Hat Function. defined on a triangulation of the domain D basis B i, i I, for piecewise linear functions 1 at center p i, 0 at other vertices p j.

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1 Hat Function defined on a triangulation of the domain D basis B i, i I, for piecewise linear functions 1 at center p i, 0 at other vertices p j B i Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

2 Hat Function approximation, determined by Lagrange data u h = i I u i B i, u i = u h (p i ) piecewise constant gradient, determined via directional derivatives grad B i (p i p j, p i p k,...) = (1, 1,...) Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

3 Assembly of a Ritz Galerkin System for Hat Functions sum contributions from each triangle T = [p i, p j, p k ] g i,k = grad B i grad B k, f i = T T T T f B i p i p j T p k Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

4 Assembly of a Ritz Galerkin System for Hat Functions compute gradients via directional derivatives grad B i grad B j ( ) p j p i p k p j = R, R = grad B k G: add submatrix, corresponding to inner vertices, det P 2 RP 1 (P t ) 1 R t F : add subvector, corresponding to inner vertices, det P 2 f (T ) } {{ } average Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

5 Hat Functions on a Regular Grid B i : piecewise linear, pyramid-like (0, 1) (1, 1) ( 1, 0) h 1 (1, 0) ( 1, 1) (0, 1) Ritz-Galerkin matrix D grad B i grad B k g i,i = 4, g (i1 ±1,i 2 ),i = g (i1,i 2 ±1),i = 1 generalized banded system 4u i u i (1,0) u i+(1,0) u i (0,1) u i+(0,1) = h 2 f i Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

6 Poisson Solver with Hat Functions intput p(k, :): vertices, first n inside t(k, 1 : 3): indices for triangles f : data at vertices output u(1 : n): solution at interior vertices Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

7 Poisson Solver with Hat Functions add contributions from triangles T = [p α, p β, p γ ] T fb κ = area(t ) f T /3 T grad B κ grad B κ = area(t ) (g κ g t κ ) area(t ) = det P /2, P = (p β p α, p γ p β ) computation of gradients g α g β g γ P = Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

8 h = 1/10; p = [0; 0]; for k=1:round(1/h); if k>1/h-1/2; n = size(p,2); end; nk = round(2*pi*k); tk = [1:nk]*2*pi/nk; p = [p (k*h)*[cos(tk); sin(tk)]]; end; Hölllig (IMNG) FEM with B-Splines 01. Januar / 10 Poisson Solver with Hat Functions % demo: poisson_hat % solution of -delta u = 1 on unit disc % with zero boundary values % Delaunay triagulation based on regularly % spaced points with approximate distance h

9 Poisson Solver with Hat Functions t = delaunay(p(1,:),p(2,:)); trimesh(t,p(1,:),p(2,:), color,[0,0,0]); axis equal;axis off;figure % fe-solution nt = size(t,1); u = poisson_hat(p,t,n,ones(nt,1)); [x,y] = meshgrid([-1:h:1]); z = griddata(p(1,1:n),p(2,1:n),u,x,y); mesh(x,y,z); Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

10 Poisson Solver with Hat Functions Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

11 Poisson Solver with Hat Functions function u = poisson_hat(p,t,n,f) % POISSON_HAT: Poisson-solver with hat-functions % p(:,k): vertices of triangulation % t(j,1:3): indices of triangles % n: number of interior vertices (first) % f(j): forces on triangles % u(1:n): solution at interior vertices % initialize Ritz-Galerkin system lp = size(p,2); G = zeros(lp,lp); F = zeros(lp,1); % hat-function gradients times triangle vectors R = [-1, 0; 1, -1; 0, 1]; Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

12 Poisson Solver with Hat Functions % loop over triangles for k = 1:size(t,1); % indices of vertices and triangle vectors indk = t(k,:); P = [p(:,indk(2))-p(:,indk(1)),... p(:,indk(3))-p(:,indk(2))]; % area of triangle dp = abs(det(p))/2; % add int_t f B_i for i in indk F(indk) = F(indk) + dp*f(k)/3; % add int_t g_i g_i for i,i in indk % compute g s from [g_a; g_b; g_c]*p = R G(indk,indk) = G(indk,indk) +... dp*r*inv(p)*inv(p )*R ; end; % solve Ritz-Galerkin sytem for relevant nodes u = G(1:n,1:n) \ F(1:n); Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

13 Solver comparision grid width unknowns iteration count: Jacobi: Gauß-Seidel (checkerboard): SOR, ω = 1.25 (checkerboard): SOR, ω = 1.5 (checkerboard): SOR, ω = 1.75 (checkerboard): SOR, ω = 1.9 (checkerboard): SOR, ω = 1.94 (checkerboard): cg: Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

14 Solving Poisson s equation with MATLAB function with boundary data no argument: number of boundary segments one argument (segment numbers): start, end, left, right values two arguments (segment and parameter): points on boundary create triangulation with initmesh function for boundary conditions solve with assempde refine with refinemesh Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

15 Solving Poisson s equation with MATLAB Step 1: function for boundary function [x,y]=circle_with_hole(bs,s) % no input: number of boundary-segments if nargin==0,x=4;return;end % boundary structure d=[ % start parameter value % end parameter value % left hand region % right hand region ]; % one input argument: structure of boundary if nargin==1,x=d(:,bs(:) );return;end Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

16 if ~isempty(s), k=find(bs==1);% boundary segment 1: circle x(k)=mcx+rc*cos((pi)*s(k));y(k)=mcy+rc*sin((pi)*s(k)); k=find(bs==2);% boundary segment 2: circle x(k)=mcx+rc*cos((pi)*s(k)+pi);y(k)=mcy+rc*sin((pi)*s(k)+pi); k=find(bs==3);% boundary segment 3: hole x(k)=mhx+rh*cos((pi)*s(k));y(k)=mhy+rh*sin((pi)*s(k)); k=find(bs==4);% boundary segment 4: hole x(k)=mhx+rh*cos((pi)*s(k)+pi);y(k)=mhy+rh*sin((pi)*s(k)+pi); end Hölllig (IMNG) FEM with B-Splines 01. Januar / 10 Solving Poisson s equation with MATLAB % two input arguments: points on boundary mcx=0;mcy=0;rc=4;% midpoint and radius of circle mhx=0;mhy=0;rh=1;% midpoint and radius of hole x=zeros(size(s));y=x; if numel(bs)==1,bs=bs*ones(size(s));end; % expand bs

17 Solving Poisson s equation with MATLAB Step 2: initial mesh [p,e,t]=initmesh( circle with hole, hmax,.5); pdemesh(p,e,t); Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

18 Solving Poisson s equation with MATLAB Step 3: boundary conditions function [q,g,h,r]=boundary(p,e,u,time) %Boundary condition data % Neumann-conditions, zero values for Dirichlet-Boundary q=zeros(1,length(e)); g=zeros(1,length(e)); % Dirichlet-Boundary: hu=r h=ones(1,2*length(e)); r=zeros(1,2*length(e)); Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

19 Solving Poisson s equation with MATLAB Step 4: solve and plot U=ASSEMPDE(B,P,E,T,C,A,F) solves div(c grad u) + au = f on triangulation with points p, edges e, and triangles t with boundary-conditionfunction b u=assempde( boundary,p,e,t,1,0,1); pdesurf(p,t,u); Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

20 Solving Poisson s equation with MATLAB Step 5: refine triangulation [p,e,t]=refinemesh( circle with hole,p,e,t); pdemesh(p,e,t); and solve again: u=assempde( boundary,p,e,t,1,0,1); Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

21 Voronoi Diagram defined for a set S of points p k R d Voronoi polygon V k : set of points which are closer to p k than to any of the other points p j S Voronoi diagram V (S): union of the subsets of hyperplanes bounding the Voronoi polygons p i Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

22 Voronoi Spheres Points p k R d with their closed Voronoi polygons V k intersecting in a single point v lie on a sphere with center v. In general, exactly d + 1 Voronoi polygons share a common point. Proof: V i and V j intersect in a a hyperplane containing v p i, p j have same distance to v, i.e., all points p k have the same distance to v Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

23 Delaunay Triangulation triangulation with vertices p k R d such that no vertex is contained in the circumscribed sphere of any simplex construction via Voronoi-diagram: in general, points p k with their closed Voronoi polygons V k intersecting in a single point v form a simplex (neighboring points define an edge) nonuniqueness, if more than d + 1 points lie on a sphere which does not contain any other points Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

24 Sphere Criterion For a Delaunay simplex with vertices p k, the circumscribed sphere does not enclose any other vertices of the triangulation. The criterion can be used to construct a Delaunay triangulation directly. Proof: none of the Voronoi polygons of the other vertices contains the center v of the circumscribed sphere Voronoi polygons of the vertices p k have the common intersection v p i in circumscribed sphere, closest to v v belongs to Voronoi polygon V i, i.e., is not the intersection of the V k s Hölllig (IMNG) FEM with B-Splines 01. Januar / 10

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