Isogeometric Analysis of Geometric Partial Differential Equations

Transcription

1 MOX-Report No. 12/2016 Isogeometric Analysis of Geometric Partial Differential Equations Bartezzagi, A.; Dedè, L.; Quarteroni, A. MOX, Dipartimento di Matematica Politecnico di Milano, Via Bonardi Milano (Italy) ttp://mox.polimi.it

2 Isogeometric Analysis of Geometric Partial Differential Equations Andrea Bartezzagi a,, Luca Dedè a, Alfio Quarteroni a,b a CMCS Cair of Modeling and Scientific Computing, MATHICSE Matematics Institute of Computational Science and Engineering, EPFL École Polytecnique Fédérale de Lausanne, Station 8, Lausanne, CH 1015, Switzerland b MOX Modeling and Scientific Computing, Department of Matematics, Politecnico di Milano, Piazza L. da Vinci 32, Milano, 20133, Italy (on leave) Abstract We consider te numerical approximation of geometric Partial Differential Equations (PDEs) defined on surfaces in te 3D space. In particular, we focus on te geometric PDEs deriving from te minimization of an energy functional by L 2 -gradient flow. We analyze two energy functionals: te area, wic leads to te mean curvature flow, a nonlinear second order PDE, and te Willmore energy, leading to te Willmore flow, a nonlinear fourt order PDE. We consider surfaces represented by single-patc NURBS and discretize te PDEs by means of NURBS-based Isogeometric Analysis in te framework of te Galerkin metod. To approximate te ig order geometric PDEs we use ig order continuous NURBS basis functions. Instead, for te time discretization of te nonlinear geometric PDEs, we use Backward Differentiation Formulas (BDF) wit extrapolation of te geometric quantities involved in te weak formulation of te problem; in tis manner, we solve a linear problem at eac time step. We report numerical results concerning te mean curvature and Willmore flows on different geometries of interest and we sow te accuracy and efficiency of te proposed approximation sceme. Keywords: Geometric Partial Differential Equation, Surface, Hig Order, Isogeometric Analysis, Mean Curvature Flow, Willmore Flow 1. Introduction Geometric Partial Differential Equations (PDEs) describe te evolution of te geometrical domain in wic tese equations are set [1]; suc problems are usually defined on surfaces in 3D and te surface itself represents te unknown of te geometric PDE. Te computational domain evolves in time, or pseudo-time, according to geometric quantities of interest, suc as te curvature of te Corresponding autor. address: andrea.bartezzagi@epfl.c (Andrea Bartezzagi) Preprint submitted to Elsevier February 26, 2016

3 Isogeometric Analysis of Geometric PDEs 2 surface, towards te (local) minimization of te associated energy functional. Problems of tis kind arise in different applications; examples are material Science, were te crystalline structure or te geometric properties of te materials can be described via matematical models [2, 3, 4], in biomembrane modeling [5, 6, 7], or, more recently, in image processing, for example for automatic contours detection or image segmentation [8, 9, 10], as well as surface reconstruction and restoration [11, 12, 13, 14, 15]. In tis work, we focus on te numerical approximation of geometric PDEs defined on 3D surfaces, and specifically on two common problems. Te first one is te mean curvature flow, for wic te considered surface moves in te direction of its mean curvature vector. Tis causes te surface to evolve towards te minimization of its area [16], and it is of fundamental interest for te study of minimal surfaces. Problems of tis kind arise, for example, wen studying grain boundary motion in alloys, or modeling pysical systems involving surface tension, suc as biological cells and membranes, bubbles, capillarity, and oters. Tis problem as been extensively studied teoretically [16, 17, 18, 19] and tackled numerically wit different approaces, e.g. by using te Finite Element Metod (FEM) in [20, 21], level set formulations in [22, 23], or pase field approaces in [1]. Ten, we focus on te Willmore flow problem [24], wic leads to te minimization of te Willmore (or bending) energy, wic appears, for example, in optimal surface modeling [25], surface restoration [15], and in pysical models for biomembranes [26, 27, 28]. Teoretical results about te existence, uniqueness, and regularity of te solutions of te Willmore flow problem can be found in [29, 30, 31, 32]. Regarding its numerical approximation ([1]), te seminal work in [33] considers a general surface evolver, wic as been applied to te Willmore energy using te FEM. Approximations based on finite difference scemes for axisymmetric solutions are proposed in [34]. In general, te numerical approximation of te Willmore flow on parametric surfaces is based on te FEM, as e.g. in [35, 36, 37]; in [38] a formulation based on te level set metod is used, wile in [39, 40] approximations of te Willmore flow for curves (also called elastic flow of curves) are studied. For a general review on te numerical approximation of geometric PDEs we refer te reader to [1]; instead, for approximating PDEs on evolving surfaces, we refer to te recent review work [41]. Neverteless, all tese approaces generally involve geometric approximations of quantities wic may lead to accuracy issues or complex numerical scemes. In tis work, we focus on geometric PDEs defined on 3D parametric surfaces represented by NURBS geometrical mappings [42]. We spatially discretize te PDEs by means of Isogeometric Analysis (IGA) [43, 44]. Developed wit te aim of filling te gap between Computer Aided Design (CAD) and FEM, IGA is a discretization metod based on te isogeometric concept for 2

4 Isogeometric Analysis of Geometric PDEs 3 wic te same basis functions are used for bot representing te geometry and constructing te approximation subspace of te PDEs solution. IGA facilitates te exact geometric representation of te computational domain, even at te coarsest level of discretization; moreover, te NURBS function spaces, in wic te approximate solution lays, can be enriced by means of refinement procedures tat preserve te geometrical representation. Furtermore, NURBS-based IGA permits a fine control over te continuity of te basis functions (over smoot surfaces), allowing te use of globally C k -continuous basis functions, wit k p 1, being p te polynomial degree, tis even on closed surfaces [44, 45]. Oter tan being particularly accurate and efficient ([46, 47, 48]), tese ig order continuous NURBS bases permit te discretization of ig order PDEs witin te framework of te standard Galerkin metod witout resorting to mixed formulations [45, 49, 50]. Te advantages of NURBS-based IGA for surface PDEs bot of second [51] and ig order PDEs are igligted in [45]. Among tese, we ave te accurate evaluation of geometric quantities as tose arising in geometric PDEs, e.g. te normal to te surface and te mean and Gauss curvatures. In tis paper, we propose NURBS-based IGA for te spatial approximation of geometric PDEs, as te mean curvature and Willmore flows. Moreover, we propose te time discretization of te nonlinear problems arising from te IGA semi-discretization by means of Backward Differentiation Formulas (BDF) [52, 53]. In particular, we treat te nonlinear terms (including te geometric ones) troug time extrapolations compatible wit te BDF sceme at and, leading to te solution at eac time step of a linear algebraic system. We provide and critically discuss several numerical results for bencmark problems described by geometric PDEs, wic sow te accuracy and efficiency of te proposed numerical sceme. Tis paper is organized as follows. In Section 2 we introduce te geometric representation and notation used trougout te rest of te paper, togeter wit te definitions of te employed differential operators defined on surfaces. In Section 3 a generic formulation of geometric PDEs is introduced, wit furter focus on te mean curvature and te Willmore flows. Our approac to te spatial and temporal discretizations of te PDEs, based on IGA and BDF scemes, is presented in Section 4. Numerical results for different test problems are provided and discussed in Section 5. Conclusions follow. 2. Geometric Representation and Differential Operators on Surfaces Let us consider a parametric domain Ω R 2 of finite and positive measure wit respect to te topology of R 2, togeter wit te parametric coordinate, a vector-valued independent variable 3

5 Isogeometric Analysis of Geometric PDEs 4 ξ = (ξ 1, ξ 2 ) R 2. From ere on we denote wit Ω a compact, connected, oriented and smoot surface in R 3, wit or witout boundary Ω, defined by te geometrical mapping X as: X : Ω Ω R 3, ξ X(ξ) = p. (2.1) By assuming it exists, we denote te inverse mapping as: X 1 : Ω Ω R 2, p X 1 (p) = ξ. (2.2) In order to define PDEs on te surface Ω, we introduce surface differential operators on te manifold Ω, e.g. te surface gradient, divergence, and Laplace-Beltrami operators; see e.g. [54, 55, 56]. We introduce some fundamental geometrical quantities associated to te mapping (2.1): in particular, te Jacobian F : Ω R 3 2 of te mapping X, defined as F i,α (ξ) := X i (ξ), for i = 1, 2, 3 ξ α and α = 1, 2, te first fundamental form or metric tensor Ĝ : Ω R 2 2, defined as Ĝ (ξ) := ( ) T ) F (ξ) F (ξ), and its determinant ĝ : Ω R, defined as ĝ (ξ) := det (Ĝ (ξ). We consider geometrical mappings (2.1) wic are invertible a.e. in Ω, i.e. we allow ĝ (ξ) = 0 only in zeromeasure subsets of Ω, tus requiring ĝ to be positive elsewere. Te geometric quantities F, Ĝ, and ĝ are terefore expressed directly on te manifold Ω by using te inverse mapping of Eq. (2.2) as: F (p) := ( F X 1 )(p), G(p) := (Ĝ X 1 )(p), and g(p) := (ĝ X 1 )(p) for p Ω. (2.3) Moreover, we denote wit n Ω : Ω R 3 te unit vector normal to te surface Ω. By proceeding as in Eq. (2.3), any sufficiently regular function defined on te manifold Ω, say φ C 0 (Ω), is expressed in te parametric domain Ω as: φ(p) = ( φ X 1 )(p) for p Ω, (2.4) wit φ (ξ) := φ(x(ξ)), for ξ Ω. By denoting wit Ω R 3 a 3D domain containing te surface Ω (a tubular region) and considering a generic function φ C 1 (Ω), we define te surface gradient of φ as te ortogonal projection of te gradient of its smoot prolongation φ on Ω onto te tangent yperplane of Ω: Ω φ := (I n Ω n Ω ) φ Ω, (2.5) wit I being te second order identity tensor. In te same way, we define te surface divergence as Ω ϕ := tr( Ω ϕ), wit ϕ [C 1 (Ω)] 3, and te Laplace-Beltrami operator as Ω φ := Ω Ω φ, 4

6 Isogeometric Analysis of Geometric PDEs 5 provided tat φ C 2 (Ω). Ten, we exploit te geometric quantities defined in Eq. (2.3) to rewrite te gradient and Laplace-Beltrami differential operators in te parametric domain Ω as ([54, 56]): ([ ] Ω φ(p) = F Ĝ 1 φ X 1) (p) for p Ω, (2.6) and respectively. Ω φ(p) = ([ 1 ĝ ( ) ] ) ĝ Ĝ 1 φ X 1 (p) for p Ω, (2.7) Te second fundamental form associated to te geometric mapping (2.1) is te second order tensor defined as: H(p) := Ω n Ω (p) for p Ω. (2.8) Te symmetric tensor H is also called sape operator [56]; it possesses a null eigenvalue associated to te eigenvector n Ω and two oter non zero eigenvalues (since we consider surfaces in R 3 ) called principal curvatures and denoted in tis work as κ i, wit i = 1, 2. curvature H as te trace of H: We define te total mean H(p) := tr(h(p)) = κ 1 (p) + κ 2 (p) for p Ω. (2.9) We consider te normal n Ω to be oriented suc tat H is positive for sperical surfaces wit te normal pointing away from teir origin. We define also te total mean curvature vector as H(p) := H(p)n Ω (p) for p Ω (2.10) and we denote wit K te Gauss curvature, defined as te product of te principal curvatures: K(p) := κ 1 (p)κ 2 (p) for p Ω. (2.11) Furtermore, we introduce te identity function x : Ω R 3 on Ω, i.e. te map x(x(ξ)) = X(ξ) for ξ Ω. (2.12) We also recall te important relation ([56]): ( Ω x)(p) = H(p) for p Ω, (2.13) wic links te surface Laplace-Beltrami operator applied to te identity function on Ω to te total mean curvature vector. 5

7 Isogeometric Analysis of Geometric PDEs 6 3. Geometric PDEs In tis section, we firstly introduce te general formulation of geometric PDEs defined on surfaces, ten we focus on te mean curvature and Willmore flow problems General formulation A geometric PDE on a surface is an equation describing te evolution of te surface itself. Starting from an initial surface Ω 0, identified by te geometrical mapping X 0 : Ω R 3, we focus on finding, for all te times t (0, T ), a family of surfaces identified by teir geometrical mapping X(t) : Ω R 3, wose evolution obeys a differential law of te form Ẋ = F(t, X, n Ω, H, K,...) on Ω t, wit X(0) = X 0, and possibly wit boundary conditions on Ω t in case te latter is not empty (tat is wen Ω t is not a closed surface). From now on, we indicate wit Ω t R 3 te surface identified by te geometrical mapping X(t) at eac time t [0, T ], and, if it exists, its boundary as Ω t ; te subscript t refers to te time dependence of te computational domain, i.e. te evolving surface. Te law F identifies te problem at and and potentially depends on several geometrical quantities associated wit te geometry. We consider geometric PDEs deriving from te minimization of an energy functional J(Ω) under certain geometrical constraints. Tis functional can be seen as te objective functional in an optimization process, were te design variable is represented by te surface Ω itself. J usually depends on geometrical quantities associated to te geometrical mapping of te surface Ω, as described in Section 2. In tis work, we focus on L 2 -gradient flows [57, 58] of energy functionals, even if oter options are available. We consider te evolving surface Ω t to be sufficiently smoot bot in space and time and described by a smoot geometrical mapping X = X(ξ, t), ξ Ω, for eac t (0, T ), wic actually depends on te form of J. Ten, following Eq. (2.12) we introduce te identity function: x : Ω t R 3 : x(x(ξ, t)) X(ξ, t), ξ Ω, for all t (0, T ). (3.1) In order to simplify te notation, encefort we drop te arguments X(ξ, t) and t, for wic we always assume tat ξ Ω and t (0, T ), for quantities defined on Ω t for all t (0, T ), as e.g. x. We remark tat Ω t is a function of te mapping x(t), as Ω t = Ω(x(t)). We igligt te fact tat, in tis work, we only consider geometric PDEs defined on parametrized surfaces: indeed, we do not aim at treating canges of topology, for wic we keep te parametric domain Ω invariant trougout te geometric evolution process. 6

8 Isogeometric Analysis of Geometric PDEs 7 We denote wit v : Ω t R 3 and wit v : Ω t R, for all t (0, T ), te velocity and te normal velocity of te surface Ω t respectively, reading: v := X t and v := v n Ω, (3.2) respectively. In general, we write te material derivative of te identity function x as ([36]): wic simplifies into: ẋ = x t + v x, (3.3) ẋ = v = X t, (3.4) since x 0 and x = I. t In order to treat te material derivative of quantities depending on volume and area integrals, we use te transport teorem [1, 56]. Let us consider a function w C 1 ( Ω t ), were we ave introduced an open set Ω t suc tat Ω t Ω t for all t (0, T ). Under tis assumption, we compute te material derivative of te function w integrated in Ω t as: d w w dω t = dt Ω t Ω t t dω t + wvh dω t + Ω t Ω t w n Ω v dω t, (3.5) were H is te mean curvature defined in Eq. (2.9). Moreover, by assuming tat eac surface Ω t is te boundary of an open bounded subset Θ t R 3, we compute te material derivative of te function w integrated in te volume Θ t as: d w w dθ t = dt Θ t Θ t t dθ t + wv dω t. (3.6) Ω t We denote wit dj(ω t )(ϕ) te sape differential of J at fixed time t (0, T ), i.e. te first variation of J corresponding to a deformation of Ω t along te direction ϕ : Ω t R 3 ([56]), as: dj(ω t )(ϕ) = dj(x)(ϕ) = d dε J(x + εϕ) ε=0 = J (x), ϕ. (3.7) } By assuming t +, te stationary points { Ω of te energy J are ten identified by: dj( Ω )(ϕ) = 0, ϕ : Ω R 3. (3.8) In order to find suc minimizers of te energy, we consider te L 2 -gradient flow of te functional J. Ten, te problem becomes: given an initial surface Ω 0 R 3, find, for all t (0, T ), Ω t suc tat: ẋ = µj (x) in Ω t, b.c.s on Ω t, x(0) = x 0 in Ω 0, (3.9) 7

9 Isogeometric Analysis of Geometric PDEs 8 were b.c.s stands for boundary conditions and µ R + represents te mobility. Eq. (3.7), problem (3.9) can be expressed in weak formulation, for all t (0, T ), as: (ẋ, ϕ) L 2 (Ω t) = µ dj(x)(ϕ) ϕ : Ω t R 3 satisfying te essential b.c.s, x(0) = x 0 in Ω 0, Tanks to (3.10) wit te natural boundary conditions taken into account in te term dj(x)(ϕ). Problem (3.10) is, in general, a igly nonlinear system of PDEs wose steady states correspond to local minima of te energy J Mean curvature flow Let us consider te energy functional J = J A defined as: J A (Ω) := 1 dω, (3.11) wic corresponds to te area of te surface Ω. Under suitable ypoteses (e.g. X C 2 ( Ω)), te first variation of J A at Ω along te direction ϕ : Ω R 3 reads ([16, 36]): dj A (Ω)(ϕ) = H ϕ dω ϕ : Ω R 3, (3.12) Ω were H is te total mean curvature vector defined in Eq. (2.10). Te mean curvature flow is te problem associated wit te minimization of te area functional J A by means of a L 2 -gradient flow. Following te prototype Eq. (3.9), te problem reads: given an initial surface Ω 0 R 3 parametrized by te mapping X 0 : Ω R 3, find Ω t parametrized by x : Ω t R 3, for all t (0, T ), suc tat: ẋ = µh(x) in Ω t, b.c.s on Ω t, (3.13) x(0) = x 0 in Ω 0, were Ω t = Ω(x(t)), as usual. Problem (3.13) models te evolution of te surface towards te local minimization of its area, as can be verified by using te transport teorem of Eq. (3.5) wit w = 1 and te normal velocity v = H, tus obtaining: were Ω t is te area of Ω t. d dt Ω t = 1 µ Ω Ω t v 2 dω t, (3.14) Te mean curvature flow problem as been extensively studied bot from te teoretical and numerical points of view. Specifically, in [18] existence of a solution is studied for te parametric 8

10 Isogeometric Analysis of Geometric PDEs 9 evolution of an initial smoot convex surface witout boundary. In [19], te evolution of nonparametric surfaces wit boundary is analyzed. Te problem as been tackled numerically initially wit FEM using linear Lagrange polynomials in [20]; ten, more advanced scemes, taking also into account te tangential motion of te mes, ave been considered, as e.g. in [21]. For a general overview of te approximation of te mean curvature flow of curves, surfaces, graps, level sets, and wit pase-field approac, we refer te interested reader to [1] Willmore flow Let us consider te Willmore energy functional J = J W, defined as: J W (Ω) := 1 2 Ω H 2 dω, (3.15) wic expresses te bending energy associated to a surface; for more details and properties about te Willmore energy, we refer to [24]. Te Willmore flow is te L 2 -gradient flow of J W. Under suitable ypoteses, te first variation of J W at Ω along ϕ : Ω R 3 is given by ([24]): dj W (Ω)(ϕ) = ( Ω H 12 ) H3 + H H 2 n Ω ϕ dω ϕ : Ω R 3. (3.16) Ω We now consider closed surfaces for tis specific problem, for wic Ω =. Following te prototype Eq. (3.9), te Willmore flow problem reads: given an initial surface Ω 0 R 3 parametrized by te mapping X 0 : Ω R 3, find Ω t parametrized by x : Ω t R 3, for all t (0, T ), suc tat: ẋ = µ ( Ωt H(x) 12 ) H3 (x) + H(x) H(x) 2 n Ωt in Ω t, x(0) = x 0 in Ω 0, (3.17) wit Ω t = Ω(x(t)). In te specific case of 3D closed surfaces, we can conveniently use te relation H 2 = κ κ 2 2 = H 2 2K, for wic we rewrite problem (3.17), for all t (0, T ), as: ( ẋ = µ Ωt H(x) + 1 ) 2 H3 (x) 2H(x)K(x) n Ωt in Ω t, x(0) = x 0 in Ω 0. (3.18) Wen te parametric domain is mono-dimensional, Ω t represents a curve and problem (3.17) corresponds to te elastic flow of curves; global existence in time of a solution for curves in R n was studied in [59] and [39], for n = 2 and n 3 respectively. For surfaces, wic represent te focus of tis paper, te Willmore flow problem as been studied analytically mainly on closed surfaces. Existence of a solution up to te finite time T < + for two-dimensional surfaces in R n, wit 9

11 Isogeometric Analysis of Geometric PDEs 10 n 3, is proven in [29], wit T depending on te curvature of te initial surface Ω 0. Existence and uniqueness of te local solution of problem (3.18) under te ypotesis tat te initial geometry Ω 0 is a compact, closed, immersed, and orientable C 2,α -surface in R 3 is proven in [32], togeter wit te global existence of te solution in time were Ω 0 is sufficiently close to a spere. In [30] global existence of solutions is sown under te assumption tat H 2 is sufficiently small, being Ω 0 H te trace-free part of te second fundamental form H. In [31] te autors proved tat if Ω 0 is a smoot immersion of a spere in R 3 and it is suc tat its Willmore energy J W (Ω 0 ) 16π, ten its Willmore flow smootly exists for all times and converges to a spere. Tis result is numerically verified in [34], were results sow tat te flow develops singularities in finite time wen Ω 0 is associated to a Willmore energy J W (Ω 0 ) > 16π. 4. Space and Time Discretizations In tis section, we describe our approac to te numerical approximation of te mean curvature and Willmore flow problems of Section 3. Let us consider te general gradient flow problem (3.9) derived from te minimization of an energy functional J(Ω). Ten, we can write te problem in weak form as follows: find, for all t (0, T ), x V g,t : m(ẋ, ϕ) + a(x)(ϕ) = 0 ϕ V 0,t x(0) = x 0, (4.1) were a( )( ) is a form wic defines te problem under consideration and m(, ) is te mass form m(ẋ, ϕ) = ẋ ϕ dω t. Te Hilbert spaces V g,t and V 0,t depend on te order of te spatial Ω t differential operators involved in te form a( )( ); in case te surface Ω t is open, V g,t accounts for te non-omogeneous essential boundary conditions and V 0,t is its omogeneous counterpart. If te problem is defined on closed surfaces, ten Ω t for all t (0, T ) and te spaces V 0,t and V g,t coincide and are identified wit te Hilbert space V t. We remark tat te forms a( )( ) and m(, ) and te function spaces depend on te current computational domain Ω t, wic itself depends on x(t), as well as te geometric quantities in te form a( )( ). In Sections 4.1 and 4.2 we introduce te spatial and time discretization tecniques adopted to approximate problem (4.1), respectively Isogeometric Analysis We spatially discretize problem (4.1) by means of NURBS-based Isogeometric Analysis (IGA) in te framework of te Galerkin metod ([44]). 10

12 Isogeometric Analysis of Geometric PDEs 11 (a) p = 1 (b) p = 2 (c) p = 3 Figure 1: Univariate B-spline basis functions of polynomial degrees p = 1, 2, and 3 obtained from te knot vectors Ξ = {{0} p+1, 15, 25, 35, 45 }, {1}p+1 ; te basis functions are globally C p 1 -continuous in Ω = (0, 1) NURBS In tis work, we specifically consider surfaces represented by NURBS, as in [45]; for furter details and properties of NURBS, see [42, 43]. NURBS surfaces reads: X (ξ) = n bf i=1 R i (ξ) P i, Ri (ξ) := n bf j=1 Te geometrical mapping (2.1) in te case of w i w j Nj (ξ) N i (ξ) for i = 1,..., n bf, (4.2) wit P i R 3, for i = 1,..., n bf, being te control points in te pysical space were te surface Ω is defined and n bf te number of basis functions. Te NURBS basis functions R i (ξ), for i = 1,..., n bf, are obtained from te B-spline basis functions N i (ξ) by projective transformations, wit weigts w i R, for i = 1,..., n bf. Tis allows te NURBS mapping to exactly represent surfaces suc as conic sections, e.g. speres and tori, wic are geometrical entities often considered wit geometric PDEs. Univariate B-spline basis functions are built using te Cox-de Boor recursion formula [42] wit te knot vectors Ξ := {ξ j } n bf +p+1 j=1, were p is te polynomial degree, ξ j R are te knots, and te intervals between te knots are called knot spans. Te multiplicity of a knot inside te knot vector controls te continuity of te basis functions across tat knot. In practice, repeating a knot k times makes te basis functions to be C p k -continuous across tat knot. In Figure 1, we report examples of B-spline basis functions wit different polynomial degrees and order of continuities. By considering a bidimensional parametric domain and denoting wit α = 1, 2 te parametric directions, te multivariate B-spline basis functions N i (ξ) are built from te tensor product of te univariate B-spline basis functions N i,α, for i = 1,..., n bf,α, built using te knot vectors Ξ α, for α = 1, 2. Te tensor product of knot vectors also defines a partition of te parametric domain Ω 11

13 Isogeometric Analysis of Geometric PDEs 12 into regions defined by te knot spans, also called mes elements NURBS-based IGA IGA is an approximation metod for PDEs based on te isoparametric concept, for wic te same basis functions used for te geometric representation are ten also used for te numerical approximation of te PDEs [43, 44]. Te NURBS computational domain, i.e. te surface Ω, at a prescribed time instance is represented at te coarsest level of discretization. From Eq. (4.2) we define te NURBS function space N over te parametric domain Ω, and te function space N over te pysical domain Ω, as: { } N := span Ri, i = 1,..., n bf respectively. and { } N := span Ri (ξ) X 1 (ξ), i = 1,..., n bf, (4.3) Tese spaces are used to build te trial function spaces for te approximation of PDEs and, since we consider a Galerkin metod, also te test function spaces. We denote wit te subscript te mes elements caracteristic size, being usually defined as te maximum diameter of te elements in te pysical space [51]. Te function spaces of Eq. (4.3) can be enriced wit several refinement procedures [43], wic maintain te geometrical representation unaffected. In particular, wit te knot insertion new knots are introduced into te knot vectors, creating possibly new knot spans and increasing te total number of mes elements and basis functions. If during te knot insertion te continuity of te basis functions is preserved, tis procedure is analogous to te -refinement of te FEM. Te order elevation procedure increases te polynomial degree p of te NURBS basis functions wile preserving te existing continuity of te basis functions, in a way analogous to te p-refinement of te FEM. Finally, k-refinement consists in te sequential application of order elevation and knot insertion procedures in order to increase te polynomial degree p wile maintaining te igest possible degree of continuity of te basis functions across te mes elements, a procedure wic does not ave an analogous counterpart in te FEM. For more details, see e.g. [42, 43, 44]. Let us consider te general gradient flow problem in weak form of Eq. (4.1). Since we deal wit surfaces defined by geometrical mappings in te form (2.1), wic are invertible a.e., we pull-back problem (4.1) into te parametric domain Ω, tus obtaining: find, for all t (0, T ), X V g : m(ẋ, ϕ) + â(x)( ϕ) = 0 ϕ V 0, X(0) = X 0, (4.4) 12

14 Isogeometric Analysis of Geometric PDEs 13 were V g and V 0 correspond to te pull-back of te function spaces V g,t and V 0,t on te parametric domain Ω, respectively; te forms m(, ) and â( )( ) result from te pull-back operation applied to m(, ) and a( )( ), respectively. In particular, we ave: m(ẋ, ϕ) = Ẋ ϕ ĝ d Ω (4.5) Ω for ϕ V 0, were ĝ is te determinant of te first fundamental form of te mapping defined in Section 2. Te form a( )( ) is pulled-back in a similar fasion, using te geometric quantities introduced in Section 2 and will be specified for te gradient flow problem under consideration. Ten, we proceed wit te spatial discretization of problem (4.4). We coose suitable NURBS function spaces for te trial and test functions, accordingly wit te NURBS mapping, for wic, for all t (0, T ), we seek solutions in te form: n bf ( x (t) = Ri X 1) P i (t) (4.6) i=1 in te pysical space, were P i (t), for i = 1,..., n bf, are te control points introduced in Section 2 wic describe te geometry and, in tis context of geometric PDEs, also play te role of vectorvalued control variables. Te semi-discretized problem reads: m(ẋ, ϕ ) + a(x )(ϕ ) = 0 ϕ V 0,t,, find, for all t (0, T ), x V g,t, : x (0) = x 0,, wic can be pulled-back into te parametric domain Ω, tus obtaining: find, for all t (0, T ), X V m(ẋ, ϕ ) + â(x )( ϕ ) = 0 ϕ V 0,, g, : X (0) = X 0, ; (4.7) (4.8) te finite-dimensional function spaces V g,t,, V 0,t,, V g, and V 0, are subsets of NURBS function spaces defined as V g,t, := V g,t [N ] 3, V 0,t, := V 0,t [N ] 3, V g, := V g [ N ] 3 and V 0, := V 0 [ N ] 3, respectively. Wile simple surfaces can straigtforwardly be represented by using C p 1 -continuous NURBS basis functions in a single patc, tis may be not te case of more complex surfaces, as te closed ones, for wic te single patc NURBS representation involves bases wic are only globally C 0 -continuous in Ω and Ω. Since we are interested in te approximation of geometric PDEs of order equal or eventually iger tan two (as te Willmore flow), we need to use trial and test function spaces wit te Galerkin metod wic guarantee ig order (at least 1) global continuity 13

15 Isogeometric Analysis of Geometric PDEs 14 of te basis functions over te wole surface. Terefore, wen dealing wit closed surfaces, we consider te construction of periodic NURBS function spaces N per, built from te original NURBS function space N, defining Ω, troug element-wise linear transformations of te basis functions and suitable constraints among te degrees of freedom [49, 45]. In particular, we apply te linear operator T per R n bf n bf to te basis functions R { } nbf := Ri defining te NURBS function space N, tus obtaining R per := T per R (ξ) and te periodic spaces per N { } := span Rper i, i = 1,..., n bf, N per N per := span i=1 and N per : { Rper i (ξ) x 1 (ξ), i = 1,..., n bf }. For more details on te periodic NURBS function spaces and error analysis about te IGA approximation of ig order PDEs defined on closed surfaces, we refer te interested reader to [45]. We remark tat, since te control points {P i } n bf i=1 describe te geometry but at te same time represent te unknown of te problem, we cannot use a subparametric approac as in [45]. Indeed, we also need to apply te same transformations to te control points in order to use te same NURBS function space for bot te solution and te geometrical representation, i.e. a pure isoparametric approac. Being P per i following relation ([49]): (4.9) R 3, i = 1,..., n bf te transformed control points, obtained wit te P per := (T per ) T P, (4.10) we stress te fact tat te representation of Ω given by te periodic NURBS basis functions R per togeter wit te control points P per is equivalent to te one given by te original NURBS basis n bf n functions R bf wit te control points P, i.e. x = R per i P per i = R i P i. Terefore, wen dealing wit closed or partially closed geometries, te quantities R per, P per and N per i=1 i=1 are substituted into te original ones in te definition of te problem to be approximated, as in Eqs. (4.6), (4.7), and (4.8). In order to simplify te notations, from now on we will drop te superscript per, referring indifferently to bot te non-transformed or te transformed NURBS function space and control points depending on te situation at and (open or closed surfaces) IGA for mean curvature flow Let us consider te mean curvature flow problem of Eq. (3.13). By assuming sufficient regularity for all te geometric quantities involved, by using te relation of Eq. (2.13) and integrating by parts te Laplace-Beltrami operator, we recast te mean curvature problem (3.13) in te general 14

16 Isogeometric Analysis of Geometric PDEs 15 formulation of Eq. (4.1) wit te form a( )( ) being defined as ([20]): a MCF (x)(ϕ) := µ Ω x : Ω ϕ dω (4.11) Ω for ϕ V 0, wit V 0 being subset of [ H 1 (Ω) ] 3. Te semi-discretized problem obtained by te NURBS-based IGA approximation of te mean curvature flow problem is in te form of Eq. (4.7), wit te function spaces V g,t, and V 0,t, being subsets of V t, := [H 1 (Ω t )] 3 N. Wen te semidiscretized problem is rewritten into te parametric domain Ω after te pull-back operation, te form of Eq. (4.8) becomes: ( ) ( ) â(x )( ϕ ) := µ F Ĝ 1 X : F Ĝ 1 ϕ ĝ d Ω (4.12) Ω for ϕ V 0,, aving used te relation in Eq. (2.6), wit te function spaces V g, and V 0, being subsets of V := [H 1 ( Ω)] 3 [ N ] IGA for Willmore flow Te Willmore flow problem of Eq. (3.18) is a nonlinear time dependent ig order PDE. For spatial discretizations based on te standard FEM wit C 0 -continuous basis functions, mixed formulations to decrease te order of te PDE are commonly used [5, 35, 36]. In addition, te term HK, wic involves bot te mean (H) and te Gauss (K) curvatures, nonlinearly depends on te principal curvatures and it is difficult to treat wit variational metods [36]; terefore, terms as K or te normal to te surface n Ω are usually avoided troug suitable manipulations in te weak formulation of te problem. Regardless of te order of te differential problem, tese considerations lead to te adoption of mixed formulations were additional unknowns are introduced, usually being te velocity v, te normal component of te velocity v along n Ω, te mean curvature H, or te mean curvature vector H [5, 35, 36, 37], for wic te resulting PDEs are of te second order. In te framework of NURBS-based IGA, one benefits bot from te exact representation of te geometry Ω, wit te possibility of calculating te geometrical quantities directly from te NURBS representation, and te ability to treat ig order surface PDE in a straigtforward manner [45]. We terefore propose te following weak formulation of te Willmore flow problem: 15

17 Isogeometric Analysis of Geometric PDEs 16 find, for all t (0, T ), x V t and v W t : v ψ dω t + µ Ω t ( Ωt x n Ωt ) Ωt ψ dω t Ω t Ω t v(0) = v 0, + µ ẋ ϕ dω t v n Ωt ϕ dω t = 0 Ω t ϕ V t, x(0) = x 0, Ω t ( ) 1 2 H2 2K ( Ωt x n Ωt ) ψ dω t = 0 ψ W t, (4.13) were V t := [ H 2 (Ω t ) ] 3, wile Wt := H 2 (Ω t ); te normal velocity v (defined in Eq. (3.2)) is also an unknown of te problem. We consider NURBS-based IGA for te approximation of (4.13). We terefore discretize te equations following te same procedure described in Section 4.1.2, seeking te trial and test functions for te unknowns x and v in te function spaces V t, := V t [N ] 3 and W t, := W t N, respectively. We remark tat, wit IGA, te evaluation of te terms involving H and K is straigtforward, since te curvatures can be computed directly and exactly from te NURBS mapping x. Problem (4.13) is rewritten into te parametric domain Ω troug a pull-back operation as described in Section and similarly to te approximation of te mean curvature flow problem of Section 4.1.3; in tis case, we also use te relation of Eq. (2.7) for te treatment of te surface Laplace-Beltrami operator in te parametric domain Ω. We remark tat, since we need to ensure tat te test and trial function spaces are subsets of H 2, we consider NURBS function spaces wit basis functions at least globally C 1 -continuous a.e. in Ω t, for all t (0, T ). Moreover, since Ω is a closed surface, we consider NURBS periodic function spaces, as mentioned in Section Time discretization Problems governed by geometric PDEs are generally nonlinear. Indeed, all te geometric quantities and differential operators involved in problem (4.1) defined in Ω depend temselves on x. In literature [5, 20, 36], suc problems are typically discretized in time wit a semi-implicit first order sceme wit an explicit treatment of te geometric nonlinear terms; in tis manner, at any given time step, all te geometric quantities and differential operators are evaluated using te solution obtained at te previous time steps. In tis paper, we propose te time discretization of geometric PDEs wit ig order implicit Backward Differentiation Formulas (BDF) [52]. We address te 16

18 Isogeometric Analysis of Geometric PDEs 17 circular dependence between te solution and te geometric quantities by treating te geometry explicitly, using te solution extrapolated from te previous time steps; see e.g. [60, 61]. Let us consider te time discretization of te spatially discretized problem (4.7) in te time interval [0, T ], wit n as time step index and t n te n-t time step, suc tat t 0 = 0 and t N = T. We introduce te approximate surface Ω n+1 as te surface defined by te NURBS mapping: n bf X n+1 (ξ) = i=1 R i (ξ) P n+1 i (4.14) from Eq. (4.2), were { P n+1 } nbf i i=1 are te control points computed at te time instance t n+1. In general, wen considering a fixed time step size t, te time discretization using a k-t order BDF sceme consists in approximating te time derivative Ẋ at time step n + 1 troug a linear combination of te mappings X at te time step n + 1 and te k previous time steps, as: ( ) Ẋ n+1 1 k α 0 X n+1 t α i X n+1 i, (4.15) for n k 1, wit te coefficients α i R, for i = 0,..., k, being cosen in a way to guarantee tat te approximation is of order k. Moreover, Ω, wic we refer as te extrapolated surface, is defined by te NURBS mapping: n bf X (ξ) = were {P i } n bf i=1 are te control points obtained from te sets { k-t order extrapolation, as ([60]): P i := i=1 k j=1 i=1 R i (ξ) P i, (4.16) } P n+1 k nbf i i=1,..., {Pn i } n bf i=1 wit a β j P n+1 j i, (4.17) for i = 1,..., n bf, wit appropriate coefficients β j R, for j = 1,..., k. By referring now to te time derivative ẋ at time step n + 1 of te identity function x, following Eqs. (4.14), (4.15), and (4.16), we approximate it wit te k-t order BDF sceme as: ẋ n+1 1 t [ α 0 x n+1 k i=1 α i ( x n+1 i ] X n+1 i ) (X ) 1, (4.18) for n k 1. For notational convenience, we define x bdf : Ω R 3 as: x bdf := k i=1 α i ( x n+1 i α 0 X n+1 i ) (X ) 1 (4.19) 17

19 Isogeometric Analysis of Geometric PDEs 18 and v n+1 : Ω R 3 as: x n+1 x bdf := α 0, (4.20) t v n+1 as well as te extrapolated solution x : Ω R 3 at time t n+1 following Eqs. (4.16) and (4.17): x := k j=1 β j ( x n+1 j ) X n+1 j (X ) 1, (4.21) for n k 1. We can terefore rewrite problem (4.7) wit respect to te unknown velocity v n+1 find, for n = 0,..., N 1, x n+1 m (vn+1, ϕ ) + t a α (vn+1 0 v 0 = v 0,, V g, :, ϕ ) = a (xbdf, ϕ ) ϕ V 0,, as follows: (4.22) were V g, and V 0, are function spaces defined in te extrapolated surface Ω, wic correspond to V g,t, and V 0,t, onto Ω, respectively, and m (, ) and a (, ) are bilinear forms in wic te differential operators and domain of integrations are defined in Ω. For example, te form m (, ) reads: Te new mapping x n+1 m (vn+1, ϕ ) := v n+1 ϕ dω. Ω (4.23) : Ω R 3 is ten obtained as: x n+1 = x bdf + t α 0 v n+1, (4.24) corresponding to te new geometrical mapping X n+1, wic defines te new surface Ω n+1 as in Eq. (4.14) and approximating Ω tn+1. Regarding te mean curvature flow, te full discrete problem is te same of Eq. (4.22), wit te form a (, ) defined as: a (zn, ϕ ) = µ Ω z n : Ω ϕ dω Ω (4.25) 18

20 Isogeometric Analysis of Geometric PDEs 19 for ϕ V0,. Instead, for te Willmore flow problem (4.13), te full discrete problem reads: find, for n = 0,..., N 1, v n+1 V ψ dω + µ t v n+1 Ω v n+1 Ω + µ t α 0 Ω α 0 and vn+1 W : ( Ω v n+1 Ω [ 1 2 (H ) 2 2K ( ) = µ Ω x bdf n Ω Ω ψ dω Ω [ 1 µ Ω 2 (H ) 2 2K ϕ dω n Ω ) Ω ψ dω ] ( Ω v n+1 ) n Ω ψ dω ] ( Ω x bdf n Ω ) ψ dω ψ W, v n+1 n Ω ϕ dω = 0 ϕ V, Ω (4.26) v 0 = v 0,, v 0 = v 0,, were V and W are te function spaces V t, and W t, built on Ω, respectively. We remark tat te problems (4.22) and (4.26) are solved by recasting tem into te parametric domain Ω. 5. Numerical Results In tis section, we present several results on te numerical approximation of te mean curvature and te Willmore flow problems on different geometries Mean curvature flow We consider te numerical approximation of te mean curvature flow problem of Eq. (3.13) using te numerical sceme (4.22) proposed in Section For all te tests we set µ = 1 (see Eq. (3.13)). Test We consider te mean curvature flow of an initial unit spere Ω 0. By recalling Eq. (3.13) and by exploiting te radial symmetry of te spere, te geometry Ω t remains a spere for eac t (0, T ) wit evolution in sperical coordinates described by te following ordinary differential equation [34]: ṙ = 2 r r(0) = r 0, for t (0, T ), (5.1) 19

21 Isogeometric Analysis of Geometric PDEs 20 (a) t = 0 (b) t = (c) t = (d) t = (e) t = Figure 2: Test Mean curvature flow of a spere. Solution at different time instances. Figure 3: Test Mean curvature flow of a spere. Evolution of te approximated area Ω n and exact area Ω tn vs. time t; NURBS of degree p = 2 and C 1 -continuous a.e. wit 220 mes elements are used. were r(t) is te radius of te spere at time t and r 0 te radius of Ω 0. Tis equation as analytical solution: r(t) = r0 2 4t for t [0, T ], (5.2) from wic it is evident tat te spere degenerates for t = r2 0. Tus, considering an initial spere 4 Ω 0 of radius r 0 = 1, we expect te solution of problem (3.13) to be represented by a srinking spere wit radius described by Eq. (5.2) and collapsing into a single point at time T = Figure 2 sows te evolution of te spere Ω n obtained by te numerical approximation of problem (4.22), at different time instances; te evolution of te area Ω n is reported in Figure 3, togeter wit te evolution of te exact area Ω tn computed wit Eq. (5.2). Te spere is represented by NURBS of degree p = 2 and C 1 -continuous a.e., for a total of 220 elements, yielding 384 DOFs 1. Time advancement is performed employing a BDF sceme of order k = 2 wit fixed 1 Te amount of DOFs reported corresponds to te size of te linear system solved at eac time step; terefore, it takes into account for te constraints set to build te periodic basis functions and te fact tat te solution is vector valued (te velocity v R 3 for eac control point). 20

22 Isogeometric Analysis of Geometric PDEs 21 (a) p = 2, C 1 -continuous a.e., 3 2 quad. pts. (b) p = 3, C 2 -continuous a.e., 4 2 quad. pts. (c) p = 2, C 1 -continuous a.e., 7 2 quad. pts. (d) p = 3, C 2 -continuous a.e., 7 2 quad. pts. Figure 4: Test Mean curvature flow of a spere. Evolution of te errors on te area err n vs time t for meses wit different NURBS basis functions (ref. 1 wit 384 DOFs and ref. 2 wit 6,144 DOFs) and using (p+1) 2 (in (a) and (b)) and 7 2 (in (c) and (d)) quadrature points per mes element. time step size t = Te linear systems arising from te full discretization of te PDEs at eac time step are solved by using te GMRES metod wit te ILUT preconditioner [52], wit te stopping criterion being te relative residual (in Euclidean norm) below a tolerance of We report in Figure 4 te beavior of te errors on te numerically approximated area vs time, say err n := Ω tn Ω n, obtained by solving problem (4.22) wit NURBS of degree p = 2 and 3, wic are C p 1 -continuous a.e., respectively. We compare te errors obtained using meses of 220 and 2,380 elements for te p = 2, C 1 -continuous NURBS basis, wile 275 and 2,555 elements for te p = 3, C 2 -continuous basis (yielding 384 and 6,144 DOF for bot p = 2 and 3). In particular, Figures 4a and 4b sow te errors obtained using te standard Gauss-Legendre quadrature rule wit (p + 1) 2 points per mes element for te approximation of te integrals, wile Figures 4c and 4d sow te errors obtained using 7 2 quadrature points per element, tus wit a significantly 21

23 Isogeometric Analysis of Geometric PDEs 22 (a) p = 2, C 1 -continuous a.e. (b) p = 3, C 2 -continuous a.e. (c) Evolution of te condition number vs. time t Figure 5: Test Mean curvature flow of a spere. Sparsity patterns (a) and (b) and evolution of te condition number κ(a) of te matrix associated to te full discrete problem (4.22) vs. time t ((c)), using NURBS basis functions of degrees p = 2 and 3, C 1 - and C 2 -continuous a.e., respectively, and two refinement levels yielding 384 and 6,144 DOFs, respectively. increased accuracy of te numerical integration. We observe tat te errors are very small in all te cases, and only increase wen te geometry tends to degenerate in a point, as expected from te exact solution of Eq. (5.2). In addition, a smooter beavior of te error err n is observed wen using a large number of quadrature nodes. Neverteless, te errors remain very small, even for te standard Gauss-Legendre quadrature formulas wit (p + 1) 2 points typically used in IGA and employed in te rest of tis work. We report in Figures 5a and 5b te sparsity patterns of te matrices A arising from te full discrete problem (4.22) wit NURBS of degree p = 2 and 3, respectively, wit 384 DOFs in bot te cases. In Figure 5c te evolutions of te condition number of te matrices associated to problem (4.22) at eac time step are reported for te NURBS already 22

24 Isogeometric Analysis of Geometric PDEs 23 (a) p = 2, C 1 -continuous a.e. (b) p = 3, C 2 -continuous a.e. (c) p = 2, C 0 -continuous a.e. (d) p = 3, C 0 -continuous a.e. Figure 6: Test Mean curvature flow of a spere. Absolute errors on te area at time tñ = errñ vs. t, for different BDF scemes (BDF of orders k = 1, 2, and 3) and NURBS basis functions (p = 2 and 3, wic are C 0 - and C p 1 -continuous a.e.). considered for te results in Figure 4; te condition number κ(a) is actually a lower bound of te 1-norm condition number of te matrix A. Te condition numbers κ increase wit te degree of te NURBS basis functions and wen te mes is refined. We remark tat, for tis specific problem, te NURBS mapping is singular at te poles of te spere, wic leads to ig values of te condition number. Moreover, te spere srinks according to te mean curvature flow, and eventually degenerates in a point. Neverteless, in te case under consideration, te condition numbers κ(a) are never ig enoug to significantly interfere wit te accuracy of te GMRES solver (for te cosen tolerance). In order to compare te performance of te proposed sceme wit BDFs of different orders wit respect to te time step size t, simulations wit BDFs of orders k = 1, 2, and 3 ave been performed, for meses composed by NURBS basis functions of degrees p = 2 and 3, wic are C 0-23

25 Isogeometric Analysis of Geometric PDEs 24 and C p 1 -continuous a.e. on te surface. Errors on te area wit respect to te exact solution are reported in Figure 6 in logaritmic scale; te reported errors are computed as errñ := Ω tñ Ωñ at fixed time tñ = 0.016, wit ñ = t ñ. Te BDF scemes are initialized wit te corresponding t numbers of exact time steps in order to bootstrap te time integration metod correctly, suc tat order of convergence of k is maintained for eac BDF used. Te meses considered are built out of NURBS basis functions of degrees p = 2 and 3, wit 220 and 275 elements, respectively; for te degree p = 2, meses wit basis functions C 1 -continuous a.e. and C 0 -continuous a.e. are considered, wit 384 and 600 DOFs, respectively; for te degree p = 3, meses wit basis functions C 2 -continuous a.e. and C 0 -continuous a.e. are considered, wit 384 and 864 DOFs respectively. We remark tat te errors corresponding to te spatial discretization are significantly small, even wen approximating te problem wit a low amount of mes elements; tis fact permits us to employ ig order temporal discretizations and recover te full rate of convergence. Also, since te smoot C p 1 -continuous basis functions lead to a smaller number of DOFs tan teir C 0 -continuous counterpart, te former generally lead to more efficient and accurate discretizations. Test Next, we study te evolution of a torus subject to mean curvature flow. We consider a family of toric surfaces Ω 0 described by te relation: ( R 0 x 2 + y 2 ) 2 + z 2 = r 2 0 (5.3) in a standard Cartesian coordinate system, were R 0 and r 0 are te torus major and minor radii, respectively, being R 0 > 0 and r 0 (0, R 0 ), of te initial configuration corresponding to Ω 0. Depending on te ratio between te two radii R 0 /r 0, te torus is evolving eiter to collapse into a circle or to self-penetrate and merge into a disk. Figures 7 and 8 sow te evolution of tori wit R 0 = 1, r 0 = 0.5 and R 0 = 1, r 0 = 0.7, respectively, subject to mean curvature flow. Te first torus as an aspect ratio R 0 /r 0 suc tat it collapses into a circle, wile te second one tends to merge into an ellipsoid; since we adopt a parametric representation of te geometry and we do not support topology canges, we let te geometry evolve until a self-intersection of te surface occurs. Te evolution of te toric areas is plotted in Figures 9a and 9b, respectively. We consider NURBS wit basis functions of degree p = 2 and globally C 1 -continuous, wit 836 elements and 1,536 DOFs for bot te cases; we used a BDF sceme of order k = 2 for integration in time wit time step size t = Test Now we consider te mean curvature flow of an open surface, in particular a cylindrical sell. We parametrize te cylinder by its radius R 0 and eigt L 0. Te bottom and 24

26 Isogeometric Analysis of Geometric PDEs 25 t = 0 t = t = t = t = Figure 7: Test Mean curvature flow of a torus wit R 0 = 1 and r 0 = 0.5. Solution at different time instances. t = 0 t = t = t = t = Figure 8: Test Mean curvature flow of a torus wit R 0 = 1 and r 0 = 0.7. Solution at different time instances. top bases of te cylinder (two circles of radius R 0 ) are fixed (i.e. we set x = x 0, on Ω), wile te lateral surface (Ω t ) is free to evolve according to te mean curvature flow. Te geometry minimizing te area depends on te aspect ratio L 0 /R 0 of te initial cylinder Ω 0. In particular, te solution may eiter be discontinuous, consisting in two circles at te bases of te cylinder, and tus involving a topology cange (known as Goldscmidt solution [62]), or exibit a catenoid as ( y local minimum, generated by rotating te catenary of equation x = a cos along te y-axis, ( ) a) L0 wit a R being solution of te nonlinear relation cos R 0 = 0. Suc catenoid as area 2a a equal to: A cat = πa 2 [sin ( ) L0 + L ] 0. (5.4) a a Figures 10 and 11 sow te evolution of two cylinders, te first wit radius R 0 = 1 and eigt 25

27 Isogeometric Analysis of Geometric PDEs 26 (a) Torus wit R = 1 and r = 0.5 (b) Torus wit R = 1 and r = 0.7 Figure 9: Test Mean curvature flow of tori. Evolution of te approximated area Ω n vs. time t for a torus wit R 0 = 1 and r 0 = 0.5 (a) and a torus wit R 0 = 1 and r 0 = 0.7 (b); NURBS of degree p = 2 and globally C 1 -continuous wit 836 mes elements, yielding 1,536 DOFs, are used. L 0 = 1, wile te second one wit R 0 = 1 and L 0 = 2. Te meses considered in te spatial approximation are bot NURBS built of basis functions of degree p = 2 and globally C 1 -continuous, wit 456 elements, for a total of 1,152 DOFs; time integration is performed employing a BDF sceme of order 2, wit time step size t = Te evolutions of te areas Ω t are plotted in Figures 12a and 12b, respectively. Te first cylinder as aspect ratio L 0 /R 0 = 1 suc tat a catenoid is a local minimum and te numerical solution effectively converges to suc geometry. Te second cylinder (for L 0 /R 0 = 2) does not present a catenoid as local minimum, terefore te minimization process continues towards te solution wit topology canges, wic we stop wen a singularity in te geometrical mapping is reaced, as indicator of a topology cange. In bot te cases, we obtain accurate solutions even by employing spatial discretizations involving a small amounts of mes elements and DOFs Willmore flow We now consider te numerical approximation of te Willmore flow problem defined in Eq. (3.18) on closed surfaces using te numerical sceme (4.13) proposed in Section For all te tests we set µ = 1 (see Eq. (3.17)). Test As initial geometry Ω 0 we consider ellipsoids described by te following relation: x 2 a y2 b z2 c 2 0 = 1 {x, y, z} R 3, (5.5) 26

28 Isogeometric Analysis of Geometric PDEs 27 (a) t = 0 (b) t = 0.05 (c) t = 0.10 (d) t = 0.15 (e) t = 0.20 Figure 10: Test Mean curvature flow of a cylinder wit R 0 = 1 and L 0 = 1. Solution at different time instances. (a) t = 0 (b) t = (c) t = (d) t = (e) t = Figure 11: Test Mean curvature flow of a cylinder wit R 0 = 1 and L 0 = 2. Solution at different time instances. (a) Cylinder wit R 0 = 1 and L 0 = 1 (b) Cylinder wit R 0 = 1 and L 0 = 2 Figure 12: Test Mean curvature flow of cylinders. Evolution of te approximated area Ω n vs. time t for a cylinder wit R 0 = 1 and L 0 = 1 (a) and wit R 0 = 1 and L 0 = 2 (b); NURBS of degree p = 2 and globally C 1 -continuous wit 456 mes elements, yielding 1,152 DOFs, are used. 27

29 Isogeometric Analysis of Geometric PDEs 28 t = 0 t = 0.49 t = 1.22 t = 3.66 t = Figure 13: Test Willmore flow of an ellipsoid wit a 0 = 4, b 0 = 4 and c 0 = 1. Solution at different time instances. were a 0, b 0, c 0 R are positive constants denoting its aspect ratio. It is known tat an ellipsoid sould converge to a spere under Willmore flow [31], wic as Willmore energy J W equal to 8π. For our numerical test, we consider te approximation of te Willmore flow applied to an initial ellipsoid Ω 0 wit parameters a 0 = 4, b 0 = 4, and c 0 = 1. Te geometry Ω 0 is represented as a NURBS surface wit basis functions of degrees p = 2 and 3, being C 1 - and C 2 -continuous a.e., respectively, wit two -refinement levels for eac degree. Te considered meses wit NURBS of degree p = 2 are made of 684 and 2,380 elements, respectively for te two refinement levels; te meses wit basis functions of degree p = 3 are instead made of 779 and 2,555 elements, respectively. Wit respect to te two -refinement levels, te total number of DOFs amounts to 2,048 and 8,192, independently of te degree p of te NURBS basis functions 2. Integration in time is performed employing te BDF sceme of order k = 2 wit a fixed time step size t = Figure 13 sows te solution obtained at different time steps, wit te mes composed of 779 elements. Te evolution in time of te Willmore energy, togeter wit te Willmore energy associated to a spere (indicated as Exact final energy), is reported in Figure 14, togeter wit te evolution in time of te area and te volume of te approximated geometry Ω n. We remark tat problem (3.18) does not involve any constraint on te area and te volume of te surface, wic are in principle free to evolve wile te Willmore energy J W is being minimized; as a matter of fact, we notice tat 2 Te number of DOFs accounts for bot a vector valued unknown (te velocity v) and a scalar unknown (te normal velocity v). 28

30 Isogeometric Analysis of Geometric PDEs 29 Figure 14: Test Willmore flow of an ellipsoid wit a 0 = 4, b 0 = 4 and c 0 = 1. Evolution of te Willmore energy J W, area and volume vs. time t (zoom) for NURBS of degrees p = 2 and C 1 -continuous a.e. wit 684 and 2,380 elements, yielding 2,048 and 8,192 DOFs, respectively, and p = 3 and C 2 -continuous a.e. wit 779 and 2,555 elements, again yielding 2,048 and 8,192 DOFs, respectively. te evolutions of te area and te volume are sensitive to te discretization under consideration. By using as stopping criterion for te Willmore flow te difference between te Willmore energy at two consecutive time steps, wic sould be under te tresold 10 5, we obtain, wit te coarsest mes built of NURBS of degree p = 2 a final error on te Willmore energy equal to (2.585%); wen refining te mes, we obtain a significant reduction of suc error, being equal to (0.675%). Instead, using NURBS of degree p = 3 yields better results, wit errors equal to (0.094%) and (0.022%) for te first and second -refinement levels, respectively. Finally, we report in Figures 15a and 15b te sparsity patterns of te matrices associated to te full discrete problem (4.26) wit NURBS of degrees p = 2 and 3, wit 2,048 DOFs for bot te cases. In Figure 15c te evolution of te condition number κ(a) of te matrices involved in problem (4.26) is reported at eac time step, for eac NURBS considered for te results of Figure 14. Te beavior of κ(a) is similar to wat experienced for Test 5.1.1, in te sense tat, te iger degree of te 29

31 Isogeometric Analysis of Geometric PDEs 30 (a) p = 2, C 1 -continuous a.e. (b) p = 3, C 2 -continuous a.e. (c) Evolution of te condition number vs. time t Figure 15: Test Willmore flow of an ellipsoid wit a 0 = 4, b 0 = 4 and c 0 = 1. Sparsity patterns in (a) and (b) and evolution of te condition number κ(a) of te matrix associated to te full discrete problem (4.26) vs. time t ((c)), using NURBS basis functions of degrees p = 2 and 3, C 1 - and C 2 -continuous a.e., respectively, and two refinement levels yielding 2,048 and 8,192 DOFs, respectively, bot for p = 2 and p = 3. NURBS basis functions and te finer te mes, te iger te condition number. Wit respect to Test 5.1.1, te condition number is generally iger, due to te ig order derivatives involved in te Willmore flow problem. Test Now, we consider te numerical approximation of te Willmore flow of a torus, described by Eq. (5.3). In particular, Clifford tori, wic are defined as aving te ratio between te outer R 0 and inner r 0 radii equal to R 0 /r 0 = 2, are stationary geometries for te Willmore flow, wit Willmore energy J W equal to 4π 2 ; tori wit different aspect ratios tend to converge to te Clifford torus. We numerically simulate te Willmore flow of a initial torus Ω 0 wit R 0 = 1 30

32 Isogeometric Analysis of Geometric PDEs 31 t = 0 t = t = t = t = Figure 16: Test Willmore flow of a torus wit R 0 = 1 and r 0 = 0.2. Solution at different time instances. and r 0 = 0.2 (i.e. for wic R 0 /r 0 = 5), represented as a NURBS surface wit basis functions of degrees p = 2 and 3, being globally C 1 - and C 2 -continuous, and two -refinement levels. Solutions at different time steps are reported in Figure 16. Time discretization uses te BDF sceme of order k = 2 and time step size t = We employed NURBS meses built wit 836 elements, yielding 2,048 DOFs, and 2,660 elements, yielding 8,192 DOFs, wit NURBS basis functions of degree p = 2 and globally C 1 -continuous, and meses wit 1,025 elements, yielding 2,880 DOFs, and 2,993 elements, yielding 9,792 DOFs, wit NURBS basis functions of degree p = 3 and globally C 2 -continuous. In Figure 17 te evolution of te Willmore energy is reported, togeter wit te Willmore energy of te Clifford torus (indicated as Exact final energy). If we compare te final Willmore energy of te approximated solution wit te Willmore energy of te Clifford torus we obtain te following errors wit te above mentioned meses, in order: (0.290%), (0.286%), (0.029%), and (0.003%). Terefore, te best compromise between accuracy and number of DOFs employed is obtained for NURBS basis functions of degree p = 3 and globally C 2 -continuous, wic guarantee a good accuracy even wit a small amount of DOFs. Finally, we report in Figures 18a and 18b te sparsity patterns of te matrices associated to te full discrete problem (4.26) wit NURBS of degrees p = 2 and 3, wit 2,048 and 2,880 DOFs, respectively. We report in Figure 18c te evolution of te condition number κ(a) at eac time step, for eac NURBS already considered in Figure 17. As usual, te condition numbers increase wit te degree of te NURBS basis functions and wit te refinement of te mes, but, for eac discretization, tese follow te same overall beavior in time. Wit respect to Test 5.2.1, te condition number tends to be smaller, since te mapping does not present any singularity. 31