How to perform numerical analysisis using the Finite Difference Method on a square plate with a corner removed.

Hi, I'm curious to know if it is possible to adapt an existing code, which is used to analyse a square plate with dimensions 'hx+1 X hy+1', so that I can use FDM on a the same plate but with the corner removed.

The idea would be that the boundary conditions could be applied to the point (1,hy).

Below is included the current script that I would like to adapt. The dimensions of the plate are defined from line 20, and the boundary conditions are defined at line 72.

 Any help would be greatly appreciated.

1>>

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2>>  %%%%%%%   An Introduction to Scientific Computing          %%%%%%%
3>>  %%%%%%%   I. Danaila, P. Joly, S. M. Kaber & M. Postel     %%%%%%%
4>>  %%%%%%%                Springer, 2005                      %%%%%%%
5>>  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
6>>  %%
7>>  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
8>>  %%
9>>  %%  Matlab Solution of exercise 2 - project 7
10>>  %%  ELAS: elastic deformation of a membrane
11>>  %%  Solution of the microphone problem (non-linear equation)
12>>  %%
13>>  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
14>>  %%
15>>  clear all; close all;
16>>  %
17>>  %    1) Construction of the linear system
18>>  %
19>>  %    number of points
20>>  nx=18;ny=18;
21>>  %    step mesh size
22>>  %    dimensions of the plate 1 mm x 1 mm (unit m) 
23>>  long=1.e-03;large=1.e-03;
24>>  hx=long/(nx+1);hy=large/(ny+1);
25>>  h2x=hx*hx;h2y=hy*hy;
26>>  h4x=h2x*h2x;h4y=h2y*h2y;h4xy=h2x*h2y;
27>>  %    points coordinates 
28>>  x=[0.:hx:long];y=[0.:hy:large];
29>>  n=nx*ny;
30>>  fprintf('\n ');
31>>  fprintf('\n Points in x direction %d',nx);
32>>  fprintf('\n Points in y direction %d',ny);
33>>  fprintf('\n Total number of points %d',n);
34>>  fprintf('\n ');
35>>  %    computing time measure 
36>>  t=cputime;   
37>>  %    matrix of realistic problem
38>>  %    plate mechanical strain 
39>>  T=100.; % in Newton/m 
40>>  %    plate Young modulus (silicium)
41>>  E=1.3e+11; % in Newton/m2
42>>  %    plate thickness
43>>  e=1.e-06; % 1 up to 10 microns  
44>>  %    Poisson coefficient
45>>  nu=0.25; %  no dimension 
46>>  c1=T;c2=(E*e*e*e)/(12.*(1.0-nu*nu));
47>>  %    Laplacian operator matrix
48>>  Ah5=ELAS_lap_matrix(hx,hy,nx,ny);
49>>  %    Bi-Laplacian operator matrix
50>>  Ah13=ELAS_bilap_matrix(hx,hy,nx,ny);
51>>  %    Test problem matrix
52>>  Ah=c2*Ah13+c1*Ah5;
53>>  %    Cholesky factorization - Matlab
54>>  Lh=chol(Ah');
55>>  %    Display of matrix structure
56>>  fs=18;
57>>  figure(10);colormap('gray');
58>>  spy(Ah);title('Matrix Ah ','FontSize',fs);
59>>  figure(11);colormap('gray');
60>>  spy(Lh');title('Matrix Lh ','FontSize',fs);
61>>  %    sampling of the right-hand side
62>>  rhs0=zeros(nx*ny,1);
63>>  for ix=1:nx
64>>  for iy=1:ny
65>>  xx=x(ix+1);
66>>  yy=y(iy+1);
67>>  uu=0.;
68>>  rhs0((iy-1)*nx+ix)=ELAS_pressure(xx,yy,uu);
69>>  end
70>>  end
71>>  %    boundary conditions 
72>>  uh=zeros(nx+2,ny+2);
73>>  %    uh is null on the boundary ==> no correction of rhs (cf ELAS_plate_ex.m)
74>>  %
75>>  %    2) Solution of the linear problem
76>>  %
77>>  vh=Lh'\rhs0;
78>>  wh=Lh\vh;
79>>  for ix=1:nx
80>>  for iy=1:ny
81>>  uh(ix+1,iy+1)=wh((iy-1)*nx+ix);
82>>  end
83>>  end
84>>  %
85>>  %    3) Display of results
86>>  %
87>>  %    capacitor thickness
88>>  h=5.e-06; % 5 microns
89>>  figure(11);
90>>  colormap('gray');surf(x,y,h-uh');
91>>  fs=18;
92>>  title('Solution for the plate problem','FontSize',fs);
93>>  %
94>>  %    4)  Solution of the nonlinear problem
95>>  %
96>>  eps0=1.0e-03;
97>>  var=1.0;
98>>  %    initialisation 
99>>  uuh=uh;                % uh = solution of linear problem
100>>  rhs=zeros(nx*ny,1);
101>>  k=0;
102>>  while ( var > eps0 )
103>>  k=k+1;
104>>  uuha=uuh;
105>>  for ix=1:nx
106>>  for iy=1:ny
107>>  xx=x(ix+1);
108>>  yy=y(iy+1);
109>>  uu=uuh(ix+1,iy+1);
110>>  rhs((iy-1)*nx+ix)=ELAS_pressure(xx,yy,uu);
111>>  end
112>>  end
113>>  vh=Lh'\rhs;
114>>  wh=Lh\vh;
115>>  for ix=1:nx
116>>  for iy=1:ny
117>>  uuh(ix+1,iy+1)=wh((iy-1)*nx+ix);
118>>  end
119>>  end
120>>  %    variation
121>>  e1=norm(uuh,2);
122>>  e2=norm(uuh-uuha,2);
123>>  var=e2/e1;
124>>  fprintf('\n Iteration  %d',k);
125>>  fprintf('\n Variation  %12.8f',var);
126>>  end
127>>  %    computing time measure 
128>>  time=cputime-t;   
129>>  fprintf('\n Computing time      %20.15f',time); 
130>>  %
131>>  %    5) Display of results
132>>  %
133>>  figure(12);
134>>  colormap('gray');surf(x,y,h-uuh');
135>>  fs=18;
title('Solution for the microphone problem','FontSize',fs);