Page 440 - Applied Numerical Methods Using MATLAB

GUI OF MATLAB FOR SOLVING PDES: PDETOOL 429

%do_fem
% for Example 9.6
clear
N = [-1 0;-1 -1;-1/2 -1;0 -1;1/2 -1; 1 -1;1 0;1 1;1/2 1; 0 1;
-1/2 1;-1 1; -1/2 -1/4; -5/8 -7/16;-3/4 -5/8;-1/2 -5/8;
-1/4 -5/8;-3/8 -7/16; 0 0; 1/2 1/4;5/8 7/16;3/4 5/8;
1/2 5/8;1/4 5/8;3/8 7/16;-9/16 -17/32;-7/16 -17/32;
-1/2 -7/16;9/16 17/32;7/16 17/32;1/2 7/16]; %nodes
N_b = 12; %the number of boundary nodes
S = [1 11 12;1 11 19;10 11 19;4 5 19;5 7 19; 5 6 7;1 2 15; 2 3 15;
3 15 17;3 4 17;4 17 19;13 17 19;1 13 19;1 13 15;7 8 22;8 9 22;
9 22 24;9 10 24; 10 19 24; 19 20 24;7 19 20; 7 20 22;13 14 18;
14 15 16;16 17 18;20 21 25;21 22 23;23 24 25;14 26 28;
16 26 27;18 27 28; 21 29 31;23 29 30;25 30 31;
26 27 28; 29 30 31]; %triangular subregions
f962 = ’(norm([x y]+[0.5 0.5])<0.01)-(norm([x y]-[0.5 0.5]) < 0.01)’;
f=inline(f962,’x’,’y’); %(E9.6.2)
g=inline(’0’,’x’,’y’);
N_n = size(N,1); %the total number of nodes
N_i = N_n - N_b; %the number of interior nodes
c = zeros(1,N_n); %boundary value or 0 for boundary/interior nodes
p = fem_basis_ftn(N,S);
[U,c] = fem_coef(f,g,p,c,N,S,N_i);
%Output through the triangular mesh-type graph
figure(1), clf, trimesh(S,N(:,1),N(:,2),c)
%Output through the rectangular mesh-type graph
N_s = size(S,1); %the total number of subregions(triangles)
x0=-1; xf=1;y0=-1; yf=1;
Mx = 16; dx = (xf - x0)/Mx; xi = x0+[0:Mx]*dx;
My = 16; dy = (yf - y0)/My; yi = y0+[0:My]*dy;
for i = 1:length(xi)
for j = 1:length(yi)
for s = 1:N_s %which subregion the point belongs to
if inpolygon(xi(i),yi(j), N(S(s,:),1),N(S(s,:),2)) > 0
Z(i,j) = U(s,:)*[1 xi(i) yi(j)]’; %Eq.(9.4.5b)
break;
end
end
end
end
figure(2), clf, mesh(xi,yi,Z)
%For comparison
bx0 = inline(’0’); bxf = inline(’0’);
by0 = inline(’0’); byf = inline(’0’);
[U,x,y] = poisson(f,g,bx0,bxf,by0,byf,[x0 xf y0 yf],Mx,My);
figure(3), clf, mesh(x,y,U)

9.5 GUI OF MATLAB FOR SOLVING PDES: PDETOOL

In this section, we will see what problems can be solved by using the GUI (graphic
user interface) tool of MATLAB for PDEs and then apply the tool to solve the
elliptic/parabolic/hyperbolic equations dealt with in Examples 9.1/9.3/9.5 and 9.6.