FEM Software Automation, with a case study on the Stokes Equations

Transcription

1 FEM Automation, with a case study on the Stokes Equations FEM Andy R Terrel Advisors: L R Scott and R C Kirby Numerical from Department of Computer Science University of Chicago March 1, 2006 Masters Presentation

2 Outline 1 2 FEM FEM 3 Numerical from 4 Numerical from

4 State of Scientific Partial Differential Equation (PDE) needs to: FEM Solve large problems Solve interesting problems Use the best methods Numerical from

5 State of Scientific Partial Differential Equation (PDE) needs to: FEM Numerical from Solve large problems Solve interesting problems Use the best methods

6 State of Scientific Partial Differential Equation (PDE) needs to: FEM Solve large problems Solve interesting problems Use the best methods Numerical from

7 The Galerkin Method Given a PDE: d 2 u dx 2 = f in (0, 1), u(0) = 0, u (1) = 0 The problem can be characterized by a weak formulation: u V such that a(u, v) = (f, v) v V, where V = {v L 2 (0, 1) : a(v, v) < and v(0) = 0 Ritz-Galerkin Approximation: FEM Numerical from u S S such that a(u S, v) = (f, v) v S where S V is any finite dimensional subspace

10 The Local Finite Element FEM A reference element, K A space of shape functions, P A basis, N Numerical from To get the global space we use a mapping to change the coordinates into the global element. Then using our given method we need to solve some matrix equations: AU = F

11 Why Automate FEM? FEM Ensure Correctness: Complicated error prone mathematical process Complicated error prone programming process Reduce Programming Hours: Gives ability to quickly change models Gives ability to quickly change elements Gives ability to quickly change methods Optimize Computation: Allow a non-expert programmer to make efficent calculations Numerical from

14 Why are we NOT Automated? FEM Different mathematical and algorithmic abstractions The local finite element is understood and automated Mathematical framework for global-local interactions needs developing. Face Directions Links to other elements Hand coding is very attractive ( If you want it done right... ) Quite difficult to switch between elements, solvers, and methods. Numerical from

18 What Are the Parts Needed for FEM? FEM Mesh Generation Function Spaces Equation Description Discrete Equation Solver Parallel Computing Support Numerical from

19 What Are the Parts Needed for FEM? FEM Mesh Generation Function Spaces Equation Description Discrete Equation Solver Parallel Computing Support uniform meshes, general geometry, adaptive meshes, unstructured meshes Numerical from

20 What Are the Parts Needed for FEM? FEM Mesh Generation Function Spaces Equation Description Discrete Equation Solver Parallel Computing Support linears, menu of options, arbitrary order, tabulator Numerical from

21 What Are the Parts Needed for FEM? FEM Mesh Generation Function Spaces Equation Description Discrete Equation Solver Parallel Computing Support menu, language, derived forms, error estimators, constraints Numerical from

22 What Are the Parts Needed for FEM? FEM Mesh Generation Function Spaces Equation Description Discrete Equation Solver Parallel Computing Support menu, library, language Numerical from

23 What Are the Parts Needed for FEM? FEM Mesh Generation Function Spaces Equation Description Discrete Equation Solver Parallel Computing Support parallel linear solve, parallel assembly, load balancing Numerical from

24 Types of FEM Simulation Engine Holds the pieces together. Tabulator Tabulates the Finite Element Linear Solver Solves the linear equation and more if you let it. Numerical from

25 Some Major Projects Simulation Engines Sundance FFC/Dolfin Deal.II ComSol Analysa FreeFEM GetDP Tabulators FIAT SyFi Linear Solvers UMFPack PETSc Trilinos FEM Numerical from

26 Math v. : The Problem Domain FEM Mathematics Distinguish what problem lies where Adaptively refine on important parts of the domain Hook up with domains of other problems effortlessly Use some mesh description. Allow coarsening (usually only uniform) Use set theoretic operators to filter different parts (Sundance) Ultimately gives some iterator for assembly process Numerical from

27 Math v. : The Problem Domain FEM Mathematics Distinguish what problem lies where Adaptively refine on important parts of the domain Hook up with domains of other problems effortlessly Use some mesh description. Allow coarsening (usually only uniform) Use set theoretic operators to filter different parts (Sundance) Ultimately gives some iterator for assembly process Numerical from

28 What meshes can we handle? None of these software packages are giving us great tools for multigrid adaptivity. FEM Numerical from

29 Math v : Problem Statement FEM Mathematics Many ways to describe a problem. Often problems are split or reformulated. Conceivably we should be able to use any well-formed formula (PDE, ODE,... ) Approaches GUI Strong Form (ComSol) The Variational Form (FFC and Sundance) The Brute Force Method (Deal.II) Numerical from

30 Math v : Problem Statement FEM Mathematics Many ways to describe a problem. Often problems are split or reformulated. Conceivably we should be able to use any well-formed formula (PDE, ODE,... ) Approaches GUI Strong Form (ComSol) The Variational Form (FFC and Sundance) The Brute Force Method (Deal.II) Numerical from

31 How about optimization problems? Use Automatic Differentiation tools on code - expensive on user side Create a symbolics engine that can give derivatives - expensive on developer side Example Problem in Microfluidic Devices To optimize flow, change channel geometry Most effective methods, use level set methods FEM Numerical from

32 Math v : Role of Symbolics The use of the variational form can be motivation for supporting a larger symbolics engine that can then be used for differentiation. FEM The Sundance Symbolics Engine Environment with large number of calculations Not symbols but numbers Graph relations to implement chain rule The SyFi Symbolics Engine Preprocessing environment, less calculations Fully symbolic. Numerical from

35 Math v : The Assembly Process FEM Mathematics Rote application of algebra and calculus The most computationally demanding process. Do I have to touch the process at all? (Declarative or procedural?) Can I get my matrix to play with before sending it to the solver? Do I leave the option of not assembling at all? Numerical from

36 Math v : The Assembly Process FEM Mathematics Rote application of algebra and calculus The most computationally demanding process. Do I have to touch the process at all? (Declarative or procedural?) Can I get my matrix to play with before sending it to the solver? Do I leave the option of not assembling at all? Numerical from

37 Math v : Solvers FEM Mathematics Just solve Au = f, what s so hard? De facto standard is to use some library. Either Trilinos, PETSc, ublas, UMFPack,... Is there more we can do here? Adaptively choose our precision. Pre-solve important blocks. Numerical from

38 Math v : Solvers FEM Mathematics Just solve Au = f, what s so hard? De facto standard is to use some library. Either Trilinos, PETSc, ublas, UMFPack,... Is there more we can do here? Adaptively choose our precision. Pre-solve important blocks. Numerical from

39 What hasn t been done? FEM In general, some important pieces are not being implemented: Usually only Lagrange Parallel assembly Adaptive/unstructured grids Error estimators or optimization loops Boundary Condition calculus or embedded geometries Numerical from

41 Introduction to Stokes Flow The Stokes equations are a model for steady incompressible flow: u + p = f u = 0 FEM Numerical from Important Features: Coupling of pressure and velocity Well studied problem Numerous methods for solving

42 Intro to Mixed Method Let V = H 1 (Ω) n and Π = {q L 2 (Ω) : Ω qdx = 0}. Given F V, find functions u V and p Π such that Where, Important Feature a(u, v) + b(v, p) = F (v) a(u, v) := b(v, q) := b(u, q) = 0 Two discrete spaces, V and Π Ω Ω v V q Π u vdx, ( v)qdx FEM Numerical from

43 Taylor - Hood Elements FEM (a) P 3 for V (b) P 2 for Π Numerical from Important Features: Available using any P k elements, extendable to 3d, Built from standard Lagrange elements, Easily extendable to arbitrary order, and Widely used

44 Crouzeix - Raviart Elements FEM Numerical from (c) Crouzeix-Raviart for V (d) P 0 for Π Important Features: Non Conforming Low Order

45 C 0 P i C 1 P i 1 Elements FEM Use a Continuous Lagrange element P i for V and a Discontinuous Lagrange element P i 1 for Π Important Features: May not satisfy inf sup condition Numerical from

46 Iterated Penalty Let r R and ρ > 0 define u n and p = w n by a(u n, v) + r( u n, v) = F(v) ( v, w n ) w n+1 = w n + ρu n Important Features One discrete spaces, V Use higher order finite elements, P 4 and above Use u n V < ɛ as stopping criteria Iteration count highly effected by choice of ρ and r, for our experiments choose ρ = r = 1.0e 3. FEM Numerical from

47 Order vs Degrees of Freedom FEM Numerical from

48 Problem statement Using a n n uniform mesh for a domain [0,1] [0,1] solve these problems. Case 2: [ ] sin(3πx) cos(3πy) u = cos(3πx) sin(3πy) p = sin(3πx) sin(3πy) FEM Numerical from

49 and Solvers Sundance FEniCS Taylor-Hood X X Crouzeix-Raviart - X C 0 P i C 1 P i 1 - X Iterated Penalty X X For each of these methods, we use UMFPACK LU Direct solver. Other iterative solvers from the Trilinos or PETSc Toolkits would also work and make the code parallel, but not the focus of this work. FEM Numerical from

51 4th Order Numbers FEM Numerical from

52 FEniCS Velocity Errors FEM Numerical from

53 FEniCS Pressure Errors FEM Numerical from

54 FEniCS Divergence Errors FEM Numerical from

55 FEniCS Velocity Errors FEM Numerical from

56 Comparisons between Packages FEM Sundance and Dolfin treat assembly very similarly FIAT a common interface for defining elements Coding time almost identical Both still very active development Numerical from

57 More Notes Notes about FEniCS A much smaller code to comprehend - hence easier to make changes if it does not currently have a feature Interface less like a scripting language in so much as development requires multiple tools to run Problems handling fancy things Notes about Sundance Seemless scripting style code Ability to handle fancier things Closer to a production quality code FEM Numerical from

58 Coding Challenges in this project Have to read the code Often documentation is either incomplet, or inakurate To see how things were really done, had to read code Possible Methods FEniCS did not have mixed methods Sundance did not have higher order methods Integration Bugs FEniCS operators did not check for underflow FEniCS Div operator is not as accurate Sundance bug with volume FEM Numerical from Iterated Penalty has typo in Brenner Scott and many papers.

62 Conclusions enables flexiblity in simulation software Mathematics Abstractions Meaningful test simulations (not just Poisson) FEM Numerical from Outlook Explore mathematical abstractions for global-local interactions Compare Grade 2 and Oldroyd-B fluid model

63 Do It Yourself Where to get the code: Sundance - FEniCS Project (FIAT, FFC, DOLFIN) - Masters Thesis - aterrel/masters FEM Numerical from Any Questions aterrel@uchicago.edu

64 Taylor-Hood Error FEM Numerical from

70 Crouzeix-Raviart P0 Error FEM Numerical from

73 C 0 P i C 1 P i 1 Error FEM Numerical from

76 Iterated Penalty Error FEM Numerical from

82 Taylor-Hood Error FEniCS TaylorHoodCase0 FEM Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Run Comp e e e e e e Application with e e e Stokes Equations e e e e e e Iteration - 2.9methods e e e Testing - 2.9Methods e e e e e e e e e Numerical from e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

83 Taylor-Hood Error FEniCS TaylorHoodCase1 FEM Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Run Comp e e e e e e Application with e e e Stokes Equations e e e e e e Iteration - 2.9methods e e e Testing - 2.9Methods e e e e e e e e e Numerical from e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

84 Taylor-Hood Error FEniCS TaylorHoodCase2 FEM Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Run Comp e e e e e e Application with e e e Stokes - 3.6Equations e e e e e e Iteration - 2.9methods e e e Testing Methods e e e e e e e e e Numerical from e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

85 Taylor-Hood Error FEM Sundance TaylorHoodCase0 Ord Mesh Vel Err Vp Pre Err Pp Div Dp Run e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Numerical from

88 Crouzeix-Raviart P0 Error FEM FEniCS Crouzeix-RaviartCase0 Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Run Comp e e e e e e e e e Numerical from e e e

89 Crouzeix-Raviart P0 Error FEM FEniCS Crouzeix-RaviartCase1 Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Run Comp e e e e e e e e e Numerical from e e e

90 Crouzeix-Raviart P0 Error FEM FEniCS Crouzeix-RaviartCase2 Ord 1-0 Mesh 4 DOFs 160 Vel Err 2.3e-01 Vp 0.00 Pre Err 2.3e+00 Pp 0.00 Div 1.3e-07 Dp 0.00 Run 0.01 Comp 2.6Testing - 1.3Methods e e e e e e Numerical from e e e e e e User Experience e e e

91 C 0 P i C 1 P i 1 Error FEniCS C0PiC-1Pi-1Case0 FEM Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Run Mathematics Comp versus e e e e e e Application with e e e e e e e e e e e e e e e e e e e e e Numerical from e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+00 -inf

94 Iterated Penalty Error FEniCS IteratedPenaltyCase0 Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Iters Run Comp FEM e e e Mathematics e e e versus e e e e e e e e e e e e e e e e e e Application4.5 with Mixed Method4.8 Formulation e e e e e e e e e Numerical 1.64 from e e e User Experience e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e+00 -inf

95 Iterated Penalty Error FEniCS IteratedPenaltyCase1 Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Iters Run Comp FEM e e e Mathematics e e e versus e e e e e e e e e e e e e e e e e e Application10.0 with Mixed Method5.0 Formulation e e e e e e e e e Numerical 2.77 from e e e User Experience e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

96 Iterated Penalty Error FEniCS IteratedPenaltyCase2 Ord Mesh DOFs Vel Err Vp Pre Err Pp Div Dp Iters Run Comp FEM e e e Mathematics e e e versus e e e e e e e e e e e e e e e e e e Application18.4 with Mixed Method4.8 Formulation e e e e e e e e e Numerical 2.75 from e e e User Experience e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

97 Iterated Penalty Error FEM Sundance IteratedPenaltyCase0 Ord Mesh Vel Err Vp Pre Err Pp Div Dp Iters Run e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Numerical from

100 What about code complexity? One bad estimate is lines of code: Dolfin + FFC 50K lines Sundance 100K lines DEAL 400K lines Some Organization Charts FEM Numerical from

101 More Detailed Dependencies An example of Dependencies for Sundance (not especially different from others) FEM Numerical from