Finite Elements

%%%%%%%%%%%%%%%%%%%%%%%%%%
    Triangulo
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% Carteciana %%%%%%%%
Syms x1 y1 L K
nn=[(-1/K)*(y+(K/L)*(x-x1-L)),(x-x1)/L,(y+y1)/K]
for i=1:3
    dn(1,i)=diff(nn(i),x);
end
for i=1:3
    dn(2,i)=diff(nn(i),y);
end

% por coordenada, primero x
paso1=dn(1,:);
paso1p=transpose(paso1);
kex=paso1p*paso1;
for i=1:3
    for j=1:3
        kx(i,j)=int(int(kex(i,j),y,y1,y1-(K/L)*(x-x1-L)),x,x1,x1+L);
    end
end
kx

% ahora coordenada y
paso2=dn(2,:);
paso2p=transpose(paso2);
key=paso2p*paso2;
for i=1:3
    for j=1:3
        ky(i,j)=int(int(key(i,j),y,y1,y1-(K/L)*(x-x1-L)),x,x1,x1+L);
    end
end
ky

for i=1:3
    for j=1:3
        kxs(i,j)=simplify(kx(i,j));
        kys(i,j)=simplify(ky(i,j));
    end
end
kxs
kys


%%%%%%%%%%%%%%%%%%%%%%%%%%
    Cuadrado
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% Cartesianas %%%%%%%
syms x y L K
N=(1/L/K)*[(x-L)*(y-K),-(y-K)*(x),(x*y),-(x-L)*(y)]

fila1=diff(N,x);
fila2=diff(N,y);


kx=int(int(transpose(fila1)*fila1,y,0,K),x,0,L);
ky=int(int(transpose(fila2)*fila2,y,0,K),x,0,L);

fe=int(int(transpose(N),y,0,K),x,0,L)
Kg=int(int(transpose(N)*N,y,0,K),x,0,L);

%%%%%%% Naturales %%%%%%%
syms e n L K
%L=K
N=(1/4)*[(1-e)*(1-n),(1+e)*(1-n),(1+e)*(1+n),(1-e)*(1+n)]

xcoor=[0,0;L,0;L,K;0,K]
Dn= [diff(N,e);diff(N,n)]

jac=Dn*xcoor;
jacinv=inv(jac);
detjac=det(jac);
B=jacinv*Dn;

% por coordenada primero x
filax=B(1,:);
Kex=int(int(detjac*transpose(filax)*filax,e,-1,1),n,-1,1)

% por coordenada segundo y
filay=B(2,:);
Key=int(int(detjac*transpose(filay)*filay,e,-1,1),n,-1,1)

f=int(int(detjac*N,e,-1,1),n,-1,1)
KG=int(int(detjac*transpose(N)*N,e,-1,1),n,-1,1)