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Amira Abdel-Rahman
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# Amira Abdel-Rahman
# (c) Massachusetts Institute of Technology 2020
#####################################FEA KE####################################################################
function lk()
E=1
nu=0.3
k=[1/2-nu/6,1/8+nu/8,-1/4-nu/12,-1/8+3*nu/8,-1/4+nu/12,-1/8-nu/8,nu/6,1/8-3*nu/8]
KE = E/(1-nu^2)*[ k[0+1] k[1+1] k[2+1] k[3+1] k[4+1] k[5+1] k[6+1] k[7+1];
k[1+1] k[0+1] k[7+1] k[6+1] k[5+1] k[4+1] k[3+1] k[2+1];
k[2+1] k[7+1] k[0+1] k[5+1] k[6+1] k[3+1] k[4+1] k[1+1];
k[3+1] k[6+1] k[5+1] k[0+1] k[7+1] k[2+1] k[1+1] k[4+1];
k[4+1] k[5+1] k[6+1] k[7+1] k[0+1] k[1+1] k[2+1] k[3+1];
k[5+1] k[4+1] k[3+1] k[2+1] k[1+1] k[0+1] k[7+1] k[6+1];
k[6+1] k[3+1] k[4+1] k[1+1] k[2+1] k[7+1] k[0+1] k[5+1];
k[7+1] k[2+1] k[1+1] k[4+1] k[3+1] k[6+1] k[5+1] k[0+1] ];
return (KE)
end
function lk_H8(nu)
A = [32 6 -8 6 -6 4 3 -6 -10 3 -3 -3 -4 -8;
-48 0 0 -24 24 0 0 0 12 -12 0 12 12 12];
k = 1/144*A'*[1; nu];
K1 = [k[1] k[2] k[2] k[3] k[5] k[5];
k[2] k[1] k[2] k[4] k[6] k[7];
k[2] k[2] k[1] k[4] k[7] k[6];
k[3] k[4] k[4] k[1] k[8] k[8];
k[5] k[6] k[7] k[8] k[1] k[2];
k[5] k[7] k[6] k[8] k[2] k[1]];
K2 = [k[9] k[8] k[12] k[6] k[4] k[7];
k[8] k[9] k[12] k[5] k[3] k[5];
k[10] k[10] k[13] k[7] k[4] k[6];
k[6] k[5] k[11] k[9] k[2] k[10];
k[4] k[3] k[5] k[2] k[9] k[12]
k[11] k[4] k[6] k[12] k[10] k[13]];
K3 = [k[6] k[7] k[4] k[9] k[12] k[8];
k[7] k[6] k[4] k[10] k[13] k[10];
k[5] k[5] k[3] k[8] k[12] k[9];
k[9] k[10] k[2] k[6] k[11] k[5];
k[12] k[13] k[10] k[11] k[6] k[4];
k[2] k[12] k[9] k[4] k[5] k[3]];
K4 = [k[14] k[11] k[11] k[13] k[10] k[10];
k[11] k[14] k[11] k[12] k[9] k[8];
k[11] k[11] k[14] k[12] k[8] k[9];
k[13] k[12] k[12] k[14] k[7] k[7];
k[10] k[9] k[8] k[7] k[14] k[11];
k[10] k[8] k[9] k[7] k[11] k[14]];
K5 = [k[1] k[2] k[8] k[3] k[5] k[4];
k[2] k[1] k[8] k[4] k[6] k[11];
k[8] k[8] k[1] k[5] k[11] k[6];
k[3] k[4] k[5] k[1] k[8] k[2];
k[5] k[6] k[11] k[8] k[1] k[8];
k[4] k[11] k[6] k[2] k[8] k[1]];
K6 = [k[14] k[11] k[7] k[13] k[10] k[12];
k[11] k[14] k[7] k[12] k[9] k[2];
k[7] k[7] k[14] k[10] k[2] k[9];
k[13] k[12] k[10] k[14] k[7] k[11];
k[10] k[9] k[2] k[7] k[14] k[7];
k[12] k[2] k[9] k[11] k[7] k[14]];
KE = 1/((nu+1)*(1-2*nu))*[ K1 K2 K3 K4;K2' K5 K6 K3';K3' K6 K5' K2';K4 K3 K2 K1'];
return KE
end
## SUB FUNCTION: elementMatVec2D
function elementMatVec2D(a, b, DH)
GaussNodes = [-1/sqrt(3); 1/sqrt(3)];
GaussWeigh = [1 1];
L = [1 0 0 0; 0 0 0 1; 0 1 1 0];
Ke = zeros(8,8);
for i = 1:2
for j = 1:2
GN_x = GaussNodes[i];
GN_y = GaussNodes[j];
dN_x = 1/4*[-(1-GN_x) (1-GN_x) (1+GN_x) -(1+GN_x)];
dN_y = 1/4*[-(1-GN_y) -(1+GN_y) (1+GN_y) (1-GN_y)];
J = [dN_x; dN_y]*[ -a a a -a; -b -b b b]';
G = [inv(J) zeros(size(J)); zeros(size(J)) inv(J)];
dN=zeros(4,8)
dN[1,1:2:8] = dN_x;
dN[2,1:2:8] = dN_y;
dN[3,2:2:8] = dN_x;
dN[4,2:2:8] = dN_y;
Be = L*G*dN;
Ke = Ke + GaussWeigh[i]*GaussWeigh[j]*det(J)*Be'*DH*Be;
end
end
return Ke
end
## SUB FUNCTION: elementMatVec3D
function elementMatVec3D(a, b, c, DH)
GN_x=[-1/sqrt(3),1/sqrt(3)]; GN_y=GN_x; GN_z=GN_x; GaussWeigh=[1,1];
Ke = zeros(24,24); L = zeros(6,9);
L[1,1] = 1; L[2,5] = 1; L[3,9] = 1;
L[4,2] = 1; L[4,4] = 1; L[5,6] = 1;
L[5,8] = 1; L[6,3] = 1; L[6,7] = 1;
# display(L)
for ii=1:length(GN_x)
for jj=1:length(GN_y)
for kk=1:length(GN_z)
x = GN_x[ii];y = GN_y[jj];z = GN_z[kk];
dNx = 1/8*[-(1-y)*(1-z) (1-y)*(1-z) (1+y)*(1-z) -(1+y)*(1-z) -(1-y)*(1+z) (1-y)*(1+z) (1+y)*(1+z) -(1+y)*(1+z)];
dNy = 1/8*[-(1-x)*(1-z) -(1+x)*(1-z) (1+x)*(1-z) (1-x)*(1-z) -(1-x)*(1+z) -(1+x)*(1+z) (1+x)*(1+z) (1-x)*(1+z)];
dNz = 1/8*[-(1-x)*(1-y) -(1+x)*(1-y) -(1+x)*(1+y) -(1-x)*(1+y) (1-x)*(1-y) (1+x)*(1-y) (1+x)*(1+y) (1-x)*(1+y)];
J = [dNx;dNy;dNz]*[ -a a a -a -a a a -a ; -b -b b b -b -b b b; -c -c -c -c c c c c]';
# display(a)
# display(b)
# display(c)
# display(dNx)
# display(dNy)
# display(dNz)
# display(J)
G = [inv(J) zeros(3,3) zeros(3,3);zeros(3,3) inv(J) zeros(3,3);zeros(3,3) zeros(3,3) inv(J)];
dN=zeros(9,24)
dN[1,1:3:24] = dNx; dN[2,1:3:24] = dNy; dN[3,1:3:24] = dNz;
dN[4,2:3:24] = dNx; dN[5,2:3:24] = dNy; dN[6,2:3:24] = dNz;
dN[7,3:3:24] = dNx; dN[8,3:3:24] = dNy; dN[9,3:3:24] = dNz;
Be = L*G*dN;
# display((G))
# display(size(dN))
# display((Be))
Ke .= Ke .+ GaussWeigh[ii]*GaussWeigh[jj]*GaussWeigh[kk]*det(J)*(Be'*DH*Be);
end
end
end
# display(Ke)
return Ke
end
function evaluateCH(CH,dens)
U,S,V = svd(CH);
sigma = S;
k = sum(sigma .> 1e-15);
SH = (U[:,1:k] * diagm(0=>(1 ./sigma[1:k])) * V[:,1:k]')'; # more stable SVD (pseudo)inverse
EH = [1/SH[1,1], 1/SH[2,2], 1/SH[3,3]]; # effective Young's modulus
GH = [1/SH[4,4], 1/SH[5,5], 1/SH[6,6]]; # effective shear modulus
vH = [-SH[2,1]/SH[1,1] -SH[3,1]/SH[1,1] -SH[3,2]/SH[2,2];
-SH[1,2]/SH[2,2] -SH[1,3]/SH[3,3] -SH[2,3]/SH[3,3]]; # effective Poisson's ratio
props = Dict("CH"=>CH, "SH"=>SH, "EH"=>EH, "GH"=>GH, "vH"=>vH, "density"=>dens);
if true
println("\n--------------------------EFFECTIVE PROPERTIES--------------------------\n")
println("Youngs Modulus:____E11_____|____E22_____|____E33_____\n")
println(" $(EH[1]) | $(EH[2]) | $(EH[3])\n\n")
println("Shear Modulus:_____G23_____|____G31_____|____G12_____\n")
println(" $(GH[1]) | $(GH[2]) | $(GH[3])\n\n")
println("Poissons Ratio:____v12_____|____v13_____|____v23_____\n")
println(" $(vH[1,1]) | $(vH[1,2]) | $(vH[1,3])\n\n")
println(" ____v21_____|____v31_____|____v32_____\n")
println(" $(vH[2,1]) | $(vH[2,2]) | $(vH[2,3])\n\n")
println("------------------------------------------------------------------------")
end
return SH
end