A Posteriori Error Estimation for Predictive Models James R. Stewart jrstewa@sandia.gov Engineering Sciences Center Advanced Computational Mechanics Architectures Dept (9143) Sandia National Laboratories Albuquerque, NM 1
Error Estimation Example Linear elastic bar with gravityloading 8-node hexahedral elements This is among the simplest problems to solve using FEA Q. Can we estimate the error accurately? A.Yes buthe acuracy depends 2
Error Estimation Example What does it depend on? The type of error estimator The mesh resolution (possibly) The engineering quantity of interest Average vertical displacement» On a side surface» On the bottom surface Average normal stress on the the bottom surface Energy norm 3
Rigorous Definition of Error Exact error is the exact solution to the math problem is the numerical solution (finite-element, finite-volume, finite-difference, etc.) 4
Non-Rigorous Definitions of Error Difference between fine and coarse meshes Difference between higher-order and lower-order meshes 5
Definitions of Error (cont.) The exact error can be written as where Remark Many (most) error estimators work directly on or with the assumption that or is small 6
More Definitions Quantity of interest Functional of solution Local engineering output» Average vertical displacement on a surface» Displacement at a point Error in quantity of interest 7
Error Estimates vs. Error Bounds Error estimates Attempt to quantify the error (notice the use of ) Quality of estimate often given by the effectivity index Error bounds Upper and lower bounds Can easily convert bounds into estimates 8
Remarks For Uncertainty Quantification, error bounds are more useful than error estimates Error estimates provide stopping criteria for adaptive mesh refinement Converting an estimate into bounds is, in general, not trivial 9
Error Estimation Example (cont.) Quantity of interest Average vertical displacement Case 1: is a side surface Case 2: is the bottom surface 10
Lower and Upper Bounds: Case 1 (Side Surface) Valid upper bound on Valid lower bound on Bounds are with respect to Quality of bounds depends on H Error estimate (average of bounds) is very good, but not exact 11
Lower and Upper Bounds: Case 2 (Bottom Surface) H 2 1 0.5 2.06e 8 2.06e 8 2.06e 8 1.0 2.06e 8 2.06e 8 2.06e 8 1. 0 2.06e 8 2.06e 8 2.06e 8 1. 0 Solution ( ) is exact for all H (note that is nodally exact) Error estimate (average of bounds) is exact for all H 12
Classes of Error Estimators Recovery methods Residual methods Extrapolation methods 13
Recovery Methods Super-convergent Patch Recovery (ZZ-SPR or ZZ) Zienkiewicz and Zhu Post process gradients of FE solution on patches of neighboring elements Gives global energy-norm estimates (under stringent assumptions) Does not lend itself to rigorous mathematical analysis Polynomial Preserving Recovery (PPR) Zhang, et al. Post process nodal values of FE solution on patches of neighboring elements Gives global energy-norm estimates Similar super-convergent properties to ZZ-SPR 14
Residual Methods Traditional categorizations Explicit residual methods» In reality, a misnomer» Provides error estimates Implicit residual methods» Can provide error estimates and bounds Original ideas of Babuska and Rheinbolt [1978-81] with extensions by Babuska and Miller [1984-87] 15
Explicit Residual Methods 16
Explicit Residual Methods Error Representation Formula Weak residual estimate: Solve dual problem for Substitute into (3) Remarks: Dual problem must be solved on either or For time-dependent problems, dual problem is backwards in time! 17
Explicit Residual Methods Strong residual estimate: where and are stability factors Remarks: (4) implies that the error can be large even if the residuals are small The stability factors are properties of the pde ISSUE: How accurately must the dual problem be solved? 18
Explicit Residual Methods Nonlinear Operators 19
Explicit Residual Methods Sample Applications Chalmers (Sweden) Group (Johnson, Eriksson, Estep, Hansbo, et al.) Advection-diffusion; General nonlinear parabolic operators Mostly global norms (done in early 90 s) Estep, Larson, and Williams Nonlinear reaction-diffusion systems Coupled parabolic pde s and (singular) ode s in time = average value in domain Heidelberg Group (Becker and Rannacher; Bangerth) DWR (Dual Weighted Residual) Method Variety of nonlinear fluid and solid mechanics problems Improved error indicators for mesh adaptivity (Time-domain) acoustic wave equation; elastic wave equation Barth and Larson Extension to finite volume methods Linear advection and nonlinear Burgers equation Nonlinear MHD 20
Implicit Residual Methods Can give error estimates and bounds Simplest form (for error estimates): Recall Substitute and rearrange; solve for Remarks: Recall ; Therefore, must be solved in a higher-order subspace For efficiency, is solved on the broken space Element residual method Subdomain residual method Estimates obtained in global norms 21
Implicit Residual Methods The Broken Space Broken space can also consist of (overlapping or non-overlapping) patches Complication: Treatment of Neumann BCs for local problems 22
Implicit Residual Methods Quantity-of-Interest Bounds Various derivations have been published (beyond the scope of this talk ), e.g., Babuska and Strouboulis Peraire and Patera Oden and Prudhomme Main result (broken space dependence made explicit) Subtract to get error bounds Requirement: (i.e., set of functions defined on also contains functions defined on ) 23
Implicit Residual Methods General Bound Procedure Following presentation of Peraire and Patera (the Truth Error Bounds) Step 0: Solve FE problem Find such that Step 1: Solve global (linear) dual problem on Find such that 24
Implicit Residual Methods General Bound Procedure Step 2: For each patch in, solve Local primal error problem: Find such that Local dual error problem: Find such that where is the jump bilinear form defined by and is the hybrid flux 25
Implicit Residual Methods General Bound Procedure Step 3: Compute the bounds and the error estimate 26
Truth Error Bounds Software Design and Implementation Quasi-statics code Adagio SIERRA design hierarchy Domain Procedure (time step control) Region A (single step of physics A) Region B (single step of physics B) Mechanics Mesh and Fields Mechanics Mesh and Fields Transfer 27
Adagio Truth Error Bounds Software Design and Implementation Observations We require solution of (1+2N) auxiliary PDE s (where N is the number of patches in )» Each has the same lhs (for linear, self-adjoint operator)» Each has, in general, a different rhs The SIERRA Framework helps manage some of this complexity 28
Adagio Truth Error Bounds Software Design and Implementation Adagio Region Mechanics Mesh and Fields Primal Patch Region Residual Fields Mesh Adagio Procedure 1. Region copy-subset 2. Solve dual problem 3. Region copy-subset For each patch in 4. Create patch mesh 5. Transfer field values 6. Solve local problems 7. Transfer field values 8. Local update inner products and norms 9. Compute bounds Global Dual Region Mechanics Mesh and Fields Dual Patch Region Residual Mesh Fields 29
Truth Error Bounds in Adagio Case 1 (Side Surface) Valid upper bound on Valid lower bound on patches are 2x2 refinements of each (coarse) H-element 30
Implicit Residual Methods Sample Applications TICAM Group (Babuska and Strouboulis; Oden and Variety of linear and nonlinear elliptic problems Elasticity problems» Local and average displacements and stress components Heat equation with nonlinear and orthotropic materials» Local temperature Burgers equation» Local velocity Incompressible Navier-Stokes» Kinetic energy of flow Helmholtz» Local amplitude Eigenvalue problems» Eigenvalues 31
Richardson Extrapolation Extrapolation Methods Applies in the asymptotic convergence region Assume is computed with two mesh sizes, 1 and, where h 2 h 1 Using a known convergence rate l l Q Q ( u ( u h 1 h 2 ) ) l Q ( u) lq( u) c ch α 1 ch α 2 α H.O.T. H.O.T. h 2 Can now eliminate to get a very accurate approximation of h where r l Q α Q ( u) l ( u1 ) r l ( u2 ) α 1 r h 1 h2 Q h h H.O.T. 32
Extrapolation Methods Richardson Extrapolation, cont. α be obtained on h If is not known (almost always), then a third solution must where e e r h 1 h 2 h l l 3 Q Q ( u ( u h 2 h 2 h 3 ) ) h 3 h 2 α More complicated if 2 l l Q Q h 1 h h log( e2 / e1 ) logr ( u ( u r h 1 h 2 ) ) constant constant (Difference in successive meshes) 33
Extrapolation Methods Converting Estimates into Bounds Grid Convergence Index (GCI) (Roache) Essentially a factor of safety F s GCI [fine grid] F s l Q e ( u RE h fine ) For two-grid extrapolation, For three-grid extrapolation, F s F s 3 1.25 (Approximate) quantity-of-interest bounds: l Q ( u h fine ) GCI l Q ( u h fine ) 34
Extrapolation Methods Ideal vs. Reality Ideal (monotonic convergence) Num Elements 35
Example Problem Elastic material (nonlinear) g Rigid Body Explicit transient dynamics (DYNA) is the displacement of interior (rigid) body 36
Extrapolation Methods Ideal vs. Reality Reality Num Elements How would you handle this? 37
Extrapolation Methods Ideal vs. Reality Reality (how it was actually handled) GCI applied to middle three points (these were monotonic) Timestep errors were ignored Num Elements 38
Areas of Active Research Extension of bounds to include modeling and uncertainty errors (Oden and Prudhomme, others ) Harder problems Nonlinear parabolic problems Hyperbolic problems Multiphysics problems Extreme anisotropic materials Bounds of exact error (Babuska and Strouboulis, Peraire, et al.) Certificates of precision Errors due to operator splitting (Estep) New extrapolation techniques (Roy, Garbey) 39
Areas of Active Research (cont.) Nearby problems (Hopkins, Roy) Discretization procedures Finite volume techniques Semi-discrete time integration (method of lines) Application to shells, other element types Stabilized methods Adaptive error control Error indicators Goal-oriented adaptive strategies 40
Battery operation Example of Hard Problem Sandia Thermal Battery and are heated above melting temperature, then cooled Need to stay melted 1 hour Problem features Highly transient Nonlinear materials (temp-dep) Nonsmooth data (read from table) Highly orthotropic materials Nonlinear BC s (radiation, convection) From error estimation view, this is a hard problem! 41
Error Estimation Risk Assessment Problem Class Sandia Code(s) Risk Research Issues Adagio Nonlinearities Elliptic Salinas (Freq dom) Calore (steady) Low Nonsmoothness Anisotropic matls Calore Time errors Parabolic Aria Fuego Medium Finite volume Turbulence Premo (subsonic) Hyperbolic Presto Salinas (Time dom) Premo (supersonic) High Explicit, lumped timestepping History-dep vars Multiphysics Calagio Fuego/Calore/Syrinx Med-High Loose coupling transfer operators 42
Current Limitations of Error Estimation Impact Can we identify anything that is limiting the impact and potential of a posteriori error estimation? (Answer: yes) Priorities (of the code customers and commercial vendors) Computational cost of the algorithms Complexity of implementation Applicability of the algorithms 43
Customer Priorities Commercial customers (code end-users) and commercial software vendors 1. Robustness 2. CPU cost 3. Memory cost 4. Accuracy Accuracy (and knowledge of accuracy) is important, but not most important In the current marketplace, customers are not willing to pay a high price for error estimation capabilities 44
Customer Priorities (cont.) ASCI V & V program has elevated the importance of error estimation» Solution verification» Model validation ASCI is driving the need to develop techniques for complex engineering problems Potential to impact the commercial sector, and change the way all engineering design and analysis is done Success is not guaranteed! Thus the need for this workshop 45
Summary Primary classes of error estimators Recovery methods Residual methods (quantity-of-interest estimates and bounds) Extrapolation methods (Quantity-of-Interest) A Posteriori error estimation is relatively mature mainly for elliptic problems Can provide both error estimates and error bounds (good for UQ) A Posteriori error estimation proven very effective for adaptive error control Remark: The optimization community has experience in dealing with time-dependent dual problems 46
Extrapolation estimators Computational Cost of the Algorithms Many (at least three) solutions required All solutions must be sufficiently resolved, which increases the cost Residual-based estimators Global dual problems required (although linear)» For estimates, must be solved on finer (h or p) mesh» For bounds, additional local (element or patch) problems must be solved for error in dual solution Time dependent problems» Dual problems are backwards in time! Remark: The optimization community has experience in dealing with time-dependent dual problems 47
Complexity of Implementation Extrapolation estimators Richardson extrapolation: Usually simple to implement» Issues mainly with ability to obtain mesh refining or coarsening tools Usually can be thought of as a post-processing tool Residual-based estimators Generally intrusive to the code Must solve additional problems (pde s)» Compute residuals, additional right-hand-sides, etc» Must have a way of handling hybrid fluxes Equilibration (very complex to implement) Bank-Weiser projection (simpler but less accurate) Use subdomains or overlapping patches (potentially costlier) Additional difficulties for finite volume discretizations 48
Extrapolation estimators Limitations on Applicability of the Algorithms Issue: applicability outside asymptotic convergence regime Residual-based estimators More algorithm research needed» Solving the backwards-in-time dual problem» Semi-discrete (method-of-lines) discretizations (instead of DG)» Hyperbolic problems» Multiphysics problems (e.g., operator splitting )» Etc. 49