniform element size, and/or elements which yield small PDE interpolation error. Several shape-based mesh optimization algorithms have been devised to yield mesh elements which satisfy various geometric requirements or which improve the PDE solution accuracy or efficiency of the PDE solver [18, 40, 41]. Mathematical study of geometric mesh quality metrics based on element shape has been performed using the Jacobian and related matrices [18, 40]. Some interpolation error-based quality metrics have been proposed [41]. Both theoretical interpolation error and extremal eigenvalue bounds have been used to develop such mesh quality metrics. Although the size-based quality metric has been less studied by researchers, elements should be smoothly graded for numerical methods to behave properly [42].

80 60 Many researchers have also proposed methods to eliminate inverted elements in a tangled mesh. Knupp [43] proposed an untangling mesh quality metric by assigning a large penalty (cost) to inverted elements and no penalty to valid elements. Despite there being multiple mesh requirements for various PDE applications, most traditional mesh optimization algorithms optimize only a single objective function and hence improve only one aspect of the mesh. Optimizing only one aspect of the mesh often results in the deterioration of other aspects of the mesh at the same time. For instance, the untangle single objective method is able to eliminate inverted elements if the initial mesh is tangled. However, this method often yields elements with poor shapes by focusing only on elimination of the inverted elements while ignoring the element quality. These poorly-shaped elements ruin the accuracy of the PDE solution, and the efficiency, stability, and conditioning of the associated finite element solver. Moreover, the shape single objective method is able to improve element shape but cannot eliminate inverted elements in the tangled mesh. The resulting mesh yields an invalid finite element solution which cannot be used in a computational simulation. Similar tradeoffs exist in regards to optimizing other aspects of the mesh, such as the element shape and element size or the element shape and the PDE interpolation error. Therefore, it should be clear that meshes obtained by sequentially optimizing two or more aspects of the mesh do not simultaneously reflect all of the competing desired properties of the mesh. Researchers have also proposed a few techniques for optimizing two or more aspects of the mesh. Escobar et al. [44] proposed a method for simultaneously untangling and improving mesh quality by combining untangling and shape-based mesh quality metrics. However, their algorithm can only be used for mesh untangling and for mesh quality improvement. In addition, it is difficult to extend their method to improvement of other mesh aspects. Recently, Franks

81 61 and Knupp proposed new quality metrics within a target matrix paradigm for simultaneous mesh untangling and quality improvement [45]. However, this method is also difficult to extend to optimize other aspects of the mesh, as it develops two specific quality metrics for simultaneous shape improvement and untangling. Combining two metrics to obtain a single objective function based on the interpolation error and condition number of the stiffness matrix has also been proposed [41]. An objective function formed in this manner needs to determine relative preferences of the quality metrics; however, it is difficult to do so in advance. We propose a multiobjective mesh optimization framework for use in simultaneous improvement of multiple aspects of the mesh with the goal of improving the accuracy, efficiency, stability, and conditioning of the associated finite element solver. Examples of such aspects of the mesh which can be simultaneously improved by techniques within our framework include the element shape, element size, associated PDE interpolation error, and the number of inverted elements. Unlike the previous approaches, we are able to simultaneously improve several aspects of the mesh. We are also able to consider new combinations of mesh aspects for optimization. The biggest strength of our multiobjective framework lies in its flexibility and extensibility to other types of mesh improvement or other types of mesh elements. Techniques within our framework combine competing objective functions into a single objective function to be solved using a multiobjective optimization method. In addition, these techniques satisfy a sufficient condition of weak Pareto optimality, which means that there is no other solution that improves all of the objective functions simultaneously [91]. Multiobjective mesh optimization methods within our framework can be classified into a priori and no articulation of preferences categories of multiobjective optimization methods.

82 62 In order to demonstrate the potential of our multiobjective mesh optimization framework, we develop several multiobjective mesh optimization methods. We focus on methods which do not require articulation of preferences (weights), as weights for different objective functions do not need to be specified, and often they are not known. The techniques which we develop include the exponential sum, objective product, and equal sum multiobjective mesh optimization methods. We test our methods on challenging real-world applications such as problems in hydrocephalus treatment and shape optimization. The contents of this chapter have been accepted for publication in [89]. 3.2 Single Objective Mesh Optimization We formulate and solve several mesh optimization problems with competing goals. These competing goals can be categorized into four groups, i.e., improving mesh qualities based on the element shape, element size, the associated PDE interpolation error, and number of inverted elements. We assume the metrics will be used in the finite element solution of Poisson s equation for the isotropic physics case [41]. A similar approach can be taken for the solution of other PDEs and for the anisotropic case Mesh Quality Metrics For each of the four goals, we describe mesh quality metrics which belong to each category. For each mesh quality metric, the mesh quality of i th element on the mesh is denoted by q i. The mesh qualities below are for the 3D case. Similar quantities can be defined in 2D.

83 63 Shape. The inverse mean ratio (IMR) quality metric considers both shape and size but focuses more on the shape [18]. The IMR defines an incidence matrix for an ideal element shape, W, and transforms the incidence matrix of a non-optimal element to W. In the isotropic case, an equilateral triangle or tetrahedron is considered to be the ideal element. The IMR metric also uses an incidence matrix, C, and a barrier function, which prevents inverted elements. Let the coordinates of the four vertices of a tetrahedron denoted by a, b, c, and d. Then, C is given by [b a, c a, d a]. For an equilateral tetrahedron in 3D, W is defined as W = (3.1) For a tetrahedral element, the IMR metric is defined as CW 1 2 F q i =, 3 det(cw 1 ) 2 3 where CW 1 F = tr((cw 1 ) T (CW 1 )) is the Frobenius norm. For the IMR metric, a lower q i value indicates a higher mesh quality. The value of the IMR metric lies between 1 and for valid (non-inverted) elements. Another shape-based quality metric, i.e., a conditioning-based scale-invariant (SI) metric, was proposed to minimize the condition number of the stiffness matrix which is associated with Poisson s equation [41]. The SI metric is defined as the reciprocal of the upper bound of the maximum eigenvalue of the stiffness matrix formulated by discretizing Poisson s equation. To

84 do so, the ideal element shape is an equilateral tetrahedron for the solution of Poisson s equation. The SI metric is defined as 64 q i = V, (3.2) A 3 2 rms where V is the volume of a tetrahedron, and A rms is the root-mean-square area of the tetrahedron s 1 faces, which is given by A rms = 4 4 i=1 A2 i. Here A i is the i th unsigned area of a tetrahedron s face. Since a smaller maximum eigenvalue and condition number indicate a higher q i, a higher q i value for the SI metric indicates a better mesh quality. The value of the SI metric lies between 0 and 6/9 for valid elements. Size. The goal of optimizing the size-based metric is to adjust vertex positions to make the size of the elements as uniform as possible. Let the volume of a tetrahedron be V. The size metric is defined as q i = V. We employ the sum of squares of q i as the objective function. The value of the size metric lies between 0 and for valid elements. More generally, elements should be smoothly graded for numerical methods to behave properly.

85 Interpolation Error. A size-and-shape interpolation-based quality metric (SS) is defined as the reciprocal of the interpolation error bounds [41]. The SS metric is defined as 65 V 4 k=1 q i = A k 1 m n 4 A m A n lmn 2, where l mn is the edge length between the m th and n th vertices. The SS metric is not scaleinvariant; hence, the value of the SS metric lies between 0 and for valid elements. Untangling. Knupp proposed an untangling mesh quality metric using a signed element area or a volume [43]. The motivation behind his metric is to give high penalty (cost) to inverted elements and no penalty to valid elements. Knupp s untangling mesh quality metric is defined as q i = 1 ( V β (V β)), (3.3) 2 where β is a user defined parameter greater than 0. The value of the untangling metric lies between 0 (for a valid element) and. Contour Plots. Contour plots show different geometric features and can be used to identify ideal shapes for each quality metric. Fig. 3.1 shows contour plots for the (a) IMR, (b) SI, (c) SS, (d) size, and (e) untangling, respectively. In the plots, blue indicates higher mesh quality, and red indicates lower mesh quality. Fig. 3.1 shows that the SS metric penalizes less a large or a small angle as compared to the IMR and SI metrics.

86 (a) IMR (b) SI (c) SS (d) Size (e) Untangling Fig. 3.1 Contour plots of the quality metric of a triangle as a function of a free vertex when the two vertices are fixed at (0,0) and (1,0). In (e), β = Single Objective Functions In this section, we improve the average mesh quality by minimizing a single objective function. We denote the number of mesh elements by E and the overall mesh quality by F. For the IMR, size, and untangling metrics, we employ the sum of squares of individual element qualities of q i : For the SS metric, F = E i=1 q 2 i. (3.4) F = E i=1 1. (3.5) We use the reciprocal of the quality metric instead of the additive inverse to optimize the mesh because the optimization process for the latter results in meshes with elements of both very good q 2 i

87 67 and very poor quality, which is not desirable [73]. Using the reciprocal results in smoothed meshes with a more even distribution of element qualities. For the SI metric, we use a function with the sum of the reciprocals of the element qualities, because inverted elements have negative q i values, and the sum of squares function in (3.4) cannot differentiate between positive and negative q i values. 3.3 Multiobjective Mesh Optimization Methods Multiobjective Optimization Problems Simultaneously optimizing more than one objective function is referred to as multiobjective optimization. We employ a multiobjective optimization framework to simultaneously optimize more than one aspect of a mesh such as the element size, shape, associated PDE interpolation error, and number of inverted elements. Different from single objective optimization problems, multiobjective optimization problems generally do not have a common minimum. Pareto optimality is a widely used concept to define an optimal solution for multiobjective optimization problems. Let F be a vector of n single objective functions which is defined as F(x) = [F 1 (x),f 2 (x),...,f n (x)] T. A point, x X, is Pareto optimal if and only if there does not exist another point, x X, such that F(x) F(x ) and F i (x) < F i (x ), for at least one function.

88 A point, x X, is weakly Pareto optimal if and only if there does not exist another point, x X, such that 68 F(x) F(x ). In this case, there is no other point that simultaneously decreases all of the objective functions. There are various techniques for solving multiobjective optimization problems. Such techniques can be categorized into three classes based on articulation of preferences. The three categories are a priori articulation of preferences, a posteriori articulation of preferences, and no articulation of preferences [91]. In methods with a priori articulation of preferences, the relative importance (weight) of the objective functions are specified in advance before the optimization problem is solved. These preferences can be viewed as specifying weights for each objective function. One of the well-known methods, which belongs to this category, is a weighted-sum of single objective functions, which is denoted by, F = n i=1 w i F i, (3.6) where w i is a weight of objective function F i. In methods with a posteriori articulation of preferences, the preferences are chosen from a set of solutions, based on which solution is most appealing to the user, without considering the relative importance of the objective functions. The one method, which belongs to this category, is the genetic algorithm. In methods with no articulation of preferences, preferences are not used in the solution of the optimization problem.

89 69 Multiobjective Mesh Optimization Framework. Our multiobjective mesh optimization framework combines n competing objective functions, F, into a single objective function, represented by F (e.g., (3.6)). Any multiobjective optimization methods, which can be combined into a single objective function, can be employed to simultaneously optimize two or more aspects of the mesh within our framework. We do not list all available methods, but, for example, the weighted sum, weighted min-max, exponential weighted, lexicographic, and Rao s methods can be employed. Multiobjective optimization methods which can be combined into a single objective function are well summarized in [91]. We develop three multiobjective mesh optimization methods within our framework, which do not require articulation of preferences, namely the exponential sum, objective product, and equal sum methods. One of the main advantages of multiobjective methods with no articulation of preferences is that weights for different objective functions do not need to be specified, and often they are not known. Exponential Sum Method. The first method we consider is the exponential sum method [90], which approximates a min-max multiobjective problem. The min-max problem is to minimize the maximum (worst) cost function. The general multiobjective form of the problem is shown in (3.7). For the multiobjective mesh optimization problem, the goal of the min-max problem is to minimize the maximum (worst) cost function. Min-max mesh optimization problems have been studied in [46, 96]. However, researchers have only optimized one aspect of the mesh by employing a single objective function. min max i F i, (3.7)

90 70 Li [90] proposed the exponential sum function to approximate the min-max problem by employing an exponential penalty function, because the min-max problem is not smooth. The exponential sum method finds a local minimum of F = cln[ n i=1 e F i/c ], (3.8) where F i is the i th objective function (described in Section 3.2.2), and c > 0 is the controlling parameter. Typical values of c are between 10 4 and Li has shown that (3.8) is an approximation of max i F i. This implies that the exponential sum method is a smooth form of the min-max problem. It was proven that the solution to (3.8) satisfies a sufficient condition of weak Pareto optimality [47]. Objective Product Method. The second method we consider is the objective product method [47], which finds a local minimum of F = n i=1 F i. (3.9) One advantage of the objective product method is that it does not require normalization of the objective function. This is a nice property since many quality metrics are on widely different scales. It is easy to see that reducing any objective function by some ratio yields the same product value as does reducing any other objective function by that ratio. It was proven that (3.9) satisfies a sufficient condition of Pareto optimality [48]. Equal Sum Method. The third method we consider is the equal sum method [47]. If w i is set to one in (3.6), the weighted-sum methods become an equal sum method. The equal sum method

91 71 finds a local minimum of F = n i=1 F i. (3.10) It was proven that (3.10) satisfies a sufficient condition of Pareto optimality [48] Nonlinear Optimization Problems In this section, we give examples of multiobjective optimization problems. By combining the mesh quality metrics in Section 3.2 with the multiobjective optimization problems described in Section 3.3.1, one can define the specific optimization problem for the exponential sum method with the untangling and SI quality metrics as F = cln ( e 1 c E i=1[ 1 2 ( V i β (V i β))] 2 + e 1 c E j=1 3 ) A 2 rms V. (3.11) Similar optimization problems can be defined for the objective product and equal sum methods using other quality metrics. An examples of the optimization problem for the objective product method using the untangling and SI quality metrics is shown in (3.12) F = ( E i=1 [ ] ) ( V i β (V i β)) E j=1 A 3 2 rms V. (3.12) An example of the optimization for the equal sum method using the same metrics is shown in (3.13). F = ( E i=1 [ ] ) ( V i β (V i β)) + E j=1 A 3 2 rms V. (3.13) The complete set of combinations can be found in Section 3.4.

92 72 Since (3.8), (3.9), and (3.10) are nonlinear optimization problems, existing nonlinear optimization methods can be employed to minimize them. We employ the nonlinear conjugate gradient method (NLCG) to solve them, since NLCG requires only the gradient of the multiobjective function and provides a fast convergence rate. Let x k be the solution (a vector of vertex positions) at the k th iteration. NLCG determines a step length, α k by using a line search technique along a search direction, p k. The step length α k minimizes F along p k. The x k is updated by computing, x k+1 = x k + α k p k. The search direction, p k, is given by p k = F(x k ) + β PR k p k 1, (3.14) where p 0 = F(x 0 ), and β PR k is a parameter given by k = F(x k ) T ( F(x k ) F(x k 1 )) F(x k 1 ) T. (3.15) F(x k 1 ) β PR At each iteration, the variation in NLCG is from the computation of βk PR. We employ the Polak- Ribière NLCG method, since we observe in our preliminary experiments that the Polak-Ribière NLCG method is more efficient in finding a local minimum for our problems.

93 Comparison among the Exponential Sum, Objective Product, and Equal Sum Methods We compare the exponential sum, objective product, and equal sum multiobjective mesh optimization methods using a toy example. This example shows the effect of various multiobjective methods on different meshes based on their element qualities. For this experiment, we initially generate a structured mesh on the rectangular domain (Fig. 3.2(a)) and perturb one interior vertex in order to generate inverted elements (Fig. 3.2(b)). We employ both the shape (SI) and untangling metric and observe the effect of using the exponential sum and objective product multiobjective methods. We perform one iteration of mesh optimization using the NLCG implemented in Mesquite [16] to find a Pareto optimal point for the problems given by (3.11), (3.12), and (3.13). Figs. 3.2(c), 3.2(d), and 3.2(e) show the resulting meshes after one iteration of the exponential sum, objective product, and equal sum multiobjective mesh optimization methods, respectively. We observe that, on this problem, the exponential sum method eliminates inverted elements, whereas the equal sum and objective product methods are not able to do so. Figs. 3.2(f), 3.2(g), and 3.2(h) show output meshes with inaccurate mesh optimization for the exponential sum and objective product multiobjective methods, respectively. Here, inaccurate mesh optimization means the mesh quality does not change up to two digits. All three methods can eliminate inverted elements, but the exponential sum and the equal sum methods yield more uniform meshes with good element quality. Here, the optimal mesh of the exponential sum and equal sum multiobjective methods are nearly the same.

94 74 Since the exponential sum method is used to solve a min-max problem, it improves the worst objective function among the F i. As a result, for this toy example, the exponential sum method succeeds in eliminating inverted elements, because it focuses on eliminating the inverted elements which correspond to the worst F i. On the other hand, the objective product method improves the overall element quality by placing equal weights on the F i, but fails to eliminate inverted elements, although it improves the overall element shapes. Similar to the objective product method, the equal sum method also equally improves two competing quality metrics. However, the equal sum method behaves a lot differently if there exist inverted elements in the mesh. This is because the q i values for the inverted elements are negative for many quality metrics (e.g., IMR and SI) but are positive for the untangling metric. If the initial mesh has inverted elements or several poor quality elements, the exponential sum multiobjective method will be effective for multiobjective mesh optimizations because it focuses on improving the worst objective function. For some tangled meshes, which include inverted elements with very small areas or volumes (close to zero), the q i values for the untangling metric in (3.3) could be very small compared with the q i for the SI (shape) metric in (3.2), which is scale-invariant. For these cases, the exponential sum method is more effective for elimination of inverted elements if the value of q i for the untangling metric is rescaled by multiplying a constant such that the maximum q i value for the untangling metric is bigger than than the maximum q i value for the SI metric. This is because the exponential sum method improves the worst objective function among the F i. The objective product multiobjective method will be more useful if the initial mesh quality has few poor quality elements and the goal is to improve the overall mesh qualities using

95 75 equal weights. If the initial mesh is close to optimal, we observe that the objective product multiobjective method sometimes converges faster to the optimal mesh than do the exponential sum and equal sum multiobjective methods. 3.4 Numerical Experiments In this section, we describe our numerical experiments used to evaluate our mesh optimization methods. Table 3.1 summarizes the experiments and goals and gives a description of the goals for each problem. For Experiment 3.4.2, we use barrier functions, i.e., A > 0 for 2D meshes and V > 0 for 3D meshes, to prevent inverted elements when using the SS metric. For Experiment 3.4.3, we start with tangled meshes and investigate whether our mesh optimization methods are able to simultaneously improve the mesh quality and eliminate inverted elements. For Experiment 3.4.4, we investigate whether our mesh optimization methods can generate meshes which optimize three aspects of the mesh, such as the element shape and size, and result in no inverted elements. For the size metric, we report root mean square (rms) values of element qualities, since meshes with small rms values indicate meshes with uniformly-sized elements. Different from previous approaches, the strength of our multiobjective mesh optimization framework lies in its ability to be extended to improve any aspects of the mesh and to untangle the mesh. Our framework is not limited to the goals which are listed in Table 3.1; rather, it can be easily applied to other categories, such as orientation and regularity [49], by simply employing a different quality metric or objective function. Other element types can be also employed (e.g., hex and quad elements) [49].

96 76 (a) Initial structured mesh on a square domain (b) Perturbed mesh with two inverted elements (c) The output mesh of the exponential sum multiobjective method after one iteration of mesh optimization (d) The output mesh of the objective product multiobjective method after one iteration of mesh optimization (e) The output mesh of the equal sum multiobjective method after one iteration of mesh optimization (f) The optimal mesh of the exponential sum multiobjective method after accurate mesh optimization (g) The optimal mesh of the objective product multiobjective method after accurate mesh optimization (h) The optimal mesh of the equal sum multiobjective method after accurate mesh optimization Fig. 3.2 Comparison among the exponential sum, objective product, and equal sum multiobjective mesh optimization methods

97 77 For each experiment, we perform numerical experiments for both single and multiobjective mesh optimization algorithms (described in Sections 3.2 and 3.3, respectively). Note that our multiobjective mesh optimization algorithms result in meshes which are weakly Pareto optimal, whereas single objective algorithms result in meshes which are locally optimal. We also report on results for the worst element quality, because the worst element quality plays a critical role and can affect the PDE solution process [41]. The machine employed for this study is equipped with an Intel Core2 Duo Processor T7500 (2.20 GHz) and 2GB of RAM. Geometric Domains. The four geometric domains considered in our experiments are shown in Fig Freefem++ [50] software was used to generate the initial meshes. Half of the interior vertices in each mesh were perturbed to create test meshes that were further from optimal. For Experiments 3.4.1, 3.4.2, and 3.4.3, we employ meshes with approximately 1,000,000 elements. For Experiment 3.4.4, we employ meshes with approximately 50,000 elements, because they are sufficient to demonstrate and test the performance of our methods. Exp. Goals Description of Goals Name Shape and size Well-shaped and uniform-sized elements Shape and interpolation error Well-shaped elements and minimal interpolation error Shape and untangling Well-shaped and noninverted elements Shape, size, and untangling Well-shaped, uniform-sized, and noninverted elements Table 3.1 Listing of numerical experiments, goals, and a description of goals.

98 78 (a) Disk (2D) (b) Barrier (2D) (c) Disk (3D) (d) Barrier (3D) Fig. 3.3 Coarse initial meshes on the disk and barrier geometric domains indicative of the actual meshes to be smoothed. Mesh Smoothing, Untangling, and Nonlinear Solver. For Experiments , a local implementation of the NLCG method (described in Section 3.3.2) was used to solve the mesh optimization problems. We employed the Polak-Ribière NLCG method in Mesquite [16] to solve each mesh optimization problem, because our preliminary experiments show that it is the fastest amongst different versions of the nonlinear conjugate gradient method. We perform inaccurate mesh smoothing as in [31, 73] such that the mesh qualities are the same up to two digits after the decimal point. We do not perform accurate mesh smoothing because accurate mesh smoothing (fully converged solution) significantly increases the smoothing time, but slightly increases the mesh quality Numerical Results for Optimizing Shape and Size For this experiment, we consider two competing goals: improving both the shape (IMR) and size metrics. Fig. 3.4 shows the average element qualities based on the (a) shape (IMR) and (b) size metrics on the 3D disk mesh, respectively. We observe that the size with the single objective mesh optimization method yields the worst result among the four different mesh optimization algorithms when the shape metric is used to measure the mesh quality. Similarly, the shape with the single objective mesh optimization method shows the worst results with respect

99 79 to the size metric. Our two multiobjective mesh optimization algorithms improve the average mesh qualities with respect to both metrics. Since the ideal element shape is an equilateral tetrahedron for optimization of the IMR metric, we can expect that the elements in the output mesh employing the IMR metric will have similar volumes. However, the size metric with the single objective method does not result in equilateral elements, because two different elements can have the same volumes although they have different shapes. Therefore, optimizing the mesh with respect to the size metric only can have a negative effect on the element shape. Fig. 3.5 shows the worst element qualities based on the (a) IMR and (b) size metrics on the 3D disk mesh, respectively. Fig. 3.5(a) shows that the size metric with the single objective method has a negative effect on the element shape of the mesh. As the number of iterations increases, the worst element quality of the size metric with the single objective method measured by the IMR metric deteriorates significantly. Fig. 3.5(b) shows that the IMR metric with the single objective method has a positive effect on the size metric for the first few iterations. Note that the optimization of the IMR metric can be beneficial for improving the size metric. This is because the ideal element shape when employing the IMR metric is an equilateral triangle or tetrahedron. Experimental results show that our exponential sum multiobjective mesh optimization algorithm improves the worst element quality by 96.6% when compared with the use of the size metric with the single objective method when measured by the IMR metric. Table 3.2 shows the smoothing times for various mesh optimization methods in terms of the number of iterations of mesh smoothing. The size metric with the single objective method shows the fastest smoothing time, because the numerical computation of the size metric is quite

100 80 simple compared with the computation for the other mesh optimization methods. For 20 iterations of mesh smoothing, our objective product mesh optimization algorithm takes approximately two times longer compared with the size metric with the single objective mesh optimization method. We also observe similar trade off between mesh quality improvement and efficiency for other quality metrics and geometric domains. Mesh optimization method Number of iterations IMR single objective Size single objective Exponential sum multiobjective (IMR and size) Objective product multiobjective (IMR and size) Equal sum multiobjective (IMR and size) Table 3.2 Smoothing time (secs) for various mesh optimization methods in terms of the number of iterations of smoothing on the 3D disk mesh Numerical Results for Optimizing Shape and Interpolation Error For this experiment, we consider two competing goals: improving both shape (IMR) and interpolation error (SS) metrics. Fig. 3.6 shows the average element qualities based on the (a) shape (IMR) and (b) interpolation error (SS) metrics on the 3D disk mesh. We observe that our multiobjective mesh optimization algorithms significantly improve the average mesh qualities based on both the IMR and SS metrics. Fig. 3.6 also shows that the SS metric with the single objective method fails to improve the IMR metric. In our previous paper [73], and in the contour plots in Section 3.2, we observed that the ideal shape when employing the SS metric is different from the one when employing the IMR metric, because the SS metric penalizes small angles less than do other mesh quality metrics. Therefore, the SS metric with the single objective method can have a negative effect on the IMR metric. However, the IMR metric with the single objective

101 81 Average element quality IMR single objective Size single objective Exponential sum multiobjective (IMR and size) Objective product multiobjective (IMR and size) Equal sum multiobjective (IMR and size) Number of iterations of mesh smoothing (a) Average element quality as measured by shape (IMR) Average element quality (rms) IMR single objective Size single objective Exponential sum multiobjective (IMR and size) Objective product multiobjective (IMR and size) Equal sum multiobjective (IMR and size) Number of iterations of mesh smoothing (b) Average element quality as measured by size (volume) Fig. 3.4 (a) Average element quality in terms of the shape metric (IMR) on the 3D disk mesh; (b) average element quality in terms of the size metric (volume) on the same mesh

102 82 Worst element quality (log scale) IMR single objective Size single objective Exponential sum multiobjective (IMR and size) Objective product multiobjective (IMR and size) Equal sum multiobjective (IMR and size) Number of iterations of mesh smoothing (a) Worst element quality as measured by shape (IMR) Worst element quality (log scale) IMR single objective Size single objective Exponential sum multiobjective (IMR and size) Objective product multiobjective (IMR and size) Equal sum multiobjective (IMR and size) Number of iterations of mesh smoothing (b) Worst element quality as measured by size (volume) Fig. 3.5 (a) Worst element quality in terms of the shape metric (IMR) on the 3D disk mesh; (b) worst element quality in terms of the size metric (volume) on the same mesh

103 83 method can be beneficial in terms of the SS metric. This is because the initial mesh has lots of poor quality elements which contain very small or large angles. Fig. 3.7 shows the worst element qualities based on the (a) IMR and (b) SS metrics. The overall trend is similar to the one with the average element qualities. Experimental results show that our exponential sum multiobjective mesh optimization algorithm improves the worst element quality by 95.4% compared with the SS metric with the single objective method according to the IMR metric. Similar to the average element qualities, the IMR metric with the single objective method can also improve the quality of the SS metric. However, after 10 iterations of mesh smoothing, the worst element quality employing the IMR metric slightly increases (deteriorates) because of the different ideal element shapes for the IMR and interpolation error metrics. Table 3.3 shows smoothing times for the various number of iterations of mesh smoothing for the mesh optimization methods. The IMR metric shows the fastest smoothing time because IMR is highly optimized in Mesquite. For the 20 iterations of mesh smoothing, our equal sum multiobjective method takes four times longer than the IMR metric with the single objective method, although it significantly improves both mesh quality metrics simultaneously. These results indicate a tradeoff between mesh quality improvement and efficiency. Mesh optimization method Number of iterations IMR single objective SS single objective Exponential sum multiobjective (IMR and SS) Objective product multiobjective (IMR and SS) Equal sum multiobjective (IMR and SS) Table 3.3 Mesh smoothing time (secs) for various mesh optimization methods in terms of the number of iterations of smoothing on the 3D disk domain.

104 84 Average element quality (log scale) IMR single objective SS single objective Exponential sum multiobjective (IMR and SS) Objective product multiobjective (IMR and SS) Equal sum multiobjective (IMR and SS) Number of iterations of mesh smoothing (a) Average element quality as measured by shape (IMR) Average element quality IMR single objective SS single objective Exponential sum multiobjective (IMR and SS) Objective product multiobjective (IMR and SS) Equal sum multiobjective (IMR and SS) Number of iterations of mesh smoothing (b) Average element quality as measured by interpolation error (SS) Fig. 3.6 (a) Average element quality as measured by the shape metric (IMR) on the 3D disk mesh; (b) average element quality as measured by the interpolation error (SS) on the same mesh

105 85 Worst element quality (log scale) IMR single objective SS single objective Exponential sum multiobjective (IMR and SS) Objective product multiobjective (IMR and SS) Equal sum multiobjective (IMR and SS) Number of iterations of mesh smoothing (a) Worst element quality as measured by shape (IMR) Worst element quality (log scale) IMR single objective SS single objective Exponential sum multiobjective (IMR and SS) Objective product multiobjective (IMR and SS) Equal sum multiobjective (IMR and SS) Number of iterations of mesh smoothing (b) Worst element quality as measured by interpolation error (SS) Fig. 3.7 (a) Worst element quality in terms of the shape metric (IMR) on the 3D disk mesh; (b) worst element quality in terms of the interpolation error (SS) metric on the same mesh

106 Numerical Results for Optimizing Shape and Untangling For this experiment, our multiobjective mesh optimization algorithms simultaneously improve the element shape and eliminate inverted elements in tangled meshes. Approximately 1% of the elements in the initial mesh are inverted such that the average mesh quality and the maximum mesh quality of the initial tangled mesh based on the untangling quality metric are and , respectively. This experiment is challenging because the SI metric without a barrier often generates inverted elements during mesh optimization. For tangled meshes, we employ the SI metric instead of the IMR metric, because the IMR metric includes a barrier to prevent inverted elements and the IMR metric in Mesquite [16] is not able to compute valid function gradients and function values for inverted elements. However, the SI metric can be used to improve the shape of the mesh elements, because it does not include the barrier in the metric. Also, the ideal element for both the IMR with W in (3.1) and SI metrics is the equilateral triangle (tetrahedron). Tab. 3.4 shows the numbers of inverted elements after employing various mesh optimization methods. The untangling metric with the single objective method eliminates all inverted elements after three iterations of mesh optimization. Our three multiobjective methods successfully eliminate all inverted elements within three iterations of mesh optimization. Note that both our exponential sum and equal sum multiobjective algorithm even eliminates inverted elements in fewer iterations of mesh optimization than does the untangling metric with the single objective method. This is because our multiobjective algorithm tries to simultaneously improve mesh qualities and eliminate inverted elements in tangled meshes.

107 87 Fig. 3.8 shows the output meshes for four different mesh optimization methods after 10 iterations of mesh optimization on the 2D barrier mesh. The initial mesh includes 543 inverted elements and exhibits poor mesh quality in terms of the SI (shape-based) metric. Since the ideal element shape for the SI metric is assumed to be an equilateral element, the multiobjective mesh optimization algorithms improve the element shape while eliminating inverted elements. Fig. 3.8 also shows that the exponential sum multiobjective mesh optimization algorithm yields the best element shape among five mesh optimization methods. The elements in the final mesh resulting from the exponential sum method are closer to equilateral compared with the elements of the meshes produced by the other three mesh optimization algorithms. These experimental results are consistent with the toy example in Section in that the exponential sum method is more effective in eliminating inverted elements than the objective product method, because it focuses on improving the worst objective function, which is the untangling objective function in this case. In terms of the mesh optimization times, all three multiobjective methods were able to simultaneously improve element qualities while eliminating inverted elements in less than one minute. Comparison with the Simultaneous Untangling and Smoothing (SUS) Algorithm of Escobar et al. on a 3D Tangled Mesh. One of the existing algorithms for simultaneously improving element shape and performing mesh untangling is due to Escobar et al. [44]. This algorithm is called Simultaneous Untangling and Smoothing (SUS) and is able to optimize only two specific aspects (element shape and untangling) of the 3D tetrahedral mesh. It cannot easily be extended to optimize other aspects of the mesh. We compare our methods with the SUS algorithm using

108 88 Mesh optimization method Number of iterations SI single objective 543 1,019 1,030 1,030 1,030 Untangle single objective Exponential sum multiobjective (SI and Untangle) Objective product multiobjective (SI and Untangle) Equal sum multiobjective (SI and Untangle) Table 3.4 Number of inverted elements as a function of the number of iterations of mesh optimization. The initial mesh has 543 inverted elements. Mesh optimization method Number of iterations SI single objective 111 1,392 1,390 1,389 1,389 1,389 Untangle single objective Exponential sum multiobjective (SI and Untangle) Objective product multiobjective (SI and Untangle) Equal sum multiobjective (SI and Untangle) Simultaneous untangling and smoothing (SUS) [44] Table 3.5 Number of inverted elements as a function of the number of iterations of mesh optimization. The initial mesh has 111 inverted elements.

109 89 (a) (b) (c) (d) (e) (f) Fig D barrier mesh: (a) the initial mesh with inverted elements (the elements circled in red); (b) the final mesh with inverted elements (the elements circled in red) when employing the SI metric with the single objective method. The number of inverted elements increases after mesh optimization; (c) the final mesh when employing the untangling metric with the single objective method; (d) the final mesh for the exponential sum multiobjective method; (e) the final mesh for the objective product multiobjective method; (f) the final mesh for the equal sum multiobjective method. The final mesh by employing the exponential sum multiobjective method yields the best element quality while eliminating all inverted elements.

110 90 Average element quality Untangle single objective Exponential sum multiobjective Objective product multiobjective Equal sum multiobjective SUS [6] Number of iterations of mesh optimization (a) Average element quality as measured by shape (SI) 10 2 Worst element quality Untangle single objective Exponential sum multiobjective Objective product multiobjective Equal sum multiobjective SUS [6] Number of iterations of mesh optimization (b) Worst element quality as measured by shape (SI) Fig. 3.9 (a) Average element quality in terms of the shape metric (IMR) on the 3D disk mesh; (b) worst element quality in terms of the size metric (volume) on the same mesh

111 91 their source codes posted in [51] on the tangled 3D disk domain, which includes 111 inverted elements. Table 3.5 shows the number of inverted elements in terms of the number of iterations of mesh optimization on the 3D disk domain. We observe that the SUS algorithm is able to eliminate all inverted elements after one iteration of mesh optimization, while our three multiobjective mesh optimization methods each take one more iteration to remove inverted elements. Fig. 3.9 shows both the average and worst element qualities based on the SI metric (shape). Both the exponential sum and equal sum multiobjective methods significantly improve the average and worst element qualities on the mesh and outperform the SUS algorithm. The SUS algorithm improves the element quality after one iteration of mesh optimization; however, the element quality does not improve and sometimes even deteriorates as the mesh optimization proceeds. The shape metrics used in this work and by the SUS algorithm are different. However, these results are also consistent with the results we obtained with the shape metric in [44] is employed to measure the element qualities. For other meshes, the SUS algorithm is able to eliminate inverted elements while improving element qualities. However, the element qualities are not as good as the element qualities obtained by our methods. Also recall that our multiobjective methods are flexible and can optimize any aspects of the mesh; however, the SUS algorithm is able to optimize only two specific aspects (element shape and untangling) of the mesh.

112 Numerical Results for Optimizing Shape, Size, and Untangling For this experiment, we consider three competing goals: improving the shape and size metrics and eliminating inverted elements. The 3D disk mesh which include 111 inverted elements in is also used as an initial mesh. Tab. 3.6 shows the number of inverted elements in terms of the number of iterations of mesh optimization. Our three multiobjective methods and the untangling metric with the single objective algorithm are able to eliminate all the inverted elements. However, only the exponential sum and the equal sum multiobjective mesh optimization methods improve the element shape while eliminating all inverted elements. Similar to Experiment 3.4.3, our exponential sum multiobjective mesh optimization algorithm eliminates all inverted elements with fewer iterations. This is because improving the SI metric also helps to eliminate inverted elements. In summary, both our exponential sum and equal multiobjective mesh optimization algorithms improve two competing mesh quality metrics while eliminating inverted elements in a tangled mesh. Fig.10 shows the average element quality based on the (a) shape (SI) and (b) size metrics for six mesh optimization algorithms on the 3D disk domain. We observe similar results for other geometric domains. Among the six mesh optimization methods, only our exponential sum and equal sum multiobjective mesh optimization algorithms are able to simultaneously improve two different mesh quality metrics while eliminating inverted elements. The size single objective method is able to simultaneously improve the element shape and size but it fails to eliminate inverted elements.

113 Average element quality SI single objective Size single objective Untangle single objective Exponential sum multiobjective Objective product multiobjective Equal sum multiobjective Number of iterations of mesh optimization (a) Average element quality as measured by shape (SI) Average element quality (rms) 2.8 x SI single objective Size single objective Untangle single objective Exponential sum multiobjective Objective product multiobjective Equal sum multiobjective Number of iterations of mesh optimization (b) Average element quality as measured by size (volume) Fig (a) Average element quality in terms of the shape metric (SI) on the 3D disk mesh; (b) Average element quality in terms of the size (volume) metric on the same mesh

114 94 Mesh optimization method Number of iterations SI single objective 111 1,392 1,390 1,389 1,326 Size single objective Untangle single objective Exponential sum multiobjective (SI, Size, and Untangle) Objective product multiobjective (SI, Size, and Untangle) Equal sum multiobjective (SI, Size, and Untangle) Table 3.6 Number of inverted elements as a function of the number of iterations of mesh optimization. The initial 3D disk mesh has 111 inverted elements. 3.5 Application of Our Multiobjective Mesh Optimization Methods to Real-World Applications In this section, we consider real-world applications and demonstrate how our multiobjective mesh optimization methods within our framework can be used to simultaneously optimize multiple aspects of the mesh. First, we study the effect of employing the multiobjective mesh optimization methods for problems in which warped meshes have created meshes with inverted elements and elements with poor qualities. For the first two experiments, we investigate whether the exponential sum, objective product, and equal sum multiobjective mesh optimization methods can simultaneously untangle the inverted mesh and improve the element qualities. We also study the effect of employing the exponential sum, objective product, and equal sum multiobjective mesh optimization methods (shape and size) on the efficiency of the finite element solution process. In this process, we consider the solution of the following Poisson s equation: u = 1 with u = 0 on the boundary. We employ the piecewise linear Galerkin finite element method as

115 95 in [73] and use PETSc [17] to compute the maximum eigenvalue, minimum eigenvalue, condition number of the stiffness matrix A, and solve the resulting linear system. The Jacobi preconditioner (P) with the minimal residual (MINRES) solver is used to solve the linear system. We consider two application problems with deforming domains in Sections and For these problems, both mesh warping and remeshing can be used to generate meshes on deformed domains. However, mesh warping is preferred to remeshing for many mesh deformation problems, because remeshing results in the accumulation of large interpolation errors between successive time steps and less accurate PDE simulation results [52]. Also, similar meshes, which have the same element connectivity, are desired between successive time steps such that the solution varies smoothly during mesh deformations [54, 88] Mesh Warping Problem on 2D Hydrocephalus Domains Hydrocephalus (also called water in the brain) is a neurological condition and is one of the most common neurological disorder treated by neurosurgeons [81]. A problem with the flow of cerebrospinal fluid (CSF) results in hydrocephalus and causes the brain ventricles to become enlarged as shown in Fig. 3.11(a). Medical treatment through CSF flow diversion by surgically implanting a CSF shunt in the brain ventricles is one common treatment of decreasing the volume of the fluid in the brain ventricles as shown in Fig. 3.11(b). The segmented images of the brain were provided by the authors in [81]. More details on the segmentation of the medical images and initial mesh generation are explained in [81]. We consider a 2D hydrocephalus mesh warping problem for tracking the evolution of the ventricles after treatment. The initial and deformed domains for the mesh warping problem are shown in Fig. 3.11(c) and Fig. 3.11(d), respectively. The initial brain mesh was provided by the

116 96 authors in [81]. The boundary vertices are known, and the deformed mesh was obtained by using FEMWARP [88] to warp the initial mesh to the target brain as shown in Fig. 3.11(d). (Note that the brain boundary (i.e., the external boundary) (not shown) was also provided by the authors in [81].) For this example, FEMWARP generates inverted elements due to large deformations as shown in Fig. 3.11(d). We do not compare with the SUS algorithm [44] on the 2D hydrocephalus domain since the SUS algorithm and source codes posted in [51] are designed for 3D tangled meshes. Figs. 3.11(e) to 3.11(h) show close-up views of the meshes before smoothing, after applying 40 iterations of the exponential sum mutiobjective method and the untangle single objective method, respectively. The initial mesh has 8,166 elements and no inverted elements. The warped mesh after employing FEMWARP has 13 inverted elements [81]. Even though smoothed mesh in Figs. 3.11(f) has no inverted elements, the elements of the meshes in Fig. 3.11(g) and Fig. 3.11(h) have better shapes and improve the worst quality element (i.e., the very skinny triangle in the middle of the mesh). Note that this example is a challenging problem because there are numerous interior vertices near the internal boundary on the deformed domain which are highly constrained due to the large deformation. Table 3.7 shows the number of inverted elements in terms of the number of iterations of mesh optimization on the 2D hydrocephalus domain (Fig. 3.11(d)). The exponential sum multiobjective method eliminates inverted elements after 9 iterations of mesh optimization, while the untangling single objective method requires 33 iterations. As discussed earlier, the exponential sum method minimizes the worst objective function, which is the untangling objective function in this case. However, the objective product method improves the overall qualities with respect

117 to both shape and untangling. Although it slowly decreases the number of inverted elements, it fails to eliminate all inverted elements on the mesh. 97 Mesh optimization method Number of iterations SI single objective Untangle single objective Exponential sum multiobjective (SI and Untangle) Objective product multiobjective (SI and Untangle) Equal sum multiobjective (SI and Untangle) Table 3.7 Number of inverted elements as a function of the number of iterations of mesh optimization. The initial mesh has 13 inverted elements. The worst element quality on the mesh highly affects the efficiency for solving elliptic PDEs (also Poisson s equation), because both the condition number of the stiffness matrix and the efficiency are closely related with the element shape on the mesh. Poorly-shaped elements increase the maximum eigenvalue and condition number of the stiffness matrix [41]. The twonorm condition number of the stiffness matrix, A, is defined as κ 2 (P 1 A) = λ max (P 1 A)/λ min (P 1 A), Table 3.8 shows the maximum eigenvalue, minimum eigenvalue, and condition number of the stiffness matrix when the above Poisson s equation is solved on meshes using various mesh optimization methods. Both the exponential sum and equal sum multiobjective methods have smaller maximum eigenvalues because they have fewer poorly-shaped elements than does the untangle single objective method. As a result, the exponential sum multiobjective method has a 19% smaller condition number and is more efficient when solving Poisson s equation compared with the untangle single objective method.

118 98 (a) Segmented image of the initial hydrocephalus domain (before treatment) [81] (b) Segmented image of the deformed hydrocephalus domain (after treatment) [81] (c) Initial mesh with no inverted elements on the hydrocephalus domain, Fig. 3.11(a) [81] (d) Deformed mesh on the hydrocephalus domain (Fig. 3.11(b)) deformed with FEMWARP. This mesh has 13 inverted elements. (e) Close-up view of the mesh in 3.11(d) with inverted elements enclosed by the oval (f) Close-up view of the mesh after applying 40 iterations of the untangling single objective method on the mesh shown in Fig. 3.11(d). This mesh has no inverted elements. (g) Close-up view of the mesh after applying 40 iterations of the exponential sum multiobjective method on 3.11(d). This mesh has no inverted elements. (h) Close-up view of the mesh after applying 40 iterations of the equal sum multiobjective method on 3.11(d). This mesh has no inverted elements. Fig Segmented images and meshes of initial and deformed domains.

119 99 Mesh optimization method λ max (P 1 A) λ min (P 1 A) κ 2 (P 1 A) Iterations SI single objective * * * * Untangle single objective e Exponential sum multiobjective (SI and Untangle) e Objective product multiobjective (SI and Untangle) * * * * Equal sum multiobjective (SI and Untangle) e Table 3.8 Maximum eigenvalue, minimum eigenvalue, condition number of P 1 A, and the number of iterations required to converge to solve Poisson s equation on the 2D hydrocephalus domain. Here, P 1 A is the preconditioned stiffness matrix. A * denotes invalid PDE solution due to inverted elements on the mesh Mesh Warping Problem on 3D Domains For this experiment, we consider a problem from mesh warping for shape optimization on 3D geometric domains [94]. For shape optimization problems, geometric parameters need to be determined automatically [94]. However, for general shape optimization problems on deforming domains, it is not simple to create mapping functions from the initial domain onto the deformed domain. Therefore, mesh warping algorithms are often used to find the new locations of vertices on deformed domains. Similar to the mesh warping problem in Sec , the boundary vertices on the deformed domain are known and the meshes are warped from the initial domain to the deformed domain. The meshes were provided by the authors in [94]. In this case, FEMWARP [88] was employed for mesh warping. The initial mesh, which is shown in Fig. 3.12(a), contains 109,535 elements; none of the elements are inverted. However, the deformed mesh, which is shown in Fig. 3.12(b), contains 109,535 elements with 13 inverted elements. Note that the computation time for using FEMWARP on these meshes is less than one second on an HP ProLiant linux workstation with two X5570 Intel quad cores with 24 GB RAM [94].

120 100 The SI single objective mesh optimization method is not able to eliminate inverted elements. However, the other three mesh optimization methods, which are the exponential sum, the objective product, and untangle single objective methods, are able to untangle the inverted elements in one iteration of mesh optimization. Table 3.9 shows the maximum eigenvalue, minimum eigenvalue, and condition number of the preconditioned stiffness matrix when Poisson s equation is solved using various mesh optimization methods. Similar to its performance on other problems, both the exponential sum and equal sum multiobjective mesh optimization methods outperform other mesh optimization methods. Similar to the comparison with the SUS algorithm in Section 3.4.3, the element qualities of the output mesh from the SUS algorithm [44] are worse than those obtained by our multiobjective methods. Therefore, the stiffness matrix generated by the SUS algorithm has a larger condition number and takes more iterations to solve Poisson s equation compared to our multiobjective methods. (a) Initial mesh with no inverted elements on the 3D bore domain [94] (b) Deformed mesh on the 3D bore domain [94] deformed with FEMWARP [88]. This mesh has 13 inverted elements which are located inside the domain. Fig Meshes of initial and deformed domains.

121 Mesh optimization method λ max (P 1 A) λ min (P 1 A) κ 2 (P 1 A) Iterations SI single objective * * * * Untangle single objective e , Exponential sum multiobjective (SI and Untangle) e , Objective product multiobjective (SI and Untangle) e , Equal sum multiobjective (SI and Untangle) e , Simultaneous untangling and smoothing (SUS) [44] e , Table 3.9 Maximum eigenvalue, minimum eigenvalue, condition number of P 1 A, and the number of iterations required to converge to solve Poisson s equation on the 3D bore domain. Here, P 1 A is the preconditioned stiffness matrix. A * denotes invalid PDE solution due to inverted elements on the mesh Effect of Simultaneously Optimizing Both Shape and Size Metrics on the Efficiency for Solving Possion s Equation We study the effect of employing shape (IMR) and size multiobjective mesh optimization methods on the condition number of the stiffness matrix and the number of iterations for solving Poisson s equation on the 2D disk domain (Fig. 3.3(a)) with 500K elements. The dominant factor which affects the condition number of the stiffness matrix is the element shape rather than element size. However, when two meshes have the same number of elements, we can expect that a more uniform mesh (more smoothly graded mesh) has a smaller condition number than a nonuniform mesh if the mesh element have the same shapes. This is because the mesh with uniform-sized elements has a larger λ min than does a nonuniform mesh. The smallest-sized element in the non-uniform mesh decreases the minimum eigenvalue of the stiffness matrix and increases the condition number [41].

122 102 Table 3.10 shows the maximum eigenvalue, minimum eigenvalue, and condition number of the preconditioned stiffness matrix when Poisson s equation is solved using various mesh optimization methods. The exponential sum multiobjective mesh optimization method has the smallest condition number based on its small λ max (P 1 A) and its large λ min (P 1 A). The size single objective mesh optimization method shows the slowest convergence rate because of its large κ 2 (P 1 A). Mesh optimization method λ max (P 1 A) λ min (P 1 A) κ 2 (P 1 A) Iterations Shape single objective e-05 49, Size single objective e-05 56, Exponential sum multiobjective e-05 48, Objective product multiobjective e-05 48, (IMR and Size) Equal sum multiobjective (IMR and Size) e-05 48, Table 3.10 Maximum eigenvalue, minimum eigenvalue, and condition number of P 1 A, and number of iterations to converge to solve Poisson s equation. Here, P 1 A is the preconditioned stiffness matrix. 3.6 Conclusions and Future Work We have proposed a multiobjective mesh optimization framework for use in simultaneous improvement of various aspects of the mesh with the goal of improving the accuracy, efficiency, stability, and conditioning of the associated finite element solver. The novelty of our framework lies in its ability to combine multiple aspects of the mesh, besides just smoothing and untangling, for mesh optimization. Specifically, we develop three methods within our framework, namely the exponential sum, objective product, and equal sum mesh optimization methods, which do not require articulation of preferences. Therefore, the relative preferences for different objective

123 103 functions do not need to be specified. Moreover, our solutions satisfy a sufficient condition of weak Pareto optimality. Our experimental results show that our multiobjective mesh optimization methods improve the worst element qualities by up to 96.6% compared with the corresponding single objective mesh optimization algorithms, respectively. Our experimental results also show that, in most cases, the exponential sum multiobjective method is the most effective out of the three methods we tested in simultaneously improving two or more aspects of the mesh. The exponential sum multiobjective method is effective in improving the worst cost objective function efficiently when the initial mesh quality is poor and the mesh contains inverted elements. The equal sum multiobjective method is simple and shows comparable performance to the exponential sum method. Experimental results show that both our exponential sum and equal sum multiobjective methods outperform the existing simultaneous untangling and smoothing (SUS) algorithm. Moreover, both our exponential sum and equal sum multiobjective methods improve two competing mesh quality metrics while eliminating inverted elements in a tangled mesh. For real-world applications, such as hydrocephalus treatment and shape optimization, our multiobjective mesh optimization methods are more effective in terms of simultaneously untangling and improving the element shape than both the untangle single objective method and SUS. In addition, our methods are more effective at decreasing the condition number of the stiffness matrices and the number of iterations needed to solve elliptic PDEs on the meshes than are the relevant single objective mesh optimization methods. In addition, for the 2D hydrocephalus treatment application, our exponential sum multiobjective mesh optimization method significantly decreases the number of iterations required to untangle all inverted mesh elements as compared to the untangle single objective method.

124 104 In this chapter, we proposed a multiobjective mesh optimization framework for mesh quality improvement and mesh untangling. In this dissertation, we have thus far focused on mesh optimization techniques for PDE problems. However, traditional PDE methods are not able to compute derivatives for discontinuous domains. However, new theory based on integral equations called a nonlocal peridynamics model has recently developed to address these issues. In the next chapter, we investigate the effects that mesh anisotropy and size (i.e., mesh refinement) have on the conditioning of the stiffness matrix for a nonlocal peridynamics model.

125 105 Chapter 4 The Effect of Anisotropy, Mesh Refinement, and Kernel Functions on the Conditioning of the Stiffness Matrix for Nonlocal Peridynamic Models 4.1 Introductions Many applications in solid mechanics such as damage, surface cracks, and fracture have discontinuities in their structures. Classical partial differential equations (PDEs) have some limitations in their modeling and computation of discontinuous domains, since they are not able to compute derivatives on such domains. In order to address this limitation of classical PDEs, a nonlocal peridynamic model was recently developed by Silling [56]. The peridynamics nonlocal theory uses integral equations rather than differential equations to model cracked surfaces and deformations. It was reported that this peridynamics model was successfully applied to other applications such as turbulence [57], nanofibers [61, 62], porous flow [60], and fracture and damage modeling of membranes [61]. The reader is referred to [63, 64] for recent surveys on nonlocal peridynamics models and applications. There have been various studies on nonlocal peridynamics models. The connection between classical PDEs (such as an elasticity theory) and

126 106 nonlocal peridynamics was investigated [65]. This study showed that a nonlocal peridynamics model is close to a classical local model when the length scale (horizon) goes to zero. The effect of various kernel functions on the nonlocal advection problem was investigated for a 1D problem [67]. Researchers have also studied finite difference and finite element discretization of nonlocal peridyanmics models [68 71]. Specifically, a posteriori error analysis and the connection between the horizon and the condition number was studied in [68]. Condition number estimates and upper bounds for the discretized linear system were studied [72]. The connections between the horizon, mesh size, and condition number were also investigated for isotropic elements [72]. On the other hand, the interactions among the mesh geometry, mesh refinement, and the condition number of the global stiffness matrix for classical PDEs have been well studied [9, 41, 73]. The connection between the anisotropy of mesh elements and the condition number for elliptic PDEs was investigated [9]. Various mesh quality metrics, interpolation error, and the condition number for elliptic, parabolic, and hyperbolic PDEs were investigated [41]. The connections between the mesh quality metric, preconditioner, and the linear solver for elliptic PDEs were computationally studied in [73]. We use the Galerkin finite element method for discretizing a linear peridynamic system and study the effect of the anisotropy of the mesh element (element shape), mesh refinement (element size), and kernel functions on the condition number for a bond-based nonlocal peridynamic model in 2D geometric domains. There exist various peridynamic models such as a bond-based model [56, 69] and a state-based peridynamic model [58]. We consider a bond-based peridynamic model which only considers central forces between particles [56, 59, 69]. We will consider both integrable and nonintegrable kernel functions and numerically show the effect of these kernel functions on the condition number for both piecewise linear and piecewise constant

127 107 basis functions. The study on the conditioning is important because it affects the accuracy of the solution and the convergence rate of solving the discretized linear system. This study is the first to explore the connections among anisotropy, mesh refinement, and the condition number for 2D meshes with various kernel functions for a nonlocal peridynamic model. This work is computationally challenging for two reasons. First, we need two different quadrature rules to approximate the double integral terms in 2D geometric domains to avoid the singularity of the denominator when we compute the condition number of the global stiffness matrix. Second, it is desirable to compute approximately the area of intersection between the horizon (δ) and the triangular element when the quadrature rule is used. We consider three different scenarios: (1) a piecewise constant basis function with an integrable kernel function, (2) a piecewise linear basis function with an integrable kernel function, and (3) a piecewise linear basis function with a nonintegrable kernel function. For each scenario, we investigate the effect of changing the anisotropy and mesh size (h) on the conditioning of the nonlocal peridynamic model. The analytical results show that the condition number is bounded by cδ 2 (c is a constant) [72] when an integrable (finite) kernel function is employed. For the first two scenarios with an integrable kernel function, we will numerically show that the condition number is almost not affected by the choice of the basis function. We will also numerically compute the constant c in the condition number bound (cδ 2 ) on uniform triangular and rectangular meshes in 2D. For a piecewise linear basis function with a nonintegrable basis function, we expect that the condition number is similar to that of general elliptic PDEs since this scenario is similar to a local model. For general elliptic PDEs, it is well-known that the condition number is proportional to h 2 when the mesh has the same anisotropy on uniform triangular and rectangular meshes in 2D [41]. Mesh anisotropy also affects the condition number for general

128 108 elliptic PDEs. For instance, let θ be the smallest angle in the right triangle, then the condition number is proportional to sin 1 (2θ) [9]. Therefore, the condition number sharply increases as θ approaches 0. For a piecewise linear basis function with a nonintegrable kernel function, we expect that the condition number is affected by the anisotropy and the mesh size similar to the general elliptic PDE case. We will investigate the effect of the anisotropy and the mesh size on the condition number and study the difference between the nonlocal peridynamic model and PDEs. The contents of this chapter will soon be submitted for review [110]. 4.2 A Bond-based Nonlocal Peridynamic Model The peridynamic model replaces a PDE with an integral equation. The bond-based peridynamic model assumes that the solid body consists of small particles and that these small particles interact with each other [59]. The interaction between particles is called a bond, and each particle interacts with other particles within a distance, δ (horizon). We assume that δ is a sphere. The equation of motion in the bond-based peridynamic model at point v = (x,y) is denoted by Lu(v) = f ( u ( v ) u(v) ) dv, (4.1) H where L is a linear integral operator, u is a displacement vector, H is a spherical neighborhood of particles which interact with a particle x, f is a pairwise peridynamic force, and dv is an infinitesimal volume related with a particle v, respectively. From Newton s Second Law, ρü(v) = Lu(v) + b(v), (4.2)

129 where ρ is the mass density, and b(v) is an external force. Combining (4.1) and (4.2) results in 109 ρü(v) = f ( u ( v ) u(v) ) dv + b(v). (4.3) H Let ξ be the distance between v and v. Then, f is 0 if ξ is larger than δ due to the definition of nonlocality. The horizon (δ) and force ( f ) between v and v over the geometric domain Ω are illustrated in Fig We only consider a microelastic material, which is commonly used in Fig. 4.1 Connection between horizon (δ), force ( f ) between v and v, inside the neighborhood, H, over the domain Ω. The point v does not interact with any points beyond the distance δ. the nonlocal peridynamic literature. For the microelastic material, the interaction between two particles is considered to be an elastic spring [56, 69]. For the bond-based nonlocal peridynamic model with a microelastic material, (4.3) can be reduced to the linearized peridynamic model which is denoted as ρü(v) = c (v v) (v v) ( ( H v v p u v ) u(v) ) dv + b(v), (4.4)

130 110 where is the Kronecker product. The reader is referred to [56, 69] for further information on properties of microelastic models. Since we assume a steady-state model, ü(v)=0. Then, (4.4) can be further reduced to H c (v v) (v v) v v p (u ( v ) u(v)) dv + b(v) = 0. (4.5) For a one-dimensional domain (e.g., v = x) with a Dirichlet boundary condition, (4.5) is expressed as 1 v+δ u(v) u(v ) δ 4 p v δ v v p dv = b(v), v Ω, where u(v) = g(v) on the boundary, Ω, The global stiffness matrix, A, is computed by using a standard Galerkin finite element method and basis functions, φ (v). The basis functions will be further discussed in Sec The M-by-M matrix A is defined as A i j = 1 ( φ j (v) φ j (v ) ) δ 4 p H v v p,φ i (v), where M is the number of mesh elements for a piecewise constant basis function and can be represented in the double integral form A i j = 1 φ j (v) φ j (v ) δ 4 p φ i (v) I H v v p dv dv. (4.6)

131 111 Here I represents all the neighborhoods of a point v where φ i (v) is nonzero. We define 1/ v v p to be the kernel function. The matrix A is a symmetric, positive-definite matrix and is sparse. Figure 4.2 shows an example of the sparsity pattern of the global stiffness matrix on a 2D geometric domain. Compared with the PDE sparsity pattern, more nonzeros appear due to nonlocality. The band of this sparse matrix is determined by the horizon length, δ. Another important difference between classical PDEs and nonlocal peridynamic models is the definition of the boundary condition. Figure 4.3 shows the difference between the classical PDE boundary condition and the nonlocal peridynamic boundary condition. For classical PDEs, the boundary condition, BΩ, is defined for boundary points. However, the boundary conditions are defined for a boundary area, BΩ, for nonlocal peridynamic models. Therefore, the additional mesh generation on the boundary area, BΩ, is required with respect to the horizon, δ. Fig. 4.2 One example of the sparsity pattern of a matrix A for the nonlocal peridynamic model. Here nz is the number of nonzeros in A.

132 Fig. 4.3 Boundary conditions for (a) classical PDE models and (b) nonlocal peridynamic models. For classical PDE problems, the boundary condition is only specified on the boundary points. For nonlocal peridynamic models, all mesh elements belonging to BΩ belong to the boundary condition. 112

133 Quadrature Rules and Basis Functions Quadrature rules for a triangle Our goal is to compute the condition number of (4.6) for 2D geometric domains. It is computationally inexpensive to compute (4.6) using quadrature rules similar to [69]. In particular, we need to choose two different quadrature rules for inner and outer integrals because the denominator term, i.e., v v, could be singular if we choose the same quadrature rule twice. The quadrature rule for integration of f over a triangle can be computed as T f (x,y)dx dy T m i=1 f (x i,y i )w i, where T is a triangle in Ω, T is the area of T, and m is the number of points used in the associated quadrature rule [74]. We employ with one point and three point quadrature rules to approximate the outer and inner integrals, respectively. For the unit triangle whose vertices are (0,0), (0,1), and (1,0), the one point quadrature rule (centroid), which has degree of precision 1, is computed as (x 1,y 1 ) = (1/3,1/3), w 1 = 1. For the unit triangle whose vertices are (0,0), (0,1), and (1,0), the three point quadrature rule (Strang [74]), which has a degree of precision 3, is computed as [74], (x 1,y 1 ) = (2/3,1/6),(x 2,y 2 ) = (1/6,2/3),(x 3,y 3 ) = (1/6,1/6), w 1 = 1/3,w 2 = 1/3,w 3 = 1/3.

134 114 Since the horizon is a circle and the mesh element is a triangle, we use another approximation to compute the intersection area of T with the horizon (circle). We consider three different cases when the intersection of T with δ is computed. The first case arises when the entire triangle is inside the horizon. For this case, the T is exactly the area of a triangle and is shown in case 1 in Fig The numerical difficulty arises when only part of a triangle is located inside the horizon as shown in cases 2 and 3 in Fig The exact intersection area is expensive to compute for cases 2 and 3. Therefore, we use approximations to compute the intersection areas of T with δ for both cases 2 and 3. For case 2, we approximately compute the intersection area inside the horizon by adding the areas of the two triangles inside the horizon as shown in Fig For case 3, we compute the intersection area of a triangle inside the horizon rather than the entire intersecting arc area inside the horizon. Fig. 4.4 Three different cases for computing T when the intersection of T with δ is computed. Here, we only compute the intersection areas inside the horizon.

135 115 Fig. 4.5 The approximation of the area of intersection of the triangular element and the horizon for case 2 in Fig Basis functions The continuous piecewise constant (PC) basis function cannot be used for classical PDEs because the derivative of zero is not defined. However, the nonlocal peridynamic model is able to use the continuous PC basis function since it uses an integral equation instead of a derivative. The PC basis function is 1 if x is on the j th element, otherwise, it is 0. The continuous PC basis function is defined as 1, if (x,y) T j φ j (x,y) = 0, otherwise. The continuous piecewise linear (PL) basis function with respect to the j th vertex is defined as [(y l y k )(x x k ) + (x k x l )(y y k )]/[(x j x k )(y j y l ) (x j x l )(y j y k )], φ j (x,y) = if (x,y) neighborhood of j th vertex 0, otherwise, where k and l are two other vertices of an element to which (x,y) belongs.

136 4.4 Connections among the horizon, mesh refinement, anisotropy of the mesh element and the condition number of the global stiffness matrix Condition number of the global stiffness matrix for a nonlocal peridynamic model The connection between the horizon, mesh size, and condition number of the global stiffness matrix is studied analytically for an integrable kernel function [72]. The condition number is bounded as follows cond(a) c min { δ 2,h 2}, where c is a constant and h is the mesh size. Since many nonlocal problems assume that δ is greater than h, this equation indicates that the condition number of the global stiffness matrix is bounded by the horizon, δ. In other words, the condition number is proportional to δ 2 when δ is greater than h. This equation also indicates that decreasing the horizon increases the upper bound of the condition number. We consider three different choices of basis functions and kernel functions in (4.6). The power of a kernel function, p, is chosen based on the following equation. In d-dimensional space (where d {1,2,3}) H 1, p d v v p dx = <, p < d. (4.7) We consider the case where d=2. If the power of a kernel function p 2, (4.6) is nonintegrable (infinite). Otherwise (if p < 2), it is integrable (finite).

137 117 Piecewise constant basis function with an integrable kernel function. For the integrable kernel function, we set p to be 1 in (4.7). For this kernel function, we expect that the condition number is bounded by cond(a) c min { δ 2,h 2}, where c is a constant and the condition number converges to a stable level for the mesh refinement and anisotropy. Since we assume that the δ is bigger than the mesh size, h, the condition number is bounded by cond(a) c δ 2. Piecewise linear basis function with an integrable kernel function. For the piecewise linear basis function with an integrable kernel function, we set p to be 1 in (4.7). Similar to the piecewise constant basis function with an integrable kernel function case, we expect that the condition number is bounded by, cond(a) c δ 2, when δ is larger than the mesh size, h. Piecewise linear basis function with a nonintegrable kernel function. For this nonintegrable kernel function, we assume that k is greater than two. For this type of peridynamic model, we expect that the condition number behavior is similar to the classical PDE behavior, since the nonlocal peridynamimcs model is close to a local model such as a PDE model for this case. This

138 means that the anisotropy and the mesh refinement increase the condition number similar to the PDE model Condition number of the global stiffness matrix for general second-order elliptic PDEs The effect of the mesh size and the mesh anisotropy on the conditioning of the global stiffness matrix for general second-order elliptic PDEs has been studied [9, 41]. Suppose we solve the general second-order elliptic PDEs on a uniform structured triangular mesh with right triangles on 2D rectangular domains. Let θ be the smallest angle in the right triangle, then the condition number is proportional to sin 1 (2θ) [9]. This equation indicates that the condition number sharply increases and approaches infinity as θ approaches to 0. If we further assume that the mesh elements have the same anisotropy, the condition number is proportional to h Numerical results on 2D rectangular meshes We investigate the effect of anisotropy of the mesh element, mesh refinement, and various kernel functions on the condition number of the global stiffness matrix of A in (4.6) on 2D structured rectangular meshes, i.e., Ω = (0,1) (0,1). We assume that δ is larger than the mesh size, h. We use the 2-norm condition number to compute the condition number of the global stiffness matrix A. Mesh generation and boundary condition. We consider structured triangular meshes on the rectangular geometric domain as shown in Fig The element sizes (edge lengths) in the initial mesh, after one level of mesh refinement, and after two levels of mesh refinement, are (a)

139 , (b) 0.05, and (c) 0.025, respectively. Additional triangular elements were generated in the area surrounding Ω inside the area where the boundary conditions are defined. One example of the anisotropic mesh, whose aspect ratio is 4, is shown in Fig 4.7. We fix the total number of elements for both isotropic and anisotropic meshes for given h and δ. The aspect ratio of a right triangle is defined as, aspect ratio = W/w [75], where w and W are shortest and the second shortest edge lengths, respectively. Figure 4.8 shows the definition of the aspect ratio of a triangle. Computation time. We use an OpenMP library to speed up the computation time. Note that we parallelize our code such that the parallelization of our code using an OpenMP library does not affect the condition number. The machine employed for this study is equipped with a 48 core AMD opteron 2.3 GHz processor. When δ=0.2 and piecewise constant basis functions are used, the total time to compute cond(a) is 1 second (h=0.1), 11 seconds (h=0.05), and 341 seconds (h=0.025), respectively. The running time for piecewise linear basis functions is much greater than the the running time for piecewise constant basis functions since we must consider all the neighborhood elements of each vertex. When δ =0.2 and piecewise linear basis functions are used, the total time to compute cond(a) is 1 second (h=0.1), 11 seconds (h=0.05), and 100,825 seconds (h=0.025), respectively.

140 120 (a) Isotropic mesh with h= 0.1 and δ=0.2 (b) Refined isotropic mesh with h= 0.05 and δ=0.2 (c) Refined isotropic mesh with h= and δ=0.2 Fig. 4.6 (a) Initial isotropic mesh with h=0.1 and δ=0.2; (b) refined mesh with h=0.05 and δ=0.2; (c) Two level-refined mesh with h=0.025 and δ=0.2

141 121 Fig. 4.7 Anisotropic mesh (aspect ratio=4) with δ=0.2. Fig. 4.8 The definition of the aspect ratio of a triangle. The aspect ratio is defined as W/w [75].

142 Piecewise constant basis function with an integrable kernel function For these experiments, we employ piecewise constant basis functions and fix the power of the kernel function, p in (4.6), to be 1 for an integrable kernel function. Table 4.1 shows the condition number of the global stiffness matrix, A, for various mesh sizes and horizons (δ) for both isotropic and anisotropic meshes. Here, the ratio is defined as the ratio of the condition number of the refined mesh (after one level of mesh refinement) divided by the condition number of the coarser mesh for the same δ. We also observe that the overall ratio decreases as δ increases for a fixed level of mesh refinement. This is because the numerical computation and the condition number for larger δ values are less sensitive than is the computation with smaller δ values. When δ is small (i.e., δ=0.2), a large portion of triangular elements within the horizon intersects the horizon (e.g., cases 2 and 3 in Fig. 4.4), and the increased ratio of intersections between the horizon and the triangular elements decreases the numerical stability. (a) Isotropic meshes with aspect ratio 1 Isotropic mesh δ =0.2 ratio δ =0.3 ratio δ =0.4 ratio 200 elements (no mesh refinement) * 8.70 * 5.65 * 800 elements (1 level of mesh refinement) elements (2 levels of mesh refinement) (b) Anisotropic meshes with aspect ratio 4 Anisotropic mesh δ =0.2 ratio δ =0.3 ratio δ =0.4 ratio 200 elements (no mesh refinement) * 8.00 * 5.53 * 800 elements (1 level of mesh refinement) elements (2 levels of mesh refinement) Table 4.1 Cond(A) of the global stiffness matrix for piecewise constant basis functions with an integrable basis kernel function (p=1). The number of elements is the number of elements in Ω. Here, the ratio is the ratio of the condition number of the current level of mesh refinement compared with that of the previous level of mesh refinement for a fixed δ. Therefore, these ratios are not defined for the 200 elements and are denoted by * for these cases.

143 123 Fix the mesh size (h) and vary the anisotropy of the elements. We compute the condition number while fixing the mesh size (h) and vary the anisotropy (aspect ratio) of the elements. Figures 4.9 and 4.10 show the connection between δ, anisotropy, and the condition number. We observe that the condition number of the global stiffness matrix A is proportional to δ 2 when δ is bigger than h, and the increasing anisotropy of the elements does not affect the condition number. The condition number is bounded by cδ 2 where c is a constant that is close to 1. Fix the anisotropy of the elements and vary the mesh size (h). We fix the anisotropy of the elements and compute the condition number for different mesh sizes ( h). Figures 4.11 and 4.12 show the connection between the horizon (δ), mesh refinement, and the condition number of the global stiffness matrix. Similar to the previous numerical results, the condition number of the global stiffness matrix A is proportional to δ 2 when δ is larger than h. We observe that the condition number of the global stiffness matrix A converges to a stable level as the number of mesh refinement levels increases (as h decreases by half). We observe that the condition number is bounded by cδ 2, where c is a constant that is close to Piecewise linear basis function with an integrable kernel function For these experiments, we use piecewise linear basis functions and fix the power of the kernel function, p, in (4.6) to be 1 for an integrable kernel function. Based on the analytical results in Sec. 4.4, we expect that the overall trend is similar to that seen for piecewise constant basis functions with an integrable kernel function. Table 4.2 shows the condition number of the global stiffness matrix for various mesh sizes and horizons (δ) for both isotropic and anisotropic

144 124 (a) h=0.05 (b) h=0.025 Fig. 4.9 Condition number of A for fixed h and varying anisotropy of the elements and δ. (a) h=0.05 (b) h=0.025 Fig Condition number of A as a function of δ 2 for fixed h and varying anisotropy of the elements and δ.

145 125 (a) Isotropic mesh (aspect ratio=1) (b) Anisotropic mesh (aspect ratio=4) Fig Condition number of A for fixed anisotropy and varying h and δ. (a) Isotropic mesh (aspect ratio=1) (b) Anisotropic mesh (aspect ratio=4) Fig Condition number of A as a function of δ 2 for fixed anisotropy and varying h and δ.

146 126 meshes. Here, the ratio is defined as the ratio of the condition number of the refined mesh (after one level of mesh refinement) divided by the condition number of the coarser mesh for the same δ. We observe that the overall ratio decreases as δ increases for a fixed level of mesh refinement in most cases. Similar to what we observed for piecewise constant basis functions, a large portion of triangles within the horizon intersects the horizon (e.g., cases 2 and 3 in Fig. 4.4) and the increased ratio of intersections increase the numerical instability. (a) Isotropic meshes with aspect ratio 1 Isotropic mesh δ =0.2 ratio δ =0.3 ratio δ =0.4 ratio 200 elements (no mesh refinement) * 6.12 * 4.00 * 800 elements (1 level of mesh refinement) elements (2 levels of mesh refinement) (b) Anisotropic meshes with aspect ratio 4 Anisotropic mesh δ =0.2 ratio δ =0.3 ratio δ =0.4 ratio 200 elements (no mesh mesh refinement) * 7.36 * 4.31 * 800 elements (1 level of mesh refinement) elements (2 levels of mesh refinement) Table 4.2 Cond(A) of a global stiffness matrix for piecewise linear basis functions with an integrable basis kernel function (p=1). The total number of elements changes with respect to the horizon, δ. The number of elements indicates a number of elements in Ω. Here, the ratio is defined as the ratio of the condition number of the current level of mesh refinement compared with that of the previous level of mesh refinement for a fixed δ. Therefore, these ratios are not defined for the 200 elements and are denoted as * for these cases. Fix the mesh size (h) and vary the anisotropy of the elements. We compute the condition number of the global stiffness matrix while fixing the mesh size (h) and varying the anisotropy of the elements. Figures 4.13 and 4.14 show the connection between the horizon (δ), anisotropy, and the condition number of the global stiffness matrix. Similar to the piecewise constant basis function with the integrable function, the condition number of the global stiffness matrix A is

147 proportional to δ 2 when δ is bigger than h. These results are consistent with the analytical results in Sec We observe that the condition number is bounded by δ Fix the anisotropy of the elements and vary the mesh size (h). We fix the anisotropy and compute the condition number for different values of h. Figures 4.15 and 4.16 show the connection between the horizon (δ), mesh refinement, and the condition number. We also observe that the condition number is proportional to δ 2 when δ is larger than h and is bounded by cδ 2. We also observe that the condition number of A converges to a stable level as the number of mesh refinement levels increases. Numerical results show that c is close to Piecewise linear basis function with the nonintegrable kernel function For these experiments, we fix the power of the kernel function, i.e., p in (4.6) to be 2.5 for a nonintegrable kernel function. Table 4.3 shows the condition number of the global stiffness matrix for various mesh sizes and horizons (δ) for both isotropic and anisotropic meshes. Here, the ratio is defined as the ratio of the condition number of the refined mesh (after one level of mesh refinement) divided by the condition number of the coarser mesh for the same δ. We observe that the condition number increases as δ decreases. Previously, we observed that the overall ratio is close to 1 when δ is 0.4. However, for these cases, the ratio is not close to 1, even if δ is 0.4 and the condition number is not bounded by the horizon. Fix the mesh size (h) and vary the anisotropy of the elements. We compute the condition number of the global stiffness matrix while fixing the mesh size, h, and varying the anisotropy

148 128 (a) h=0.05 (b) h=0.025 Fig Condition number of A for fixed h and varying anisotropy of the elements and δ.

149 129 (a) h=0.05 (b) h=0.025 Fig Condition number of A for fixed h and varying anisotropy of the elements and δ.

150 130 (a) Isotropic mesh (aspect ratio=1) (b) Anisotropic mesh (aspect ratio=4) Fig Condition number of A for fixed anisotropy and varying h and δ.

151 131 (a) Isotropic mesh (aspect ratio=1) (b) Anisotropic mesh (aspect ratio=4) Fig Condition number of A for fixed anisotropy and varying h and δ.

152 (a) Isotropic meshes with aspect ratio 1 Isotropic mesh δ =0.2 ratio δ =0.3 ratio δ =0.4 ratio 200 elements (no mesh refinement) * 7.80 * 6.07 * 800 elements (1 level of mesh refinement) elements (2 levels of mesh refinement) (b) Anisotropic meshes with aspect ratio 4 Anisotropic mesh δ =0.2 ratio δ =0.3 ratio δ =0.4 ratio 200 elements (no mesh refinement) * * 8.54 * 800 elements (1 level of mesh refinement) elements (2 levels of mesh refinement) Table 4.3 Cond(A) of a stiffness matrix for piecewise linear basis functions with a nonintegrable basis kernel function (p=1). The number of elements indicates the number of elements in Ω. Here, the ratio is defined as the ratio of the condition number of the current level of mesh refinement compared with that of the previous level of mesh refinement for a fixed δ. Therefore, these ratios are not defined for the 200 element case and are denoted as * for such cases. of the elements. Figure 4.17 shows the connection between the horizon (δ), anisotropy, and the condition number of the global stiffness matrix. Figure 4.17 shows that the condition number of the global stiffness matrix A increases when the element shape is not isotropic. However, further increase in anisotropy of the elements does not increase the condition number, which means the effect of anisotropy is limited. Recall that the condition number sharply increases and approaches infinity as θ approaches 0 for general elliptic PDEs. Fix the anisotropy of the elements and vary the mesh size (h). We fix the anisotropy of the elements and compute the condition number for various mesh sizes, h. Figure 4.15 shows the connection between the horizon (δ), mesh refinement, and the condition number. Figure 4.11 shows that the condition number of the global stiffness matrix A is proportional to h 1. Note that for general elliptic PDEs, the condition number of the global stiffness matrix is proportional to h 2 if the mesh has the same aniostropy. Therefore, the mesh refinement for a nonlocal peridynamic model increases the condition number less compared with the one for general elliptic

153 133 (a) h=0.05 (b) h=0.025 Fig Condition number of A when h=0.05 for fixed h and varying anisotropy and δ.

154 134 (a) Isotropic mesh (aspect ratio=1) (b) Anisotropic mesh (aspect ratio=4) Fig Condition number of A for fixed anisotropy (aspect ratio=1 and 4) and varying h and δ.

155 135 (a) h=0.1 (no mesh refinement) (b) h=0.05 (after mesh refinement) Fig Condition number of A for fixed h and anisotropy and various power of a kernel function, p.

156 136 PDEs. Fix the anisotropy of the elements and mesh size (h), and vary p. We fix the anisotropy of the elements and h but vary p and see the effect of various values of p on the condition number. We observe that as p increases, the condition number rapidly increases. Simple regression results show that the condition number exponentially increases as p 2, which means the kernel function is nonintegrable in this range. 4.6 Conclusions We have investigated the effects of anisotropy, mesh refinement, and various kernel functions on the condition number for a 2D nonlocal bond-based peridynamic model. This is the first study to examine the connections between the anisotropy and the conditioning on 2D geometric domains. We employ the Galerkin finite element method to discretize the nonlocal peridynamic model and investigate the effects that various choices of basis functions have on the condition number of the global stiffness matrix. We observe consistent numerical results with the analytical results when we employ an integrable kernel function. As far as the integrable kernel function is concerned, the condition number is bounded by cδ 2 (where c is a constant) and is not affected by the choice of the basis function, when δ is bigger than the mesh size (h). Our numerical results show that the constant c is close to 1 on the 2D uniform triangular and rectangular meshes on geometric domains for an integrable kernel function. For these scenarios, mesh anisotropy and the mesh refinement affect the condition number very little. For piecewise linear basis functions with a nonintegrable kernel function, we observe that the condition number is proportional

157 137 to h 1 when we fix the mesh anisotropy. We observe that anisotropic mesh elements increase the condition number compared to the isotropic mesh elements when we fix h. However, the effect of mesh anisotropy is limited since further increase in the mesh anisotropy does not increase the condition number. We also observe that the condition number exponentially increases as the power of a kernel function increases. Our results can be used to generate meshes with a small condition number for a bond-based nonlocal peridynamic model. For future research, we will consider other peridynamic models on unstructured geometric domains. The sparsity pattern of a nonlocal peridynamic model is different from PDEs since the sparsity pattern is affected by the horizon radius. We also plan to develop a preconditioner to decrease the condition number for a bond-based nonlocal peridynamic model. In this dissertation, we have thus far focused on mesh quality improvement techniques for PDEs and nonlocal peridynamic models. In the next part of dissertation, we will focus on optimization-based meshing techniques for mesh deformation problems and shape matching problems. In the next chapter, we will propose a hybrid deformation algorithm using the direction of boundary motion and multiobjective mesh optimization. We find the optimal interior vertex positions on the deformed domain when the boundary deformations are known.

158 Introduction There are numerous applications where discretized geometric domains vary with respect to time such as the solution of Arbitrary-Lagrangian-Eulerian (ALE) flow simulations [76], deformation of the human face in computer graphics [78], deformation of sequences of medical images [81, 82], and deformations in biomedical applications [79, 80]. When these domain deformations occur, the meshes approximating the domain should be also updated appropriately with respect to time such that the mesh remains valid. This mesh deformation (update) method is called mesh warping [88, 95] (mesh morphing [77] or moving meshes [92]) with respect to the applications. There exist many mesh deformation methods; however, we focus on mesh deformation algorithms which recompute interior mesh vertex positions after the mesh boundary has been deformed. Our algorithm requirements are twofold: to produce meshes elements of good element quality including no inverted elements, and second to maintain similarity during mesh deformation. An untangled deformed mesh is important since tangled meshes with inverted elements result in physically invalid solutions when standard finite element shape functions are used. Similarity means identical mesh connectivities and topologies in the deformed mesh as were present in the initial mesh along with the preservation of general mesh characteristics (e.g., shape, size, anisotropy) of the original mesh in the deformed mesh. This is important because many applications need a smooth variation of meshes between each time step or iteration [43], and the initial mesh may have been constructed with anisotropies designed with the particular simulation in mind. For deforming domains, both mesh deformation and remeshing can be used to generate meshes on deformed domains. However, mesh deformation is preferred

159 139 to remeshing, because remeshing results in the accumulation of large interpolation error between successive time steps and less accurate PDE simulation results [52]. Also, remeshing is inefficient compared to updating meshes via mesh deformation. Researchers have proposed various mesh deformation algorithms based on Laplace equation methods, e.g., finite element-based mesh warping (FEMWARP) [84, 88], weighted Laplacian smoothing [97], biharmonic partial differential equations [76], elasticity based approaches [92, 93, 95], by utilizing an inverse distance weighting function [83], and by combining vertex movement with other techniques which alter mesh topology [85 87]. FEMWARP creates elements with good quality for small or moderate boundary deformations and is exact for affine transformations, is easy to implement and scales well with mesh size. However, FEMWARP ignores information about the nature of the deformation and often generates many inverted elements for anisotropic deformations. The authors in [88] proposed three modifications of FEMWARP, a small-step FEMWARP, mesh refinement, and combining FEMWARP with an optimizationbased untangler to address this problem. Small-step FEMWARP can be used to generate untangled meshes on the deformed domain by introducing a pseudo time-stepping of the boundary deformation with steps small enough to prevent the creation of inverted elements in each FEMWARP solution. Although this small-step FEMWARP preserves element connectivities and is able to prevent inverted elements, it does not preserve the original mesh element shapes on the deformed domain. Also, it may not be practical to use this boundary pseudo time-stepping for some applications since it requires a continuous function from the old to new boundary conditions, which is not always available [88]. The mesh refinement strategy is not always practical since the mesh generation source code is not always available. In addition, more refinement alter the mesh topology. The combined approach with FEMWARP and mesh untangling is also

160 140 successful in eliminating inverted elements on the deformed mesh. However, it is also unable to preserve the initial mesh element shapes on the deformed mesh. The biharmonic operator method to solve mesh warping problems [76] controls both the normal mesh spacing and the boundary discretization. However, the method is computationally expensive and also does not maintain similarity on the deformed domain. Finally, it does not include a mechanism for eliminating inverted elements if the deformed mesh includes any inverted elements. The elasticity equation-based methods described in [92, 93] provide a mechanism for decreasing the creation of inverted elements by choosing stiffness coefficients in a problem dependent fashion. Another related elasticity based approach called Untangling Before Newton (UBN) is proposed for handling large boundary deformations by using the iterative stiffness method [95]. Recently, the log-barrier approach for the worst element mesh quality improvement and untangling is also proposed and shows better untangling performance compared with UBN [96]. UBN is also not able to preserve similar element shapes on the deformed domain. There is another category of optimization-based mesh warping algorithms which use the target matrix paradigm (TMP) [43]. These methods are capable of maintaining similarity between initial and deformed meshes for various aspects of the mesh (e.g., shape, size, shape and size). However, these methods often have poor convergence characteristics if the mesh undergoes large boundary deformations. All of the methods mentioned thus far suffer from at least one of the following problems: a tendency to produce tangled elements for large boundary deformations, or an inability to preserve features of the initial mesh in the deformed mesh. In this chapter, we examine techniques for robustly warping meshes subject to anisotropic boundary deformations which address both of these problems. In Section 4.8, we review FEMWARP which our method generalizes.

161 141 Section 4.9 introduces our two-step hybrid algorithm. The first step is to estimate the internal vertex positions of the deformed mesh using knowledge of the nature of the boundary deformation. We choose the relative weighting of neighbor vertices in our anisotropic FEMWARP method based on the direction of the boundary deformation. We prove that for some simple mesh configurations and certain types of anisotropic deformations, the method we propose will maintain desired properties of the initial mesh independent of the magnitude of the boundary deformation. In more complicated cases the mesh produced in the first step often possesses some inverted elements, so, as a second step, we propose a multiobjective mesh optimization of element shape and element untangling. This step is designed to produce meshes with no inverted elements while maintaining element similarity with the initial mesh. In Section 4.10, we demonstrate that for several complicated boundary deformations the first step of our method produces meshes with fewer inverted elements than other existing methods. We also demonstrate that our multiobjective optimization is capable of producing meshes with no inverted elements and preserving characteristics of the initial mesh. The contents of this chapter will soon be submitted for review [111]. 4.8 Background In this section, we briefly describe the FEMWARP algorithm and motivate our anisotropic FEMWARP which uses anisotropic PDE coefficients. The reader is referred to [84, 88] for further information on FEMWARP.

162 FEMWARP The warping computed by FEMWARP is equivalent to the application of Laplacian smoothing on mesh deformation problems. We describe the FEMWARP algorithm as applied to 2D geometric domains. The equations are similar for 3D. FEMWARP solves Laplace s equation with a Dirichlet boundary condition which is denoted as u = 0 on Ω, where u = u 0 on the boundary, Ω. FEMWARP updates interior vertex positions given a boundary deformation discretized using piecewise linear finite elements [88]. Let B and I be the sets of boundary and interior vertices, respectively. Let N I and N B be the numbers of interior and boundary vertices, respectively. FEMWARP is composed of three steps and represents each interior vertex as an affine combination of its neighbors. The first step of FEMWARP forms the (N I + N B ) (N I + N B ) global stiffness matrix A by assembling the element stiffness matrices on the undeformed domain. By ordering the interior unknowns first and the boundary unknowns second, the matrix A can be partitioned into boundary and interior sub-matrices A I A B A =, (4.8) (A B ) T C where A I is the (N I N I ) submatrix of A which denotes the connections of interior vertices to interior vertices; A B is the (N I N B ) submatrix denoting the connections between interior and boundary vertices, and C is the (N B N B ) submatrix representing the connections among boundary vertices. Let (x I, y I ) and (x B, y B ) be the coordinates of the interior and boundary

163 vertices on the initial undeformed domain, respectively. Then, since linear functions are in the null space of A, (4.8) implies that 143 A I x I + A B x B = 0 A I y I + A B y B = 0. (4.9) Let ( ˆx I, ŷ I ) and ( ˆx B, ŷ B ) be the coordinates of interior and boundary vertices on the deformed domain, respectively. The second step is to apply a known boundary deformation to (4.9), i.e., (x B,y B ) ( ˆx B,ŷ B ). The final step is to compute the coordinates of interior vertices on the deformed domain ( ˆx I, ŷ I ) by solving the linear system A I ˆx I + A B ˆx B = 0 A I ŷ I + A B ŷ B = 0, (4.10) for ( ˆx I,ŷ I ) FEMWARP and anisotropic boundary deformation FEMWARP has several advantages over other mesh warping algorithms in that it is simple to implement and is exact for affine boundary deformations [88]. It was reported that FEMWARP is successful in yielding noninverted elements during the mesh updating process for some applications such as a beating canine heart from an atrial pacing experiment [97] and also for shape optimization problems [94]. However, FEMWARP often generates a large number

164 144 of inverted elements if deformations are not affine and anisotropic deformation occurs in one direction. This is because FEMWARP is not designed to consider the direction of boundary deformation and equally considers all vertex neighbors even if the deformation occurs in only one direction. We present a motivating example which was used in [76]. The initial mesh and the deformed mesh using FEMWARP are shown in Figure The initial mesh is uniform and structured on a rectangular domain. Here, the the deformation only occurs in the y direction. Figure 4.20 shows the deformed mesh produced by FEMWARP includes 20 inverted elements and has several valid elements with poor quality. This mesh also fails to preserve the element spacing of the initial mesh on the deformed domain. We point out that these inverted elements with poor element shapes occur because FEMWARP equally considers all neighboring vertices when it computes the stiffness matrix and does not consider the direction of the boundary deformation. For this kind of one-directional deformation, it is obvious that we should only connect neighbors aligned in the y direction. In the next subsection, we will show that similarity (here, uniform spacing on the deformed domain) can be achieved using our anisotropic FEMWARP method with judicious choice of coefficients. 4.9 Hybrid Mesh Deformation Algorithm Our hybrid mesh deformation algorithm is composed of two steps. First, we apply anisotropic FEMWARP for determining an initial guess for the interior vertex positions on the deformed domain. The goal of this first step is to preserve the element spacing and to reduce the number of inverted elements on the deformed domain. For this step, we consider the direction

165 145 (a) Mesh on initial domain (b) Mesh on deformed domain Fig (a) Initial mesh on the rectangular domain. (b) Deformed mesh using FEMWARP. This mesh has 20 inverted elements. of deformation when choosing the PDE coefficients. Similar to other mesh deformation algorithms, anisotropic FEMWARP may produce inverted elements. So as a second step, we apply the multiobjective mesh optimization using a Target Matrix Paradigm (TMP) shape term and an untangling term to find the optimal interior vertex positions on the deformed domain. Note that meshes with noninverted elements are important since meshes with inverted elements yield physically invalid finite element solutions Step 1: Anisotropic FEMWARP using anisotropic PDE coefficients We propose an anisotropic FEMWARP method to solve the mesh deformation problem which is better suited to cases where the initial mesh is anisotropic and/or the deformation is aligned with one coordinate axis. However, this idea is easily extended to other kind of deformations (such as a diagonal deformation). For anisotropic FEMWARP in 2D, the PDE used to

166 generate the linear system defining the connection between the interior and boundary vertices is 146 α 2 u x 2 β 2 u = 0 on Ω, (4.11) y2 where u = u 0 on the boundary of Ω, i.e., Ω. Here, we assume that α > 0 and β > 0. From here, we consider 2D meshes to simplify the notation; however, the extension to 3D meshes is quite natural, and we will explain how to extend our anisotropic FEMWARP to 3D. Anisotropic FEMWARP has a different stiffness matrix, A, from FEMWARP because different PDE coefficients are employed. It follows the same three steps as FEMWARP which are explained in Sec after the stiffness matrix is formulated. Similar to FEMWARP, anisotropic FEMWARP also represents each interior vertex as an affine combination of neighbors, but anisotropic FEMWARP adaptively changes its weights for each interior vertex with respect to the PDE coefficients, α and β. We will show that this adaptive strategy significantly helps to reduce the number of inverted elements compared with FEMWARP, which always fixes α=1, β=1. We will also show that for extreme cases, if either α or β is zero, we only consider neighbors aligned in x-axis or y-axis when we formulate A. How to preserve similar element spacing on the deformed domain? We begin with a discussion of how to maintain similar element spacing between the initial and deformed domains by appropriately choosing the PDE coefficients in (4.11). We show that for some simple mesh configurations and certain types of anisotropic deformations, setting one of the PDE coefficients to one and other zero in (4.11) is optimal to maintain element spacing. In all our results in this

167 147 section, we assume that the initial domain is rectangular and axis aligned and the mesh is a triangulated structured grid with the grid lines also axis aligned. We begin with the simplest case, i.e., that of uniform vertex spacing equal to h in x and k in y. Referring to the generic triangle i h, j 1 k C k T ij A h B Fig Generic element in structured triangular mesh with uniform elements. in this assumed initial mesh shown in Figure 4.21, it is a straightforward exercise to derive the element stiffness matrix for (4.11) discretized by linear finite elements. αk 2h + βh 2k αk 2h αk 2h βh 2k 0 βh 2k αk 2h 0 βh 2k. (4.12) Proposition 1. If α = 0 and β = 1 in (4.11), then the positions of the interior vertices on the deformed domain are determined only by the boundary vertices which had the same x coordinate in the initial domain.

168 148 Proof. Since α = 0 we can see that the element stiffness matrix in (4.12) has only non-zeros in the positions connecting vertices A and C, those vertices with the same x coordinate. Since each element stiffness matrix is structured so that only vertices with the same x coordinate in the original mesh are connected, this characteristic also holds in the global stiffness matrices A I and A B. With the same assumptions for the domain and mesh characteristics and with α = 0 and β = 1 in (4.11), we also have Proposition 2. The mesh spacing in the y direction is uniform in the deformed mesh. Proof. As explained earlier, anisotropic FEMWARP basically follows the same three steps as does FEMWARP in Sec By (4.10), A I ŷ I +A B ŷ B = 0. Proposition 1 allows us to rearrange the linear system A I ŷ I = A B ŷ B into a block tridiagonal system with each independent block consisting of unknown interior vertex coordinates with the same x coordinate and each block ordered by increasing mesh y coordinate. After appropriate scaling, each block then has the form ŷ 1 ŷ 2. ŷ N 3 ŷ N 2 = ŷ ŷ N 1. (4.13)

169 149 The solution to this linear system is ŷ j = ŷ 0 + j ) (ŷn 1 ŷ 0. N 1 Thus the y coordinates are equally spaced in the deformed mesh. If we relax our grid assumptions slightly and allow variable grid spacing in y as a form of anisotropy, then our proposed method still satisfies our desired properties. We assume that the grid spacing in y is given by k l, l = 0...N 2. Therefore, the undeformed grid y coordinates are y j = y 0 + j 1 l=0 k l. Proposition 3. The FEMWARP method with α = 0 and β = 1 will maintain the relative grid spacing in y of the initial undeformed mesh in the deformed mesh. Proof. First introduce a parameter γ = (ŷ N 1 ŷ 0 )/(y N 1 y 0 ) to carry the relationship between the deformed mesh y-axis extents and the original mesh extents. The element stiffness matrix and global stiffness matrix construction from Proposition 2 carry though with minor modifications for the varying grid spacing in y and the block tridiagonal system to solve for each block of interior vertices with the same x coordinates is ( ) k0 k1. 1 k k 1 ( ) 1 k k 2 1 k ( ) k N 4 k N k N 3 1 k N 3 ( ) k N 3 k N k N 2. ŷ 1 ŷ 2. ŷ N 3 ŷ N 2 = 1 k 0 ŷ k N 2 ŷ N 1. (4.14)

170 150 The solution to this system is ŷ j = ŷ 0 + γ j 1 k l. l=0 Therefore the deformed mesh has the same relative grid spacing in y as the undeformed mesh. Symmetric arguments naturally cover the case of x-axis deformations by setting α = 1 and β = 0. The observation we make is that the choice of setting one coefficient to 0 and the other to 1 in (4.11) in anisotropic FEMWARP will preserve the original mesh spacing in that direction. Furthermore, for these axis aligned meshes, the decoupling that occurs in the global stiffness matrix construction allows for anisotropic one-dimensional boundary deformations to be easily treated while still preserving the original relative mesh spacing along each grid line in that dimension. Anisotropic FEMWARP is exact for affine boundary transformations. One of the main advantages of FEMWARP is that this method is exact for affine boundary transformations [88]. We will show that anisotropic FEMWARP is also exact for affine boundary transformations. Proposition 4. Let A B and A I be the submatrices generated using anisotropic FEMWARP (see Sec ). We define [x B,y B ] and [x I,y I ] to be the known boundary and interior coordinates on the initial domain, respectively. We also define [ ˆx B,ŷ B ] and [ ˆx I,ŷ I ] to be the known boundary and unknown interior coordinates on the deformed domain, respectively. Suppose the affine transformation occurs on the each boundary coordinate, k B, then there exist a nonsingular matrix M and a vector w such that,

171 151 ˆx k = M ŷ k x k y k + w. Then, each interior vertex, l I, [ ˆx l,ŷ l ], can be represented as an affine transformation such that ˆx l = M ŷ l x l y l + w. Proof. Since anisotropic FEMWARP follows the same three steps as does FEMWARP explained in Sec , the interior vertex positions of anisotropic FEMWARP can be computed as A I ˆx I + A B ˆx B = 0 A I ŷ I + A B ŷ B = 0. This means that [ ˆx I,ŷ I ] = (A I ) 1 A B [ xˆ B, yˆ B ]. (4.15) Since affine boundary deformation occurs, ˆx B = Lx B + w ŷ B = Ly B + w, (4.16) where L is a nonsingular matrix. Substituting (4.16) into (4.15) results in [ ˆx I,ŷ I ] = (A I ) 1 A B ([x B,y B ]L T + e B w T ) (4.17)

172 where e B is a vector of size N B with all 1 s. In order to show that interior vertices are exact, we need to show that 152 [ ˆx I,ŷ I ] = [x I,y I ]L T + e I w T (4.18) where e I is a vector of size N I with all 1 s. By combining (4.17) and (4.18), we need to show that A I ([x I,y I ]L T + e I w T ) = A B ([x B,y B ]L T + e B w T ). Similar to FEMWARP, the weights of each interior vertex sums to one, i.e., A I e I + A B e B = 0. Also from (4.9), A I x I = A B x B A I y I = A B y B. This concludes the proof. Choosing appropriate PDE coefficients for boundary deformations. Previously, we observed that setting α=0 and β=1 in anisotropic FEMWARP is a good strategy for preserving element spacing for an anisotropic boundary deformation aligned with the y-axis. Similarly, anisotropic FEMWARP with α=1 and β=0 is desired when a boundary deformation occurs which is aligned

173 153 with the x-axis. We now apply this idea to our motivating example shown in Figure 4.20 and compare with FEMWARP, which fixes α=1 and β=1. Figure 4.22 shows a comparison of the meshes produced by FEMWARP and anisotropic FEMWARP, respectively. For this example, we chose α = 0 and β = 1 for anisotropic FEMWARP, since the deformation occurs only in the y direction. We observe that the deformed mesh using anisotropic FEMWARP does not have inverted elements, whereas the deformed mesh generated by FEMWARP has 20 inverted elements. Also, we observe that anisotropic FEMWARP is able to preserve similar element spacing on the deformed domain. (a) FEMWARP (b) Anisotropic FEMWARP Fig (a) Deformed mesh using FEMWARP. This mesh has 20 inverted elements. (b) Deformed mesh using anisotropic FEMWARP with coefficients α=0 and β =1 in (4.11). This deformed mesh using anisotropic FEMWARP does not have any inverted elements. We now consider more general cases where the boundary deformation occurs not aligned with the x-axis or y-axis (e.g., a diagonal deformation). We adaptively change α and β values with respect to the direction of boundary deformation. We have demonstrated that α controls the strength of the x-axis coupling between adjacent vertices, and β controls the strength of

174 154 the y-axis coupling. Therefore, increasing α corresponds to putting more weights on neighbors aligned with the x-axis than on neighbors aligned with the y-axis when we formulate the global stiffness matrix A. This strategy is appropriate when more deformation occurs in the x-axis than in the y-axis. Similarly, β should be bigger than α when more deformation occurs along the y-axis than the x-axis. We compute the cumulative boundary vertex displacements in the x and y directions and observe in which direction more deformation occurs by computing the relative ratio. If the cumulative boundary vertex displacement in the x direction is larger than the one in the y direction, we say that more deformation occurs along the x-axis and use the relative ratio of these deformations to compute α and β. The relative ratio can also be understood as the angle of the direction of the deformation. More generally, α and β are defined as α = N B k=1 ( xˆ k x k ),k B β = N B k=1 ( yˆ k y k ),k B, (4.19) where N is a total number of boundary vertices on the mesh. Also, recall that [x k,y k ] are the known boundary coordinates on the initial domain, and [ xˆ k, yˆ k ] are the known boundary coordinates on the deformed domain. Figure 4.23 summarizes our ideas and shows the connection between PDE coefficients (α and β) and the direction of boundary motion. Here, a and b in Fig 4.23.(c) are α and β in (4.19), respectively. Extension to 3D meshes. The extension to 3D meshes is natural since we just add one more coefficient term, i.e.,γ, for the z-axis. The PDE used to generate the linear system for anisotropic FEMWARP in 3D is

175 155 Fig Connection between the PDE coefficients (α and β) and the direction of motion. If the deformation occurs (a) aligned with the x-axis or (b) aligned with the y-axis, we only consider neighbors aligned with the same axis. Here, the cross-out indicates that we do not consider those neighbors. If deformation occurs that is not aligned with either the x or y axes (c), we use the angle of direction of the deformation to choose the appropriate PDE coefficients. Here, a and b are α and β in (4.19), respectively. α 2 u x 2 β 2 u y 2 γ 2 u = 0 on Ω, (4.20) z2 where u = u 0 on the boundary, Ω. Here, we assume that α > 0, β > 0, and γ > 0. Similar to the 2D cases, we set only one of the PDE coefficients as 1 and the other two coefficients as 0 if deformation occurs that is aligned with the x-,y-, or z-axis in order to preserve similar element spacing. For example, if the deformation only occurs in z-axis, we set α = 0, β = 0, and γ = 1. If the deformation that occurs is not aligned with any of the axes, we follow the same ideas in (4.19), i.e., α = N B k=1 ( xˆ k x k ), β = N B k=1 ( yˆ k y k ), k B k B (4.21) γ = N B k=1 ( zˆ k z k ), k B.

176 Step 2: Multiobjective mesh optimization with shape and untangling Although anisotropic FEMWARP is able to preserve similar element spacing on the deformed domain and produces fewer inverted elements on the deformed domain than does FEMWARP, large deformations still often cause element inversion on the deformed domain. We propose a multiobjective mesh optimization of element shape and untangling to correct this problem. The idea of using multiobjective mesh optimization for mesh quality improvement and mesh untangling was proposed in our previous paper [73]. We employ this multiobjective mesh optimization framework and apply this framework to mesh deformation problems. We focus on preserving similar element shape by employing a TMP shape metric [54]. Let A i be the Jacobian matrix of the mapping from the reference element to the i th mesh element. The fundamental object used to construct TMP quality metrics is a dimensionless Jacobian matrix denoted T i defined by T i = (A def ) i (A init ) 1 i, where (A def ) i and (A init ) i are the Jacobians of the mappings from the reference element to the actual elements in the deformed and initial domains, respectively. The scale- and rotation-invariant TMP shape metric is defined as T i (adj(t i ) t )) 2 F q i =, T R2 R 2 T i 3F T i (adj(t i ) t ) 2 F, T R3 R 3. (4.22) The TMP shape metric is zero when the quality of the deformed element is the same as in the initial mesh, which means the element shape is identical. Note that other similarity (e.g., size

177 157 and shape and size) could be preserved by simply changing (4.22) into other TMP metrics such as TMP size or TMP shape and size metric, which are defined in [16]. In order to eliminate inverted elements, we employ the untangling beta quality metric which is defined q j = V j β (V j β), where V j is the area of the j th element, and β is a user-defined parameter greater than 0. The untangling beta metric is zero when the mesh element is not inverted. The overall mesh quality computed using the TMP shape metric is F 1 = E i=1 q 2 i, where E is the number of mesh elements. Similarly, the overall mesh quality computed by the untangling metric is F 2 = E q 2 j. j=1 Our goal is to simultaneously optimize these two objective functions to simultaneously untangle and improve the element shape on the deformed domain. We find the maximum cost function between F 1 and F 2 and also minimize the worst cost function between F 1 and F 2. Then, the min-max problem for our mesh deformation optimization problem is min max i {F i }. (4.23)

178 158 However, (4.23) is not smooth and has neither a Jacobian nor a Hessian. Therefore, we use the exponential sum multiobjective method to approximate the solution to (4.23). The exponential sum multiobjective function approximates the min-max problem by employing the exponential penalty function and is defined F = c ln [ 2 e F i/c i=1 ], (4.24) where c is a controlling parameter and is typically chosen between 10 4 and It was proven that the solution to (4.24) satisfies a sufficient condition of Pareto optimality [91]. One of the main strengths of employing this exponential sum function is it does not require any articulation of preferences between the two objective functions. Similar to [89], we employ the nonlinear conjugate gradient method to find a local optimal point, since this is the default NLCG method in Mesquite. Here, the local optimal point means the optimal vertex locations on the deformed domain Numerical Experiments We present more realistic unstructured mesh examples in this section. For our second step, i.e., multiobjective mesh optimization, we use Mesquite (version 2.99) [16]. We first consider three examples where anisotropic deformation occurs aligned with the x or y axis. We also apply our hybrid mesh deformation algorithms when deformation occurs that is not aligned with either the x or y axis (e.g., diagonal deformation). Finally, we apply our hybrid algorithm to the 3D mesh deformation example.

179 159 We make comparisons to several existing mesh warping methods and compare three key features: (1) the number of inverted elements produced, (2) the time to untangle the deformed mesh using our multiobjective mesh optimization with both FEMWARP and anisotropic FEMWARP, and (3) the similarity of the deformed mesh to the original mesh as measured by the TMP shape metric for three different algorithms: FEMWARP, our hybrid algorithm (anisotropic FEMWARP + multiobjective mesh optimization), and UBN [95]. In (1), we will show that the anisotropic FEMWARP significantly helps to reduce the number of elements as a first step and our second step, i.e., multiobjective mesh optimization, is able to eliminate inverted elements from tangled meshes both with FEMWARP and anisotropic FEMWARP. In (2), we will show that anisotropic FEMWARP reduces the number of inverted elements which significantly helps the convergence time of our second step, i.e., multiobjective mesh optimization. Finally in (3), we show that our hybrid algorithm is able to preserve the element shape by showing the similarity measure by the TMP shape metric. Note that when we compare with UBN, we only use the iterative stiffening aspect of UBN which comes after FEMWARP in the UBN code. The machine employed for this study is equipped with an AMD Quad-core Opteron processor (2.3 GHz) and 32GB of RAM; we use a single core on this machine Moving cylinder domain for anisotropic boundary deformation aligned in x-axis We first consider a moving cylinder example in a channel as [76] for testing anisotropic deformation. We set the coefficients of anisotropic FEMWARP in (4.11) as α=1 and β =0 for this problem since deformation only occurs which is aligned with the x-axis. Figure 4.24 shows the initial mesh, the deformed mesh generated by FEMWARP, the deformed mesh generated by anisotropic FEMWARP (after the first step of hybrid algorithm), and the optimized mesh (after

180 160 the second step of the hybrid algorithm) on the deformed domain. Figure 4.24 shows that the deformed mesh for FEMWARP has 38 inverted elements around the inner boundary, while our hybrid algorithm does not have any inverted elements. Finally, Fig. 4.24(d) demonstrates that our hybrid algorithm is able to recover similar element shapes on the deformed domain. Figure 4.25 compares element qualities measured by the TMP shape metric, which measures similarity between the deformed and the initial elements. A smaller value indicates more similarity with zero meaning the shapes are identical. The quality measure of the worst element using our hybrid algorithm is 80.4% less than the one using UBN and 45.5% less than the one produced by FEMWARP. Figure 4.26(a) shows the number of inverted elements for various amounts of translation and (b) time to untangle inverted elements using multiobjective mesh optimization. This figure shows the number of inverted elements is reduced up to 91.2% when we employ anisotropic FEMWARP compared with FEMWARP. Figure 4.26(b) shows the time to untangle meshes (from (a)) using multiobjective mesh optimization. We observe that anisotropic FEMWARP reduces the running time up to 90.6% over that of FEMWARP, when we employ our multiobjective mesh optimization to untangle meshes with inverted elements. This is because anisotropic FEMWARP requires fewer iterations to untangle inverted meshes than does FEMWARP Moving bar domain for anisotropic boundary deformation aligned in y-axis We consider a moving bar example derived from a mechanical engineering problem [92] on unstructured meshes. We choose the coefficients for anisotropic FEMWARP to be α=0 and β=1 in (4.11) for this problem since deformation occurs which is aligned with the y-axis. Figure 4.27 shows the initial mesh, the deformed mesh generated by FEMWARP, the deformed mesh

181 161 (a) Mesh on initial domain (b) Mesh on deformed domain with FEMWARP (c) Mesh on deformed domain with anisotropic FEMWARP (d) Optimized mesh with anisotropic FEMWARP followed by multiobjective optimization Fig Moving cylinder domain for anisotropic boundary deformation aligned with the x-axis: (a) Initial mesh for a moving cylinder in a channel. (b) Deformed mesh using FEMWARP for a moving cylinder. This mesh has 38 inverted elements. (c) Deformed mesh using anisotropic FEMWARP for a moving cylinder. This mesh has zero inverted elements. (d) Optimized mesh on the deformed domain with no inverted elements using anisotropic FEMWARP followed by multiobjective mesh optimization with TMP shape and untangling.

182 162 Fig Mesh quality measured by the TMP shape metric on the deformed cylinder domain. A smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 80.4% less than the quantity of the one using UBN [95]. Note that FEMWARP includes 38 inverted elements, while the deformed meshes resulting from both our hybrid algorithm and UBN do not have any inverted elements. (a) Number of inverted elements after initial guess (b) Time (sec) to untangle inverted elements using multiobjective mesh optimization Fig Number of inverted elements with respect to the translation of inner boundary after applying (a) FEMWARP (α=1 and β=1) and anisotropic FEMWARP (α=1 and β=0). The inner boundary shifts right. (b) Time to untangle inverted elements using multiobjective mesh optimization. This figure shows the time to untangle inverted meshes using multiobjective mesh optimization.

183 163 generated by anisotropic FEMWARP (after the first step of hybrid algorithm), and the optimized mesh (after the second step of hybrid algorithm) on the deformed domain. These figures clearly show our hybrid algorithm simultaneously preserves element shape and eliminates inverted elements on the deformed domain. Also, our anisotropic FEMWARP (first step) has significantly fewer inverted elements than does FEMWARP and helps to decrease the optimization time in the second step of our hybrid algorithm. Figure 4.28 shows element qualities on the deformed domain for our hybrid algorithm and for FEMWARP. For this domain, UBN is not able to find the deformed domain due to the singular stiffness matrix which arises when numerical tolerances are reached. More specifically, this is the result of numerical tolerance issues related to elements with nearly zero area around the inner bar. The worst element quality measure produced by our hybrid algorithm is 93.4% smaller than that produced by FEMWARP. FEMWARP produced 118 inverted elements, while our anisotropic FEMWARP produced only 17. Figure 4.29 shows the number of inverted elements with respect to amount of translation and the time to untangle inverted elements using multiobjective mesh optimization, respectively. Similar to the cylinder domain, our anisotropic FEMWARP has significantly (up to 94.2%) fewer inverted elements after the initial guess step (step 1) and resulted in taking less time (up to 63.7%) to remove the inverted elements using multiobjective mesh optimization Moving gate domain for y axis aligned anisotropic boundary deformation We consider a moving gate domain similar to one described in [92] where deformation occurs on a portion of the outer boundary. We choose coefficients of anisotropic FEMWARP of α=0 and β =1 in (4.11) for this anisotropic boundary deformation, since deformation only occurs that is aligned with the x-axis. Figure 4.30 shows the initial mesh, the deformed mesh generated

184 164 (a) Mesh on initial domain (b) Zoomed-in mesh on the initial domain (c) Mesh on deformed domain with FEMWARP (d) Zoomed-in mesh on the deformed domain with FEMWARP (e) Mesh on deformed domain with anisotropic FEMWARP (α = 0 and β = 1) (f) Zoomed-in mesh on the deformed domain with anisotropic FEMWARP (g) Optimized mesh on the deformed domain with anisotropic FEMWARP followed by multiobjective mesh optimization (h) Zoomed-in mesh on the deformed domain with anisotropic FEMWARP followed by multiobjective mesh optimization Fig Moving bar domain for anisotropic boundary deformation aligned in y-axis:(a) Initial mesh and (b) zoomed-in on the bar domain. (c) Deformed mesh with FEMWARP and (d) zoomed in mesh with FEMWARP. Deformed mesh with FEMWARP has 118 inverted elements. (e) Deformed mesh using anisotropic FEMWARP and (f) zoomed-in mesh with anisotropic FEMWARP. The deformed mesh with anisotropic FEMWARP has 17 inverted elements. (g) Optimized mesh and (h) zoomed-in mesh with anisotropic FEMWARP followed by multiobjective mesh optimization. This optimized mesh does not have any inverted elements.

185 165 Fig Mesh quality measured by the TMP shape metric on the deformed bar domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 93.5% less than the quality of the mesh obtained from the use of FEMWARP [95]. Note that FEMWARP includes 118 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements. For this domain, UBN fails to find the deformed mesh due to numerical tolerance issues. by FEMWARP, the deformed mesh generated by anisotropic FEMWARP (after the first step of hybrid algorithm), and the optimized mesh (after the second step of the hybrid algorithm) on the deformed domain. This example clearly shows that anisotropic FEMWARP outperforms FEMWARP. A very large number of inverted elements is generated around the center corner (as shown in Fig. 4.30(b)), since FEMWARP connects vertices isotropically when the stiffness matrix is formulated. Anisotropic FEMWARP decreases the number of inverted elements by preferentially connecting neighbors aligned with the y axis. Similar to previous examples, our hybrid algorithm is successful in preserving element shapes and outperforms FEMWARP as shown in Fig Similar to the moving bar example, UBN fails to find the deformed mesh due to numerical tolerance issues. This is the result of numerical issues related to refined elements with an approximately zero area around the center corner.

186 166 (a) Number of inverted elements with respect to the amount of translation (b) Time (sec) to untangle inverted elements using multiobjective mesh optimization Fig Number of inverted elements with respect to the translation of inner boundary after applying (a) FEMWARP (α=1 and β=1) and anisotropic FEMWARP (α=0 and β=1). The inner bar moves down. (b) We apply multiobjective mesh optimization to simultaneously smooth and untangle inverted elements. This figure shows timing to untangle inverted elements using multiobjective mesh optimization. Our multiobjective mesh optimization is able to improve element qualities while eliminating inverted elements but anisotropic FEMWARP takes up to 63.7% less time to eliminate inverted elements compared with FEMWARP since anisotropic FEMWARP has fewer iterations to untangle inverted elements than FEMWARP Cylinder in a channel domain subject to a diagonal deformation We consider a moving cylinder example similar to that of Section ; however, now the inner cylinder moves diagonally along the vector [0.6,0.1] T. Based on our strategies for setting the coefficients of anisotropic FEMWARP described in Fig. 4.23, we choose the coefficients of α=6 and β=1. Figure shows the initial, the deformed mesh generated by FEMWARP, the deformed mesh generated by anisotropic FEMWARP (after the first step of the hybrid algorithm), and the optimized mesh (after the second step of the hybrid algorithm) on the deformed domain. Compared with FEMWARP, the number of inverted elements is reduced by 86%. Similar to the previous examples, our hybrid algorithm results in meshes with better mesh quality with no inverted elements and similar element shapes as shown in Fig The worst element quality of the deformed mesh which results from our hybrid algorithm is 79.2% less than the one

187 167 (a) Mesh on initial domain (b) Zoomed-in mesh on the initial domain (c) Mesh on deformed domain with FEMWARP (d) Zoomed-in mesh on the deformed domain with FEMWARP (e) Mesh on deformed domain with anisotropic FEMWARP (α = 0 and β = 1) (f) Zoomed-in mesh on the deformed domain with anisotropic FEMWARP (g) Optimized mesh on the deformed domain with anisotropic FEMWARP followed by multiobjective mesh optimization (h) Zoomed-in mesh on the deformed domain with anisotropic FEMWARP followed by multiobjective mesh optimization Fig Moving gate domain for anisotropic boundary deformation aligned in y-axis: (a) Initial mesh and (b) zoomed-in on the gate domain. (c) The deformed mesh with FEMWARP and (d) zoomed-in mesh with FEMWARP. Deformed mesh with FEMWARP has 103 inverted elements. (e) The deformed mesh using anisotropic FEMWARP and (f) zoomed-in mesh with anisotropic FEMWARP. The deformed mesh with anisotropic FEMWARP has 14 inverted elements. (g) The optimized mesh and (h) zoomed-in optimized mesh on the deformed domain with no inverted elements using anisotropic FEMWARP followed by multiobjective mesh optimization with TMP shape and untangling.

188 168 Fig Mesh quality measured by the TMP shape metric on the deformed gate domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 9.85% less than the one using FEMWARP [95]. Note that FEMWARP includes 103 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements. For this domain, UBN fails to find the deformed mesh due to numerical tolerance issues. (a) Number of inverted elements (b) Time to untangle inverted elements Fig Number of inverted elements with respect to the translation of outer boundary after applying (a) FEMWARP (α=1 and β=1) and anisotropic FEMWARP (α=0 and β=1). The corner in the middle moves down. (b) We apply multiobjective mesh optimization to simultaneously smooth and untangle inverted elements. This figure shows the time to untangle inverted elements using multiobjective mesh optimization. Our multiobjective mesh optimization is able to improve element qualities while eliminating inverted elements but anisotropic FEMWARP takes up to 90.6% less time to eliminate inverted elements compared with FEMWARP.

189 using UBN and 41.8% less than the one using FEMWARP. Note that FEMWARP also includes 103 inverted elements on the deformed mesh Bending bar domain for nonlinear 2D deformation We consider a bending bar deformation which is introduced in [92]. The initial mesh is same as Fig , but the bar undergoes a bending deformation. For this nonlinear deformation, we basically follow the same idea shown in Fig We compute the cumulative vertex displacements in the x and y directions and use this information to choose values of the coefficients for α and β. For this problem, our methodology results in coefficients of α = 1 and β = 4. Figure 4.35 shows the initial mesh, the deformed mesh generated by FEMWARP, the deformed mesh generated by anisotropic FEMWARP (after first step of hybrid algorithm), and (d) the optimized mesh (after second step of hybrid algorithm) on the deformed domain. Compared with FEMWARP, the number of inverted elements is reduced to 63.2%. Figure 4.36 shows that our hybrid algorithm significantly improves mesh quality (by greater than 96%) compared with FEMWARP. These results are consistent with other displacement magnitudes of bending. Similar to the moving bar and gate domains, UBN is not able to find the deformed domain due to numerical tolerance issues D moving sphere domain for z axis aligned anisotropic boundary deformation We consider a moving sphere in a cube example on 3D unstructured meshes. The inner sphere is moving along the z axis. We choose the coefficients of anisotropic FEMWARP to be α=0, β=0, and γ=1 in (4.20) for this problem since deformation only occurs aligned with

190 170 (a) Mesh on initial domain (b) Mesh on deformed domain with FEMWARP (c) Mesh on deformed domain with anisotropic FEMWARP (d) Optimized mesh on the deformed domain with anisotropic FEMWARP followed by multiobjective mesh optimization Fig Moving cylinder domain for anisotropic diagonal boundary deformation: (a) Initial mesh for a moving cylinder in a channel. (b) The deformed mesh using FEMWARP for a diagonal deformation. It has 101 inverted elements. (c) The deformed mesh using anisotropic FEMWARP for a diagonal deformation. It has 15 inverted elements. (d) The optimized mesh on the deformed domain with no inverted elements using anisotropic FEMWARP followed by multiobjective mesh optimization with TMP shape and untangling.

191 171 Fig Mesh quality measured by the TMP shape metric on the deformed cylinder domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 79.2% less than the one using FEMWARP [95]. Note that FEMWARP includes 103 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements. the z-axis. Figure 4.37 shows the initial surface mesh, the initial volume mesh, and the deformed volume mesh (after the second step of the hybrid algorithm) on the deformed domain. Table 4.38 shows mesh quality statistics on the deformed domain using our hybrid algorithm and FEMWARP, respectively. Here, a smaller value indicates more similarity with zero meaning identical shape. We observe significant quality improvements of our hybrid algorithm over FEMWARP. The worst element quality of the deformed mesh which results from our hybrid algorithm is 86.2% less than the one using FEMWARP. Figure 4.39 shows the number of inverted elements for various amounts of translation and the time to untangle inverted elements using multiobjective mesh optimization. The overall trend is similar to those seen for 2D examples. Anisotropic FEMWARP significantly helps to reduce the number of inverted elements as a first step. The reduced number of inverted elements decreases the time needed to eliminate inverted elements as a second step. Note that our second step, multiobjective mesh optimization, is able

192 172 (a) Mesh on initial domain (b) Zoomed-in mesh on the initial domain (c) Mesh on deformed domain with FEMWARP (d) Zoomed-in mesh on the deformed domain with FEMWARP (e) Mesh on deformed domain with anisotropic FEMWARP (α = 1 and β = 4) (f) Zoomed-in mesh on the deformed domain with anisotropic FEMWARP (g) Optimized mesh on the deformed domain with anisotropic FEMWARP followed by multiobjective mesh optimization (h) Zoomed-in mesh on the deformed domain with anisotropic FEMWARP followed by multiobjective mesh optimization Fig Bending bar domain for anisotropic boundary deformation:(a) Initial mesh and (b) zoomed-in mesh on the bar domain. (c) Deformed mesh with FEMWARP and (d) zoomedin mesh with FEMWARP. Deformed mesh with FEMWARP has 159 inverted elements. (e) Deformed mesh using anisotropic FEMWARP and (f) zoomed-in mesh with anisotropic FEMWARP. Deformed mesh with anisotropic FEMWARP has 48 inverted elements. (g) Optimized mesh and (h) zoomed-in mesh with anisotropic FEMWARP followed by multiobjective mesh optimization. This optimized mesh does not have any inverted elements.

193 173 Fig Mesh quality measured by the TMP shape metric on the deformed bar domain. The smaller value indicates more similarity with zero meaning identical shape. The worst element quality of the deformed mesh which results from our hybrid algorithm is 92.5% less than the one using FEMWARP [95]. Note that FEMWARP includes 159 inverted elements while the deformed mesh resulting from our hybrid algorithm does not have any inverted elements. For this domain, UBN fails to find the deformed mesh due to numerical tolerance issues. to simultaneously untangle inverted elements and improve qualities on the deformed domain for both FEMWARP and anisotropic FEMWARP Conclusions We have presented a new hybrid mesh deformation algorithm for computing the interior vertex positions of unstructured meshes subject to anisotropic boundary deformation. The aim of this method is to overcome the shortcomings of existing mesh warping methods, namely the propensity for creating inverted elements when the boundary deformation is large and the inability to maintain qualities of the initial mesh in the deformed mesh. Our first step is an anisotropic FEMWARP method which computes an initial guess for the interior vertex positions and has proven to exactly preserve initial mesh characteristics for a restricted class of meshes

194 174 (a) Surface mesh on the initial domain (b) Volume mesh on the initial domain (c) Optimized volume mesh on the deformed domain Fig Moving sphere in a 3D cube domain: (a) Surface mesh on the initial domain (b) Volume mesh on the initial domain (c) Optimized volume mesh using our hybrid algorithm on the deformed domain. This optimized mesh does not have any inverted elements.

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