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Weight2D.m
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Weight2D.m
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function [w, dwdx, dwdy] = Weight2D(type, para, x,y,xI,yI,dmI)
% EVALUATE WEIGHT FUNCTION
%
% SYNTAX: [w, dwdr, dwdrr] = GaussWeight(type, para, di, dmi)
%
% INPUT PARAMETERS
% type - Type of weight function
% para - Weight function parameter
% x,y - gauss point coordinates matrix
% xI,yI - nodal point coordinate
% dmI - Support size
% OUTPUT PARAMETERS
% w - Value of weight function at r
% dwdx - Value of first order derivative of weight function with respect to x at r
% dwdy - Value of first order derivative of weight function with respect to y at r
r = sqrt((x-xI).^2+(y-yI).^2)/ dmI;
nnodes_x=length(x);
nnodes_y=length(y);
w=zeros(nnodes_x,nnodes_y);
dwdx=zeros(nnodes_x,nnodes_y);
dwdy=zeros(nnodes_x,nnodes_y);
drdx=zeros(nnodes_x,nnodes_y);
drdy=zeros(nnodes_x,nnodes_y);
dwdr=zeros(nnodes_x,nnodes_y);
dwdr=zeros(nnodes_x,nnodes_y);
for j=1:nnodes_y
for i=1:nnodes_x
if(r(i,j)==0)
drdx(i,j)=0;
drdy(i,j)=0;
else
drdx(i,j) = x(i,j)/(dmI.^2*r(i,j));
drdy(i,j)=y(i,j)/(dmI.^2*r(i,j));
end
% EVALUATE WEIGHT FUNCTION AND ITS FIRST AND SECOND ORDER OF DERIVATIVES WITH RESPECT r AT r
if (type == 'GAUSS')
[w(i,j),dwdr(i,j)] = Gauss(para,r(i,j));
elseif (type == 'CUBIC')
[w(i,j),dwdr(i,j)] = Cubic(r(i,j));
elseif (type == 'SPLI3')
[w(i,j),dwdr(i,j)] = Spline3(r(i,j));
elseif (type == 'SPLI5')
[w(i,j),dwdr(i,j)] = Spline5(r(i,j));
elseif (type == 'SPLIB')
[w(i,j),dwdr(i,j)] = BSpline(dmI/2,r(i,j));
elseif (type == 'power')
[w(i,j),dwdr(i,j)] = power_function(para,r(i,j));
elseif (type == 'CRBF1')
[w(i,j),dwdr(i,j)] = CSRBF1(r(i,j));
elseif (type == 'CRBF2')
[w(i,j),dwdr(i,j)] = CSRBF2(r(i,j));
elseif (type == 'CRBF3')
[w(i,j),dwdr(i,j)] = CSRBF3(r(i,j));
elseif (type == 'CRBF4')
[w(i,j),dwdr(i,j)] = CSRBF4(r(i,j));
elseif (type == 'CRBF5')
[w(i,j),dwdr(i,j)] = CSRBF5(r(i,j));
elseif (type == 'CRBF6')
[w(i,j),dwdr(i,j)] = CSRBF6(r(i,j));
else
error('Invalid type of weight function.');
end
end
end
dwdx = dwdr .* drdx;
dwdy= dwdr .* drdy;
% shape function
function [w,dwdr] = Gauss(beta,r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
b2 = beta*beta;
r2 = r*r;
eb2 = exp(-b2);
w = (exp(-b2*r2) - eb2) / (1.0 - eb2);
dwdr = -2*b2*r*exp(-b2*r2) / (1.0 - eb2);
end
function [w,dwdr] = Cubic(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
w = 1-6*r^2+8*r^3-3*r^4;
dwdr = -12*r+24*r^2-12*r^3;
end
function [w,dwdr] = Spline3(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
elseif (r<=0.5)
w = 2/3 - 4*r^2 + 4*r^3;
dwdr = -8*r + 12*r^2;
else
w = 4/3 - 4*r + 4*r^2 - 4*r^3/3;
dwdr = -4 + 8*r -4*r^2;
end
function [w,dwdr] = Spline5(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
w = 1-10*r^3+15*r^4-6*r^5;
dwdr = -30*r^2 + 60*r^3-30*r^4;
end
function [w,dwdr] = BSpline(h,r)%h is the distance between two nodes.
if (r>1.0)
w = 0.0;
dwdr = 0.0;
elseif (r<=0.5)
w = 1/(pi*h^3)*(1-6*r^2+6*r^3);
dwdr = 1/(pi*h^3)*(-12*r+18*r^2);
else
w = 2/(pi*h^3)*(1-r)^3;
dwdr = -6/(pi*h^3)*(1-r)^2;
end
function [w,dwdr] = power_function(arfa,r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
a2 = arfa*arfa;
r2 = r*r;
w = exp(-r2/a2);
dwdr = (-2*r/a2)*exp(-r2/a2);
end
function [w,dwdr] = CSRBF2(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
w = (1-r)^6*(6+36*r+82*r^2+72*r^3+30*r^4+5*r^5);
dwdr = 11*r*(r+2)*(5*r^3+15*r^2+18*r+4)*(r-1)^5;
end
function [w,dwdr] = CSRBF1(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
w = (1-r)^4*(4+16*r+12*r^2+3*r^3);
dwdr = -4*(1-r)^3*(4+16*r+12*r^2+3*r^3)+(1-r)^4*(16+24*r+9*r^2);
end
function [w,dwdr] = CSRBF3(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
w = 1/3+r^2-4/3*r^3+2*r^2*log(r);
dwdr = 4*r-4*r^2+4*r*log(r);
end
function [w,dwdr] = CSRBF4(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
w = 1/15+19/6*r^2-16/3*r^3+3*r^4-16/15*r^5+1/6*r^6+2*r^2*log(r);
dwdr = 25/3*r-16*r^2+12*r^3-16/3*r^4+r^5+4*r*log(r);
end
function [w,dwdr] = CSRBF5(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
w = (1-r)^6*(35*r^2+18*r+3);
dwdr =-6*(1-r)^5*(35*r^2+18*r+3)+(1-r)^6*(70*r+18);
end
function [w,dwdr] = CSRBF6(r)
if (r>1.0)
w = 0.0;
dwdr = 0.0;
else
w = (1-r)^8*(32*r^3+25*r^2+8*r+1);
dwdr =-8*(1-r)^7*(32*r^3+25*r^2+8*r+1)+(1-r)^8*(96*r^2+50*r+8);
end