Einführung in die Finite Element Methode Projekt 2

Transkript

1 Einführung in die Finite Element Methode Projekt 2 Juri Schmelzer und Fred Brockstedt Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

2 Project 2 -Tasks 1 Study Section 3.4 of the lecture notes 2a) Write conforming mesh generator 2b) Write non conforming mesh generator 3 Rewrite the solver for hanging nodes 4 Convergence analysis Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

3 Agenda 1 Tasks 2 Mesh generator 3 Adapting the finite element solver 4 Convergence analysis 5 Demo 6 Questions Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

4 Mesh generator Task Write a conforming/non-conforming mesh generator. Definition (Mesh) A Mesh M of a computational domain Ω R 2 is a collection of cells {K k } M k=1 und M = M. Algorithm 1 Generate nodes 2 Find the corresponding elements (triangles) 3 Find hanging nodes Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

5 Conforming mesh generator Series 4 In series 4 we already wrote a mesh generator for conforming meshes, with the help of Mesh2d. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

6 Adaptation of a conforming mesh generator Problem With every uniform refinement, refine.m squares the amount of nodes. Goal Mesh with fewer nodes. Refine towards the origin using Mesh2d. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

7 Implementation M = size(t,1); % find the triangle at the origin t = sortrows(t, 1); ti = false(m,1); ti(1) = true; [p t] = refine(p,t,ti); % refine towards the % triangle at the origin Problem Mesh2d will malfunction. So patch line 129 in Mesh2d/refine.m with t e ( s p l i t 1, : ) = s o r t ( t e ( s p l i t 1, : ) ) ; Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

8 Non conforming mesh generator Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

9 Non conforming mesh generator Definition (hanging node) A node of a mesh M that is located in the interior of a geometric face of one of its cells is known as hanging node. Definition (non conforming mesh) A non conforming mesh is a mesh with hanging nodes. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

10 Non conforming mesh generator Special case for unit square refined towards the origin handle special nodes (origin, first diagonal point) that are always present. find the corresponding triangles in the lower left and upper right. find the hanging nodes on the subdiagonal. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

11 Non conforming mesh generator Implementation r = r +1; % amount o f r e f i n e m e n t s nodes = 1. / 2. ^ 0 : r ; % nodes on the d i a g o n a l w i t h o u t % 0 and 1 i n r e v e r s e o r d e r p = [ 0 0 ] ; % s t a r t at the o r i g i n %% c r e a t e the t h r e e s p e z i a l nodes, % numbered c o u n t e r c l o c k w i s e p ( end +1, : ) = [ nodes ( end ) 0 ] ; p ( end +1, : ) = [ nodes ( end ) nodes ( end ) ] ; p ( end +1, : ) = [ 0 nodes ( end ) ] ; %% c r e a t e the t r i a n g l e s t = [ ] ; Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

12 Non conforming mesh generator the first onion ring Create the 5 new points and the 6 new triangles. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

13 Non conforming mesh generator %% c r e a t e 5 new nodes p ( end +1, : ) = [ nodes ( k ) 0 ] ; % boundary x a x i s node % s a v e the p o i s i t i o n o f the f i r s t new % node f o r our t r i a n g l e c r e a t i o n nodeindex = l e n g t h ( p ) ; p ( end +1, : ) = [ nodes ( k ) nodes ( k + 1 ) ] ; p ( end +1, : ) = [ nodes ( k ) nodes ( k ) ] ; p ( end +1, : ) = [ nodes ( k+1) nodes ( k ) ] ; p ( end +1, : ) = [ 0 nodes ( k ) ] ; Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

14 %% w r i t e down the 6 new t r i a n g l e s % the 2 bottom r i g h t t r i a n g l e s t ( end +1, : ) = [ nodeindex 5 nodeindex nodeindex 3]; t ( end +1, : ) = [ nodeindex nodeindex+1 nodeindex 3]; % the 2 top r i g h t t r i a n g l e s t ( end +1, : ) = [ nodeindex 3 nodeindex+1 nodeindex +3]; t ( end +1, : ) = [ nodeindex+1 nodeindex+2 nodeindex +3]; % 2 t r i a n g l e s to the top l e f t t ( end +1, : ) = [ nodeindex 1 nodeindex 3 nodeindex +4]; t ( end +1, : ) = [ nodeindex 3 nodeindex+3 nodeindex +4]; %% s a v e the hanging nodes hn ( end +1, : ) = [ nodeindex+1 nodeindex nodeindex +2]; hn ( end +1, : ) = [ nodeindex+3 nodeindex+2 nodeindex +4]; Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

15 Non conforming mesh generator Algorithm If the above points and elements have been generated, apply the following algorithm to refine the mesh. 1 take point on the diagonal [ 1 2 n, 1 2 n ] 2 make boundary node and hanging node to the left and the bottom 3 create the corresponding triangles 4 save the hanging nodes and their neighbours after the last iteration remove the two hanging nodes Generating the non conforming mesh Repeat for r refinements. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

16 Memory usage Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

17 Definition P is a hanging Node if the following conditions are fullfilled: P is part of a triangle in the 2D mesh P lies on a side of an other triangle and is not on of its corner Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

18 Definition P is a hanging Node if the following conditions are fullfilled: P is part of a triangle in the 2D mesh P lies on a side of an other triangle and is not on of its corner In our case they are the middlepoint of two non-hanging Nodes. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

19 Definition P is a hanging Node if the following conditions are fullfilled: P is part of a triangle in the 2D mesh P lies on a side of an other triangle and is not on of its corner In our case they are the middlepoint of two non-hanging Nodes. The assembling part of stiffness, mass, load and neumannload needs to be rewritten for it. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

20 Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

21 How to compute the T-matrix? Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

22 The T-Matrix can be computed with the following steps: Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

23 The T-Matrix can be computed with the following steps: 1 If P j (K) is not hanging Node, than T i,j = 1 if P j (K) = P i (M) 2 If P j (K) is a hanging Node, than get the neighbours of P j (K) i.e. Q 1, Q 2 and set T i,j = 1 2 if P(Q 1) = P i (M) P(Q 2 ) = P i (M) with P(Q i ) describing the index of Q i in the mesh 3 Any other entry is set to 0. With this we get a sparse matrix with 5 entries at most for our mesh. Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

24 Code defining the coordinates of the triangle, initialising the T-Matrix and searching for hanging nodes in the triangle and which of them is a hanging Node f o r k=1:m c o o r d i n a t e s = p ( t ( k, : ), : ) ; containhangnode=ismember ( t ( k, : ), hang ( :, 1 ) ) ; T=z e r o s ( l e n g t h ( p ), 3 ) ; m=f i n d ( containhangnode ) ; Q= [ 1, 2, 3 ] ;Q(m) = [ ] ; Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

25 Code If triangle without hanging node than creating the usual T-Matrix i f ( abs ( containhangnode )==0) T( t ( k, 1 ), 1 ) = 1 ; T( t ( k, 2 ), 2 ) = 1 ; T( t ( k, 3 ), 3 ) = 1 ; Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

26 Code setting 1 for nonhanging nodes in the triangle and 1 2 on the neighbours of the hanging Nodes. e l s e f o r l =1:3 max( s i z e (m) ) T( t ( k,q( l ) ),Q( l ))=1; end f o r l =1:max( s i z e (m) ) [~, hangindex ]= ismember ( t ( k,m( l ) ), hang ( :, 1 ) ) ; T( hang ( hangindex, 2 ),m( l ))=1/2; T( hang ( hangindex, 3 ),m( l ))=1/2; end end Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

27 Code Calculating K T K B K TK T = B and removing of hanging node rows and columns A=A+s p a r s e (T) e l e m S t i f f n e s s P 1 ( c o o r d i n a t e s )... s p a r s e (T ) ; end A( hang ( :, 1 ), : ) = [ ] ; A ( :, hang ( :, 1 ) ) = [ ] ; Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

28 Analytic solution Problem Task u + u = f in Ω = [0, 1] 2 u n = g Choose f and g such that u := e ( γ(x+y)) Solution f = e ( ( γ(x+y)) 2γ 2 e γ(x+y) e g = (x+y)γ ) γ e (x+y)γ n γ on Ω Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

29 Exact solution of u Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

30 Analytic solution Task Choose γ > 0 and compute the discretization error in the energy norm. Remember Series 6 e n e := a(u u n, u u n ) u is known and u n is the solution of the solver. a(u u n, u u n ) = fudx Ω }{{} solve analyticaly Solution 1 0 fu n dx Ω }{{} u f 1 ( 1 0 f (x, y)u(x, y)dxdy = 4γ (1 2γ 2 )(e 2γ 1) 2) 2 Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

31 Convergence Analysis Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

32 Convegence Analysis nodes conform hanging uniform Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29

33 Questions Questions, remarks and comments! Juri Schmelzer und Fred Brockstedt Einführung in die Finite Element Methode Projekt / 29