Building a finite element program in MATLAB Linear elements in 1d and 2d

Transcription

1 Building a finite element program in MATLAB Linear elements in 1d and 2d D. Peschka TU Berlin Supplemental material for the course Numerische Mathematik 2 für Ingenieure at the Technical University Berlin, WS 2013/2014 D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 1 / 32

2 Content 1 Introduction 2 Decomposition and elements 3 Reference element 4 Basis and shape functions 5 The global matrix D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 2 / 32

3 Plan 1 Introduction 2 Decomposition and elements 3 Reference element 4 Basis and shape functions 5 The global matrix

4 Introduction Let V a Hilbert space, e.g. H0 1 (Ω). For a bilinear form a : V V R and linear form f : V R solve the following problem: Definition (variational problem) Find u V such that for all v V. a(u, v) = f (v), Example (variational problem for c 0) a(u, v) = f (v) = Ω Ω u(x) v(x) + c u(x)v(x) dx, f (x)v(x) dx, is the weak form of the elliptic problem u + cu = f. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 3 / 32

5 Introduction By choosing a finite dimensional subspace V h V we introduce the Galerkin approximation: Definition (Galerkin approximation) Find u h V h such that for all v h V h. a(u h, v h ) = f (v h ) We have a set of functions w i which form a basis of V h. Then every element of V h (and in particular the solution) can be written u(x) = dim V h i=1 α i w i (x) D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 4 / 32

6 Introduction Then we may rewrite the Galerkin approximation: Definition (Galerkin approximation) Find α i R for i = 1... dim V h such that dim V h i=1 a(w i, w j )α i = f (w j ) for all j = 1... dim V h. Equivalently we may write where A ij = a(w j, w i ), f i = f (w i ). Aα = f D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 5 / 32

7 Introduction Motivation: The finite element method (FEM) is a particular method to systematically generate the finite-dimensional subspace V h of V and generate the Galerkin matrix A. Therefore the method generates a decomposition of Ω = k Ω k, defines basis functions w i which for each i are only non-zero on a few Ω k, maps these to a reference domain/shape functions. These basis function are usually piecewise polynomials and globally continuous. The resulting matrix A is sparse. Here we are going to construct a finite-dimensional subspace of the Hilbert space H 1 (Ω) or H 1 0 (Ω). Many concepts shown here are still valid for H r (Ω) and r > 1. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 6 / 32

8 Introduction Finite elements Loosely speaking Lagrange finite elements consist of geometric objects (triangles, quadrilaterals, tetrahedrons,...), the set of piecewise polynomials P on each geometric object, the set of unknowns, i.e. values of basis functions at certain points p i (on vertices, edges, elements,...) i = 1... dim V h. The construction of FE basis functions w i P is such that the relation between the set of polynomials and the unknowns is and this condition gives unique w i s. w i (p j ) = δ ij D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 7 / 32

9 Plan 1 Introduction 2 Decomposition and elements 3 Reference element 4 Basis and shape functions 5 The global matrix

10 Decomposition and elements: 1d definition Definition (admissible decomposition 1d) For domain Ω R the decomposition Ω = nelement k=1 is called admissible, if Ω k Ω l is either the full element only for k = l, a common vertex, or otherwise empty. Ω k D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 8 / 32

11 Decomposition and elements: 2d definition Definition (admissible triangulation/decomposition 2d) For domain Ω R 2 with polygonal boundary the decomposition Ω = nelement k=1 is called admissible triangulation, if Ω k Ω l is either the full triangle element only for k = l, a common vertex, a (full) common edge, or otherwise empty. Ω k D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 9 / 32

12 Decomposition and elements: 1d and 2d Question: How do we represent a triangulation in practice? Answer 1d: x, e2p, npoint, nelement where npoint is the number of points/vertices, nelement is the number of elements (intervals), x R npoint is the collection of vertices of triangles, e2p R nelement 2 the element-to-point (or vertex) map. Answer 2d: x, e2p, npoint, nelement where npoint is the number of points/vertices, nelement is the number of elements (triangles), x,y R npoint is the collection of vertices of triangles, e2p R nelement 3 the element-to-point (or vertex) map. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 10 / 32

13 Decomposition and elements: Remark Some programs might provide additional information, e.g. in 2d number of edges, identification of boundary points, identification of boundary edges, neighboring elements, special functions operating on triangulation, e.g. see DelaunayTri class in MATLAB. Despite from being very useful, most of these things can be derived from x,y,e2p. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 11 / 32

14 Decomposition and elements: 1d example points 1 2 x 0 x 1 3 x 2 4 x 3 5 x 4 elements npoint=5, nelement=4, x=[0 1/4 2/4 3/4 1], e2p(1:4,1)=1:4, e2p(1:4,2)=2:5, Ω k = [x k 1, x k ] for k = 1..nelement. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 12 / 32

15 Decomposition and elements: 1d MATLAB code 1 npoint = 5; % # points in decomposition 2 nelement = npoint - 1; % # elements / intervals 3 4 x = linspace (0,1, npoint ); % create vertices 5 6 e2p (1: nelement,1) = 1: npoint -1; % create e2p, part 1 7 e2p (1: nelement,2) = 2: npoint ; % create e2p, part plot (x,0*x,'b-o',' MarkerFaceColor ','r') % draw decomposition D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 13 / 32

16 Decomposition and elements: 2d example npoint=16, nelement=18, x=[ ]/3, y=[ ]/3, e2p R Ω k = conv{x e2p(k,1), x e2p(k,2), x e2p(k,3) }. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 14 / 32

17 Decomposition and elements: 2d example Where the convex-hull of points is defined { } 3 3 conv(x 1, x 2, x 3 ) := x R 2 : x = ξ α x α ; ξ α 0; ξ α = 1, α=1 α=1 or alternatively for a point as x = x 1 x 2 x 3 y y 1 y 2 y 3 for the barycentric coordinates ξ 1, ξ 2, ξ 3 0. ξ 1 ξ 2, ξ 3 Note: The generalization to 3d gives a tetrahedron, the restriction to 1d gives an interval. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 15 / 32

18 Decomposition and elements: 2d example e2p = D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 16 / 32

19 Decomposition and elements: 2d MATLAB code 1 n = 4; % parameter 2 [x,y]= meshgrid ( linspace (0,1,n )); % 2D array of vertices 3 x=x (:); y=y (:); % array -> vector 4 e2p = delaunay (x,y); % Delaunay triangulation 5 npoint = size (x,1); % # points 6 nelement = size (e2p,1); % # elements 7 8 % plot the decomposition 9 triplot (e2p,x,y,'o-','color ','b',' MarkerFaceColor ','r') D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 17 / 32

20 Plan 1 Introduction 2 Decomposition and elements 3 Reference element 4 Basis and shape functions 5 The global matrix

21 t Reference element: Definition 1d: Ω ref = (0, 1), p 1 = 0, p 2 = 1 2d: Ω ref = {(s, t) R 2 : s, t > 0; 0 < s < 1 t}, p 1 = (0, 0), p 2 = (1, 0), p 3 = (0, 1) s D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 18 / 32

22 Reference element: Map to element For each element F k we have a map F k : Ω ref Ω k which is affine linear so that the following identity holds F k (p m ) = x i for i=e2p(k,m) and x i is the ith vertex. The map is a linear polynomial, i.e., F k (p) = A k p + B k where (1d) A k, B k R or (2d) A k R 2 2, B k R 2. We have and (1d) and (2d) B k = x e2p(k,1) A k = x e2p(k,2) x e2p(k,1) R A k = ( x e2p(k,2) x e2p(k,1), x e2p(k,3) x e2p(k,1) ) R 2 2 D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 19 / 32

23 Reference element: Map to element Note: Note that F k = A k MATLAB code fragment to determine B k for a given k 1 B (1)= x(e2p (k,1)); % first component of B 2 B (2)= y(e2p (k,1)); % second component of B MATLAB code fragment to determine (A k ) 22 for a given k 1 A (2,2)= y(e2p (k,3)) -y(e2p (k,1)); % 22 component of A D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 20 / 32

24 Reference element Some extensions are useful: transformation F k using polynomial of higher degree e.g. isoparametric elements. 1 quadrilateral elements (2d), tetrahedrons (3d) 1 More complicated since F k depends on space and we need to introduce numerical integration techniques. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 21 / 32

25 Plan 1 Introduction 2 Decomposition and elements 3 Reference element 4 Basis and shape functions 5 The global matrix

26 Basis and shape functions For a given decomposition/triangulation Ω k we define basis functions w i by w i (x j ) = δ ij, w i is piecewise polynomial (linear) on every Ω k, w i is continuous, so that w i are in the finite element space w i V h V. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 22 / 32

27 Note: Red circles indicate the position of vertices. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 23 / 32 Basis and shape functions: 1d With npoint=5 and V = H 1 (0, 1) we have the following w i : 1 w 5 w 4 w 3 w 2 w x x x x x For V = H 1 0 (0, 1) the (three) basis functions of V h are w 1 (x) = w 2 (x), w 2 (x) = w 3 (x), w 3 (x) = w 4 (x).

28 Basis and shape functions: 2d With npoint=16, Ω = (0, 1) 2 and V = H 1 (Ω) we have 16 basis functions w i. For example i = 1, i = 6, i = 15: D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 24 /

29 Basis and shape functions: 2d With npoint=16, Ω = (0, 1) 2 and V = H 1 (Ω) we have 16 basis functions w i. For example i = 1, i = 6, i = 15: D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 24 / 32

30 Basis and shape functions: 2d With npoint=16, Ω = (0, 1) 2 and V = H 1 (Ω) we have 16 basis functions w i. For example i = 1, i = 6, i = 15: D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 24 /

31 Basis and shape functions: basis shape The basis functions are maps w i : Ω R for i = 1... dim V h. The shape functions are maps φ n : Ω ref R for n = 1... nphi. Definition (shape functions) Using the map F k we have the identity w i ( Fk (p) ) = φ n (p) for i=e2p(k,n) for every p Ω ref, k = 1... nelement, n = 1... nphi. This is the key-property ensuring continuity of w i and thereby w i V h H 1 (Ω). Beyond linear Lagrange elements we need to extend e2p(1:nelement,1:nphi), where each n = 1... nphi is identified with a point p n at 1d) i) a vertex, ii) an element 2d) i) a vertex, ii) an edge, iii) an element, 3d) i) a vertex, ii) an edge, iii) a face, iv) an element. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 25 / 32

32 Basis and shape functions: basis shape Example Linear intervals in 1d: nphi=2, Linear triangles in 2d: nphi=3, Linear tetrahedrons in 3d: nphi=4, Quadratic intervals in 1d: nphi=3, Quadratic triangles in 2d: nphi=6, Quadratic tetrahedrons in 3d: nphi=10. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 26 / 32

33 Basis and shape functions: Shape functions Example (1d linear interval, nphi=2) φ 1 (s) = 1 s, φ 2 (s) = s p 1 = 0, p 2 = 1. Example (2d linear triangle, nphi=3) φ 1 (s, t) = 1 s t, φ 2 (s, t) = s, φ 3 (s, t) = t, p 1 = (0, 0), p 2 = (1, 0), p 3 = (0, 1), Example (2d quadratic triangle, nphi=6) φ n (s, t) = a n 1 + a n 2 s + a n 3 t + a n 4 st + a n 5 s 2 + a n 6 t 2 where the 36 coefficients are determined by the condition φ n (p m ) = δ nm p 1 = (0, 0), p 2 = (1, 0), p 3 = (0, 1), p 4 = ( 1 2, 0), p 5 = ( 1 2, 1 2 ), p 6 = (0, 1 2 ). D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 27 / 32

34 Basis and shape functions: Matrices for shape functions In principle we want to compute integrals such as M = Ω w i (x)w j (x) dω = nelement k=1 Ω k w i (x)w j (x) dω Since we have the identity w i (F k (p)) = φ n (p) we can compute this integral over Ω k = F k (Ω ref ) using integration by substitution f (x) dx = f ( F k (p) ) det( F k ) dp F k (Ω ref ) Ω ref and get w i (x)w j (x) dω = φ m (p)φ n (p) det(a k ) dp Ω k Ω ref with the identity i = e2p(k,m), j = e2p(k,n). A similar equation holds for S = Ω w i(x)w j(x) dx. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 28 / 32

35 Plan 1 Introduction 2 Decomposition and elements 3 Reference element 4 Basis and shape functions 5 The global matrix

36 The global matrix How to build the matrix M in 1d: nelement M ij = φ m (p)φ n (p) det(a k ) dp Ω ref k=1 where i = e2p(k,m), j = e2p(k,n). 1 %% prepare fem stuff 2 p = 7; 3 nphi = 2; 4 x = linspace (0,1,2ˆp+1); 5 [ nelement, npoint, e2p ] = generateelements (x); 6 %% build matrices 7 ii = zeros (nelement,nphi ˆ2); % sparse i- index 8 jj = zeros (nelement,nphi ˆ2); % sparse j- index 9 mm = zeros ( nelement, nphi ˆ2); % entry of Galerkin matrix 10 %% build global from local 11 for k =1: nelement % loop over elements 12 [edet, dfinv ] = generatetransformation (k,e2p,x); % compute map 13 mloc = localmass ( edet ); % element mass matrix 14 % compute i,j indices of the global matrix 15 ii( k,: ) = [e2p (k,1) e2p (k,2) e2p (k,1) e2p (k,2)]; % local -to - global 16 jj( k,: ) = [e2p (k,1) e2p (k,1) e2p (k,2) e2p (k,2)]; % local -to - global 17 % compute a(i,j) values of the global matrix 18 mm( k,: ) = mloc (:); 19 end 20 % create sparse matrices 21 M= sparse (ii (:), jj (:), mm (:)); D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 29 / 32

37 The global matrix MATLAB code generateelements.m 1 % Generate infrastructure for FEM. 2 function [ nelement, npoint, e2p ] = generateelements (x) 3 npoint = length (x); 4 nelement = npoint -1; 5 e2p = zeros ( nelement,2); 6 e2p (1: nelement,1)=1: npoint -1; 7 e2p (1: nelement,2)=2: npoint ; MATLAT code generatetransformation.m 1 % Compute the determinant of the Jacobian on every triangle. 2 function [edet, dfinv ]= generatetransformation (k,e2p,x) 3 edet = x(e2p (k,2)) -x(e2p (k,1)); 4 dfinv = 1/ edet ; MATLAB code localmass.m 1 % Local mass matrix M = \ int phi_i phi_j dx 2 function m= localmass ( det ) 3 m=det *[1/3 1/6;1/6 1/3]; MATLAB code localstiff.m 1 % Local stiffness matrix S = \ int phi '_i * phi '_j dx 2 function S=localstiff (det, dfinv ) 3 ndim = 1; 4 nphi = 2; 5 6 gradphiloc (1,1) = -1; 7 gradphiloc (2,1) = +1; 8 9 dphi (1: nphi,1: ndim ) = gradphiloc (1: nphi,1: ndim )* dfinv (1: ndim,1: ndim ); 10 S=( dphi (:,1)* dphi (:,1) ') * det ; D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 30 / 32

38 The global matrix When computing S we need partial derivatives of w i, which can be represented for instance by x w i (x, y) = x φ m (F 1 k (x, y)) where F k = A k p + B k = (x, y) with p = (s, t). Then x w i = φ m s y w i = φ m s s x + φ m t t x s y + φ m t t y so that using p = A 1 k ((x, y) B k ), we get ( ) ( ) ( ) x w w i = i x s = x t s φ m y w i y s y t t φ m so that w i w j = φ m A 1 k A k φ n. = (A 1 k ) φ m. D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 31 / 32

39 End D. Peschka (TU Berlin) FEM with MATLAB Num2 WS13/14 32 / 32