MLS2D.m

% two-DIMENSIONAL MLS APPROXIMATION
% by Jin Jia

clc
clear all

   



% SET UP NODAL COORDINATES 
[xI,yI] = meshgrid(-2: 0.5 : 2);
nnodes = length(xI)*length(yI);

% SET UP COORDINATES OF EVALUATION POINTS
[x,y] =meshgrid (-2 : 0.1: 2);
npoints = length(x)*length(y);
scale = 3;
% DETERMINE RADIUS OF SUPPORT OF EVERY NODE
dmI = scale *0.5* ones(1, nnodes);

% Evaluate MLS shape function at all evaluation points x
[PHI, DPHIx, DPHIy] = MLS2DShape(3, nnodes, xI,yI, npoints, x,y, dmI, 'GAUSS', 3.0 ); 




% Curve fitting. y = peaks(x,y)
ZI  =xI.*exp(-xI.^2- yI.^2);    % Nodal function values 
z  =x.*exp(-x.^2- y.^2);% Exact solution
Zpoints=zeros(1,npoints);
for i=1:npoints
Zpoints(1,i)=z(i);
end
Znodes=zeros(1,nnodes);
for i=1:nnodes
    Znodes(1,i)=ZI(i);
end                        %conveerse two-dimension data to one-dimension data 
zh = PHI *Znodes';  % Approximation function
err = norm(Zpoints' - zh) / norm(Zpoints) * 100;% Relative error norm in approximation function

 %converse one-dimension data to two-dimension data 
Zponintsh=reshape(zh,length(y),length(x)); 
%

 [dZx,dZy]= gradient(z,0.1,0.1);
ddzx=reshape(dZx,1,npoints);
ddzy=reshape(dZy,1,npoints);

dZhx  = DPHIx * Znodes';  % First order derivative of approximation function
errd = norm(ddzx' - dZhx) / norm(ddzx) * 100;  % Relative error norm in the first order derivative

dZponintshx=reshape(dZhx,length(y),length(x));




dZhy  = DPHIy * Znodes';  % Second order derivative of approximation function
errdd = norm(ddzy' - dZhy) / norm(ddzy) * 100;  % Relative error norm in second order derivative

dZponintshy=reshape(dZhy,length(y),length(x));