Automating the Finite Element Method

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1 Automating the Finite Element Method Anders Logg Toyota Technological Institute at Chicago Johan Hoffman, Johan Jansson, Claes Johnson, Robert C. Kirby, Matthew Knepley, Ridgway Scott Simula Research Laboratory, Oslo August p. 1

2 Automating the finite element method The FEniCS project automates the following aspects of the finite element method: Automatic generation of finite elements (FIAT) e = (K, P, N ) Automatic evaluation of variational forms (FFC) a(v, u) = Ω v u dx Automatic assembly of linear systems (DOLFIN) for all elements e T Ω A += A e Simula Research Laboratory, Oslo August p. 2

3 $ ( ' & % $ # "! +, * Simula Research Laboratory, Oslo August p. 3 Benchmark results: impressive speedups " )

4 Outline FEniCS and the Automation of CMM FIAT, the finite element tabulator FFC, the form compiler DOLFIN, a PSE for differential equations Benchmark results Future directions for FEniCS Simula Research Laboratory, Oslo August p. 4

5 FEniCS and the Automation of CMM Simula Research Laboratory, Oslo August p. 5

The FEniCS project A free software project for the Automation

The Toyota Technological Institute at Chicago The University

Sweden) NADA, KTH (Stockholm, Sweden) Argonne National

6 The FEniCS project A free software project for the Automation of Computational Mathematical Modeling (ACMM), developed at The Toyota Technological Institute at Chicago The University of Chicago Chalmers University of Technology (Göteborg, Sweden) NADA, KTH (Stockholm, Sweden) Argonne National Laboratory (Chicago) Simula Research Laboratory, Oslo August p. 6

7 Main goal: the Automation of CMM Au = f frag replacements U u TOL > 0 Input: Model Au = f and tolerance TOL > 0 Output: Solution U u satisfying U u TOL Produce a solution U satisfying a given accuracy requirement, using a minimal amount of work Simula Research Laboratory, Oslo August p. 7

8 A key step: the automation of discretization frag replacements Au = f F (x) = 0 (V, ˆV ) Input: Model Au = f and discrete representation (V, ˆV ) Output: Discrete system F (x) = 0 Produce a discrete system F (x) = 0 corresponding to the model Au = f for the given discrete representation (V, ˆV ) Simula Research Laboratory, Oslo August p. 8

9 Basic algorithm: the (Galerkin) finite element method frag replacements a(v, u) = L(v) F (x) = 0 (V, ˆV ) Input: Variational problem a(v, u) = L(v) for all v and discrete representation (V, ˆV ) Output: Discrete system F (x) = 0 Simula Research Laboratory, Oslo August p. 9

10 FEniCS development Basic principles: Focus on automation Focus on efficiency Focus on consistent user-interfaces Focus on applications Implementation: Organized as a collection reusable components A rapid and open development process Modern programming techniques Novel algorithms Simula Research Laboratory, Oslo August p. 10

11 FEniCS components FIAT, the finite element tabulator Robert C. Kirby FFC, the form compiler Anders Logg DOLFIN, a PSE for differential equations Hoffman, Jansson, Logg, et al. FErari, optimized form evaluation Kirby, Knepley, Logg, Scott, Terrel Ko, simulation of mechanical systems Johan Jansson Puffin, educational version for Octave/MATLAB Hoffman, Logg (PETSc) Simula Research Laboratory, Oslo August p. 11

12 FIAT, the finite element tabulator Simula Research Laboratory, Oslo August p. 12

13 FIAT: FInite element Automatic Tabulator Automates the generation of finite element basis functions Simplifies the specification of new elements Continuous and discontinous Lagrange elements of arbitrary order Crouzeix Raviart (CR) elements Nedelec elements Raviart Thomas (RT) elements Brezzi Douglas Marini (BDM) elements Brezzi Douglas Marini Fortin (BDM) elements In preparation: Taylor-Hood, Arnold-Winther,... Simula Research Laboratory, Oslo August p. 13

Implementation Use orthonormal basis on triangles and tets (Dubiner) Express basis functions in terms of the orthonormal basis Translate conditions on function

14 Implementation Use orthonormal basis on triangles and tets (Dubiner) Express basis functions in terms of the orthonormal basis Translate conditions on function space into linear algebraic relations Implemented in Python Efficient linear algebra Efficient tabulation mode Simula Research Laboratory, Oslo August p. 14

15 Basic usage >>> from FIAT.Lagrange import * >>> from FIAT.shapes import * >>> element = Lagrange(TETRAHEDRON, 3) >>> basis = element.function_space() >>> v = basis[0] >>> v((-1.0, -1.0, -1.0)) 1.0 >>> from FIAT.quadrature import * >>> quadrature = make_quadrature(tetrahedron, 2) >>> points = quadrature.get_points() >>> table = basis.tabulate_jet(2, points) Simula Research Laboratory, Oslo August p. 15

16 FFC, the form compiler Simula Research Laboratory, Oslo August p. 16

17 FFC: the FEniCS Form Compiler Automates a key step in the implementation of finite element methods for partial differential equations Input: a variational form and a finite element Output: optimal C/C++ placements a(v, u) = Ω u(x) v(x) dx FFC Poisson.h >> ffc [-l language] poisson.form Simula Research Laboratory, Oslo August p. 17

18 Design goals Any form Any element Maximum efficiency Possible to combine generality with efficiency by using a compiler approach: Generality Efficiency Compiler Simula Research Laboratory, Oslo August p. 18

19 Features Command-line or Python interface Support for a wide range of elements (through FIAT): Continuous scalar or vector Lagrange elements of arbitrary order (q 1) on triangles and tetrahedra Discontinuous scalar or vector Lagrange elements of arbitrary order (q 0) on triangles and tetrahedra Crouzeix Raviart on triangles and tetrahedra Others in preparation Efficient, close to optimal, evaluation of forms Support for user-defined formats Primary target: DOLFIN/PETSc Simula Research Laboratory, Oslo August p. 19

20 Basic usage 1. Implement the form using your favorite text editor (emacs): 2. Compile the form using FFC: >> ffc poisson.form This will generate C++ code (Poisson.h) for DOLFIN Simula Research Laboratory, Oslo August p. 20

21 Components of FFC Simula Research Laboratory, Oslo August p. 21

22 Basic example: Poisson s equation Strong form: Find u C 2 (Ω) with u = 0 on Ω such that u = f in Ω Weak form: Find u H0 1 (Ω) such that v(x) u(x) dx = v(x)f(x) dx Ω Ω for all v H 1 0 (Ω) Standard notation: Find u V such that a(v, u) = L(v) for all v ˆV with a : ˆV V R a bilinear form and L : ˆV R a linear form (functional) Simula Research Laboratory, Oslo August p. 22

23 Obtaining the discrete system Let V and ˆV be discrete function spaces. Then a(v, U) = L(v) for all v ˆV is a discrete linear system for the approximate solution U u. With V = span{φ i } M i=1 and ˆV = span{ ˆφ i } M i=1, we obtain the linear system Ax = b for the degrees of freedom x = (x i ) of U = M i=1 x iφ i, where A ij = a( ˆφ i, φ j ) b i = L( ˆφ i ) Simula Research Laboratory, Oslo August p. 23

24 Computing the linear system: assembly Noting that a(v, u) = e T a e(v, u), the matrix A can be assembled by A = 0 for all elements e T A += A e The element matrix A e is defined by A e ij = a e ( ˆφ i, φ j ) for all local basis functions ˆφ i and φ j on e Simula Research Laboratory, Oslo August p. 24

25 Multilinear forms Consider a multilinear form a : V 1 V 2 V r R with V 1, V 2,..., V r function spaces on the domain Ω Typically, r = 1 (linear form) or r = 2 (bilinear form) Assume V 1 = V 2 = = V r = V for ease of notation Want to compute the rank r element tensor A e defined by A e i = a e (φ i1, φ i2,..., φ ir ) with {φ i } n i=1 the local basis on e and multiindex i = (i 1, i 2,..., i r ) Simula Research Laboratory, Oslo August p. 25

26 Tensor representation of forms In general, the element tensor A e can be represented as the product of a reference tensor A 0 and a geometric tensor G e : A e i = A 0 iαg α e A 0 : a tensor of rank i + α = r + α G e : a tensor of rank α Basic idea: Precompute A 0 at compile-time Generate optimal code for run-time evaluation of G e and the product A 0 iα Gα e Simula Research Laboratory, Oslo August p. 26

27 The (affine) map F e : E e x 3 replacements X 3 = (0, 1) e x = F e (X) x 2 F e x 1 E X F e (X) = x 1 Φ 1 (X) + x 2 Φ 2 (X) + x 3 Φ 3 (X) X 1 = (0, 0) X 2 = (1, 0) Simula Research Laboratory, Oslo August p. 27

28 Example 1: the mass matrix Form: a(v, u) = Ω v(x)u(x) dx Evaluation: A e i = e φ i1 φ i2 dx = det F e E Φ i1 (X)Φ i2 (X) dx = A 0 i G e with A 0 i = E Φ i 1 (X)Φ i2 (X) dx and G e = det F e Simula Research Laboratory, Oslo August p. 28

29 Example 2: Poisson Form: a(v, u) = Evaluation: A e i = φ i1 (x) φ i2 (x) dx e = det F e X α1 X α2 x β x β Ω v(x) u(x) dx E Φ i1 X α1 Φ i2 X α2 dx = A 0 iαg α e with A 0 iα = E Φ i1 X α1 Φ i2 X α2 dx and G α e = det F e X α1 x β X α2 x β Simula Research Laboratory, Oslo August p. 29

30 An algebra for forms Basic elements: basis functions and their derivatives Define addition, subtraction and multiplication with scalars to generate a vector space Define multiplication between elements of A to generate an algebra: A = {v : v = c φ ( ) / x ( ) } A is the algebra of linear combinations of products of basis functions and their derivatives A is closed under addition, subtraction, multiplication and differentiation Consider only multilinear forms which can be expressed as integrals over the domain Ω of elements of A Simula Research Laboratory, Oslo August p. 30

31 Evaluation of forms For any v = c φ ( ) / x ( ), we have A e i = a e (φ i1, φ i2,..., φ in ) = v i dx e = ( c ) φ ( ) / x ( ) dx = c α e ( E = A 0 iαg α e, ) Φ ( ) / X ( ) dx i iα where A 0 iα = ( E Φ ( ) / X ( ) dx ) iα and Gα e = c α Simula Research Laboratory, Oslo August p. 31

32 Optimization Each entry of the element tensor is given by a scalar product Find dependencies between entries: Colinearity Coplanarity Edit distance Optimization handled by FErari (in preparation) Operation count can sometimes be reduced to less than one multiply-add pair per entry Alternative approach: BLAS level 2 or 3 Simula Research Laboratory, Oslo August p. 32

33 Reference tensor (P 2 Poisson in 2D) Simula Research Laboratory, Oslo August p. 33

34 Minimum spanning tree (P 2 Poisson in 2D) Simula Research Laboratory, Oslo August p. 34

35 DOLFIN, a PSE for differential equations Simula Research Laboratory, Oslo August p. 35

36 DOLFIN Provides a simple, consistent and intuitive API to FEniCS Provides solvers for a collection of standard PDEs Implemented as a C++ library Simula Research Laboratory, Oslo August p. 36

37 Basic usage #include <dolfin.h> #include "Poisson.h" using namespace dolfin; int main() {... Mesh mesh("mesh.xml.gz"); Poisson::BilinearForm a; Poisson::LinearForm L(f); Matrix A; Vector x, b; FEM::assemble(a, L, A, b, mesh, bc); GMRES solver; solver.solve(a, x, b); } return 0; Simula Research Laboratory, Oslo August p. 37

38 Basic concepts (classes) Linear algebra Vector, Matrix, VirtualMatrix LinearSolver, Preconditioner Geometry Mesh, Boundary, MeshHierarchy Vertex, Cell, Edge, Face Finite elements FiniteElement, Function BilinearForm, LinearForm Ordinary differential equations ODE, ComplexODE, ParticleSystem, Homotopy Other: File, Progress, Event, Parameter Simula Research Laboratory, Oslo August p. 38

39 Components of DOLFIN Simula Research Laboratory, Oslo August p. 39

40 Components of DOLFIN Simula Research Laboratory, Oslo August p. 40

41 Replacing the backend of DOLFIN Simula Research Laboratory, Oslo August p. 41

42 Example: updated elasticity/plasticity Simulations/animations by Johan Jansson, Computational Technology, Chalmers Simula Research Laboratory, Oslo August p. 42

43 Benchmark results Simula Research Laboratory, Oslo August p. 43

44 H Y Q L U T S R Q P O N M L K J I B < X9 W X [Z Z 4<A < >?01 Simula Research Laboratory, Oslo August p. 44 = 81 ;<3 : Impressive speedups G BC O V F BC E BC D BC 2 3 / 01 1

45 u ] ] f ~ y ~ } { z y x w v o a ^ ^ e i f ˆ ain a m i kl]^ Simula Research Laboratory, Oslo August p. 45 j e^ hi` g a edf c ] Impressive speedups t op ƒ s op r op q op b ` _ ` \ ]^ ^

46 Test case 1: the mass matrix Mathematical notation: FFC implementation: a(v, u) = Ω vu dx v = BasisFunction(element) u = BasisFunction(element) a = v*u*dx Simula Research Laboratory, Oslo August p. 46

47 Ÿ ž š Ž Œ Š Ÿ ž š Ž Œ Š ª Results Simula Research Laboratory, Oslo August p. 47 œ «œ

48 Test case 2: Poisson Mathematical notation: a(v, u) = Ω v u dx = Ω d i=1 v u dx x i x i FFC implementation: v = BasisFunction(element) u = BasisFunction(element) a = v.dx(i)*u.dx(i)*dx Simula Research Laboratory, Oslo August p. 48

49 Ã Â Á À ½ ¼» ³ ² ± Ã Â Á À ½ ¼» ² ± Í Simula Research Laboratory, Oslo August p. 49 Results º É Å ÊÈ Ç ÈÆ Ä ÅÆ ÌË Ë ¹ ¾ µ É Å ÊÈ Ç ÈÆ Ä ÅÆ ÌË Ë Î º ¹ ¾

50 Test case 3: Navier Stokes Mathematical notation: a(v, u) = Ω v(w u) dx = Ω d i=1 d j=1 v i w j u i x j dx FFC implementation: v = BasisFunction(element) u = BasisFunction(element) w = Function(element) a = v[i]*w[j]*u[i].dx(j)*dx Simula Research Laboratory, Oslo August p. 50

51 à ß Þ Ý Ú Ù Ø Ó Ò Ï à ß Þ Ý Ú Ù Ø Ó Ò Ï ê Simula Research Laboratory, Oslo August p. 51 Ñ Ò Ð Ñ Ò Ð Results æ â çå ä åã á âã éè è ÔÏ Ö ÔÏ Ü Û Õ ÔÏ Ñ ÏÐ í ÔÏ æ â çå ä åã á âã éè è ì ÔÏ ë ÔÏ Ü Û ÔÏ Ö ÔÏ Õ ÔÏ Ñ ÏÐ

52 Test case 4: Linear elasticity Mathematical notation: 1 a(v, u) = Ω 4 ( v + ( v) ) : ( u + ( u) ) dx = d d Ω i=1 j=1 1 4 ( v i x j + v j x i )( u i x j + u j x i ) dx FFC implementation: v = BasisFunction(element) u = BasisFunction(element) a = 0.25*(v[i].dx(j) + v[j].dx(i)) * \ (u[i].dx(j) + u[j].dx(i)) * dx Simula Research Laboratory, Oslo August p. 52

53 ÿ þ ý ü ù ø ò ñ î ÿ þ ý ü ù ø ò ñ î Simula Research Laboratory, Oslo August p. 53 ð ñ ï ð ñ ï Results ö óî õ óî û ú ô óî ð îï óî óî óî û ú ö óî õ óî ô óî ð îï

54 Speedup Form q = 1 q = 2 q = 3 q = 4 q = 5 q = 6 q = 7 q = 8 Mass 2D Mass 3D Poisson 2D Poisson 3D Navier Stokes 2D Navier Stokes 3D Elasticity 2D Elasticity 3D Impressive speedups but far from optimal Data access costs more than flops Solution: build arrays and call BLAS (Level 2 or 3) Simula Research Laboratory, Oslo August p. 54

55 ,, * & & * & &,,,, # "! Simula Research Laboratory, Oslo August p. 55 Code bloat * + ( ) %' & $ %& * + ( ) %' & $ %&, /. -., / & 5 (. * 21 )43 0 % 6 2& 5 (. * 21 )43 0 % * ( : (*9 78%& * ( : (*9 78%&

56 Future directions for FEniCS Simula Research Laboratory, Oslo August p. 56

57 Future directions for FEniCS New parallel mesh component (05) Completely parallel assembly/solve (05) Optimize code generation through FErari (05/06) Optimize code generation through BLAS (05/06) Optimize compiler to reduce compile time (05/06) Complete support for non-standard elements (05/06): Crouzeix Raviart, Raviart Thomas, Nedelec, Brezzi Douglas Marini, Brezzi Douglas Fortin Marini, Arnold Winther, Taylor Hood,... Automatic generation of dual problems (06) Automatic generation of error estimates (06) Manuals, tutorials, mini-courses (05/06) Simula Research Laboratory, Oslo August p. 57

58 References A compiler for variational forms Kirby/Logg, submitted to TOMS (2005) Optimizing the evaluation of finite element matrices Kirby/Knepley/Logg/Scott, SISC (2005) Topological optimization of the evaluation of finite element matrices Kirby/Logg/Scott/Terrel, submitted to SISC (2005) FIAT: A new paradigm for computing finite element basis functions Kirby, TOMS (2004) Optimizing FIAT with the Level 3 BLAS Kirby, submitted to TOMS (2005) Evaluation of the action of finite element operators Kirby/Knepley/Scott, submitted to BIT (2004) Simula Research Laboratory, Oslo August p. 58

59 Additional slides Simula Research Laboratory, Oslo August p. 59

60 The Automation of CMM Au = f (iv) U u Ãũ = f cements (i) F (x) = 0 (ii) X x U u TOL (iii) (V, ˆV ) tol > 0 U u Simula Research Laboratory, Oslo August p. 60

61 Example 3: Navier Stokes Form: a(v, u) = Ω v (w )u dx Evaluation: A e i = φ i1 (w )φ i2 dx e = det F e X α3 w α2 Φ i1 [β]φ Φ i 2 [β] α2 [α 1 ] x α1 X α3 E dx = A 0 iαg α e with A 0 iα = E Φ i 1 [β]φ α2 [α 1 ] Φ i 2 [β] X α3 dx and G α e = det F e X α3 x α1 w α2 Simula Research Laboratory, Oslo August p. 61

62 Complexity of form evaluation Basic assumptions: Bilinear form: i = 2 Exact integration of forms Notation: q: polynomial order of basis functions p: total polynomial order of form d: dimension of Ω n: dimension of function space (n q d ) N: number of quadrature points (N p d ) n C : number of coefficients n D : number of derivatives Simula Research Laboratory, Oslo August p. 62

63 Complexity of tensor contraction Need to evaluate A e i = A0 iα Gα e Rank of G α e is n C + n D Number of elements of A e i is n2 Number of elements of G α e is n n C dn D Total cost: T C n 2 n n C d n D (q d ) 2 (q d ) n C d n D q 2d+n Cd d n D Simula Research Laboratory, Oslo August p. 63

64 Complexity of quadrature Need to evaluate A e i at N pd quadrature points Total order of integrand is p = 2q + n C q n D Cost of evaluating integrand is n C + n D d + 1 Total cost: T Q n 2 N(n C + n D d + 1) (q d ) 2 p d (n C + n D d + 1) q 2d (2q + n C q n D ) d (n C + n D d + 1) Simula Research Laboratory, Oslo August p. 64

65 Tensor contraction vs quadrature Speedup: T C q 2d+n Cd d n D T Q q 2d (2q + n C q n D ) d (n C + n D d + 1) T Q /T C (2q + n Cq n D ) d (n C + n D d + 1) q n Cd d n D Rule of thumb: tensor contraction wins for n C = 0, 1 Mass matrix (n C = n D = 0): T Q /T C (2q) d Poisson (n C = 0, n D = 2): T Q /T C (2q 2) d (2d + 1)/d 2 Not clear that tensor contraction wins for the stabilization term of Navier Stokes: (w )u (w )v Need an intelligent system that can do both! Simula Research Laboratory, Oslo August p. 65

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