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eso_frequency.m
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eso_frequency.m
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%% cleaning
clc; clear; close all
%% coordinate matrix
nx = 51; % nodes number in x
ny = 11; % nodes number in y
Lx = 5; % length in x [m]
Ly = 1; % length in y [m]
ex = nx - 1; % element number in x
ey = ny - 1; % element number in y
coordx = linspace(0,Lx,nx); % coordinate in x
coordy = linspace(0,Ly,ny); % coordinate in y
[X,Y] = meshgrid(coordx,coordy);
X = X';
Y = Y';
num = 1:nx*ny; % number of nodes
coord = [num(:),X(:),Y(:)]; % coordinate matrix
%% incidence matrix
inci = zeros(ex*ey,6); % incidence matrix pre-location
A = coord(:,1);
A = reshape(A,[nx,ny]);
k = 1;
for j = 1:ny-1
for i = 1:nx-1
if k > ex*ey/2
mat = 1;
else
mat = 1;
end
inci(k,:)=[mat, A(i,j),A(i+1,j),A(i+1,j+1),A(i,j+1),k];
k = k + 1;
end
end
%% plot
figure()
patch('Faces',inci(:,2:5),'Vertices',coord(:,2:3),'FaceColor','blue')
axis equal
title('Otimização Topológica Evolucionária')
ylabel('y')
xlabel('x')
% chr = int2str(coord(:,1));
% text(coord(:,2),coord(:,3),chr)
%% boundary conditions matrix
bc = zeros(2*ny,3);
k = 1;
for i = 1:ny
bc(2*i-1,:) = [k, 1,0];
bc(2*i,:) = [k, 2,0];
k = k + nx;
end
k = nx;
for i = ny+1:2*ny
bc(2*i-1,:) = [k, 1,0];
bc(2*i,:) = [k, 2,0];
k = k + nx;
end
%% plot of nodes
% figure()
% scatter(coord(:,2),coord(:,3),'x')
% axis equal
% hold on
% scatter(coord(bc(:,1),2),coord(bc(:,1),3),'o','green')
%% material matrix
%E rho nu
material = [200e9 7000 0.3; %steel [Pa] [kg/m³] []
70e9 2700 0.27]; %aluminium
%% solver
nel = size(inci,1); % element number
nnodes = size(coord,1);% nodes number
alldof = 1:nnodes*2; % degrees of freedom
kg = zeros(2*nnodes); % global stiffness matrix pre-location
mg = zeros(2*nnodes); % global mass matrix pre-location
for i = 1:nel
no1 = inci(i,2); % first node element
no2 = inci(i,3); % second node element
no3 = inci(i,4); % third node element
no4 = inci(i,5); % fourth node element
E = material(inci(i,1),1); % young's module
rho = material(inci(i,1),2); % density
nu = material(inci(i,1),3); % poisson module
D = E*[1/(1-nu^2), nu/(1-nu^2), 0;
nu/(1-nu^2), 1/(1-nu^2),0;
0, 0, 1/(2*(1+nu))];
line = inci(1,:);
posxy = coord(line(2:length(line)-1),2:3);
% Gauss points
np=2;
pxi=[-1/sqrt(3),1/sqrt(3)];
peta=[-1/sqrt(3),1/sqrt(3)];
wi=1;
wj=1;
Nfun=4;
ke=zeros(2*Nfun);
Bs=zeros(3,2*Nfun);
teste = zeros(4);
teste2 = zeros(4);
teste3 = zeros(4);
for j=1:np
xi=pxi(j);
for k=1:np
eta=peta(k);
N=[1/4*(1-xi)*(1-eta), 1/4*(1+xi)*(1-eta), 1/4*(1+xi)*(1+eta), 1/4*(1-xi)*(1+eta)];
dNxi=[eta/4 - 1/4,1/4 - eta/4,eta/4 + 1/4,- eta/4 - 1/4];
dNeta=[xi/4 - 1/4,- xi/4 - 1/4,xi/4 + 1/4,1/4 - xi/4];
J=[dNxi;dNeta]*posxy;
detJ=det(J);
iJ=J^-1;
p = 1:2:8;
B = iJ*[dNxi;dNeta];
Bs(1,p) = iJ(1,1)*dNxi + iJ(1,2)*dNeta;
Bs(2,p+1) = iJ(2,1)*dNxi + iJ(2,2)*dNeta;
Bs(3,p) = Bs(2,p+1);
Bs(3,p+1) = Bs(1,p);
ke = ke + 0.1*Bs'*D*Bs*det(J)*wi*wj; % Elementar stiffness matrix
teste = teste + B(1,:)'*B(1,:)*detJ;
teste2 = teste2 + B(2,:)'*B(2,:)*detJ;
teste3 = teste3 + N'*N*detJ;
end
end
a = coord(no2,2) - coord(no1,2);
b = coord(no3,3) - coord(no2,3);
Area = a*b;
me = 0.1*rho*Area/36*[4 0 2 0 1 0 2 0;
0 4 0 2 0 1 0 2;
2 0 4 0 2 0 1 0;
0 2 0 4 0 2 0 1;
1 0 2 0 4 0 2 0;
0 1 0 2 0 4 0 2;
2 0 1 0 2 0 4 0;
0 2 0 1 0 2 0 4];
loc = [no1*2-1 no1*2 no2*2-1 no2*2 no3*2-1 no3*2 no4*2-1 no4*2]; % localization vector
kg(loc,loc) = kg(loc,loc) + ke; % global stiffness matrix assemble
mg(loc,loc) = mg(loc,loc) + me; % global stiffness matrix assemble
end
freedof = alldof';
for k = 1 : size(bc,1)
freedof(2*bc(k,1)-(2-bc(k,2))) = 0;
end
%% eso frequency optimization
count = 0; % iteration count
x=ones(nel,1); % elements presence
mode = 1; % vibration mode
alpha_max = zeros(1000,1);
frequency = zeros(1000,4);
nel_del = 4;
while 1
count = count + 1; % iteration count
kg_aux = sparse(kg(logical(freedof),logical(freedof))); % column & rows elimination
mg_aux = sparse(mg(logical(freedof),logical(freedof))); % column & rows elimination
[eig_vector,eig_value] = eigs(kg_aux,mg_aux,30,'SM');
eig_value = sum(eig_value);
eig_aux = [eig_value;eig_vector]';
eig_aux = sortrows(eig_aux,1);
omega = eig_aux(mode,1);
eig_vector = eig_aux(mode,2:end)';
eig_vector_aux = zeros(size(kg,1),1);
eig_vector_aux(logical(freedof),1) = eig_vector;
alpha = zeros(nel,2);
frequency(count,1) = sqrt(omega)/(2*pi);
frequency(count,2) = sqrt(eig_aux(2,1))/(2*pi);
frequency(count,3) = sqrt(eig_aux(3,1))/(2*pi);
frequency(count,4) = sqrt(eig_aux(4,1))/(2*pi);
for i = 1:nel
no1 = inci(i,2); % first node element
no2 = inci(i,3); % second node element
no3 = inci(i,4); % third node element
no4 = inci(i,5); % fourth node element
element = [no1*2-1 no1*2 no2*2-1 no2*2 no3*2-1 no3*2 no4*2-1 no4*2];
u = eig_vector_aux(element,1);
alpha(i,1) = i;
alpha(i,2) = u'*(omega*me-ke)*u;
end
if sum(alpha(:,2)) ~= 0
print('Soma diferente de zero')
end
alpha_max(count,1) = max(alpha((x == 1),2));
alpha = alpha((x == 1),:);
alpha_aux = sortrows(alpha,-2);
remove_element_aux = alpha_aux(1:nel_del,1);
x(remove_element_aux) = 0;
inci_aux = inci(remove_element_aux,:);
patch('Faces',inci_aux(:,2:5),'Vertices',coord(:,2:3),'FaceColor','white')
pause(1e-7)
for j = 1:nel_del
i = remove_element_aux(j);
no1 = inci(i,2); % first node element
no2 = inci(i,3); % second node element
no3 = inci(i,4); % third node element
no4 = inci(i,5); % fourth node element
E = material(inci(i,1),1); % young's module
nu = material(inci(i,1),3); % poisson module
D = E*[1/(1-nu^2), nu/(1-nu^2), 0;
nu/(1-nu^2), 1/(1-nu^2),0;
0, 0, 1/(2*(1+nu))];
line = inci(1,:);
posxy = coord(line(2:length(line)-1),2:3);
% Gauss points
np=2;
pxi=[-1/sqrt(3),1/sqrt(3)];
peta=[-1/sqrt(3),1/sqrt(3)];
wi=1;
wj=1;
Nfun=4;
ke=zeros(2*Nfun);
Bs=zeros(3,2*Nfun);
for j=1:np
xi=pxi(j);
for k=1:np
eta=peta(k);
N=[1/4*(1-xi)*(1-eta), 1/4*(1+xi)*(1-eta), 1/4*(1+xi)*(1+eta), 1/4*(1-xi)*(1+eta)];
dNxi=[eta/4 - 1/4,1/4 - eta/4,eta/4 + 1/4,- eta/4 - 1/4];
dNeta=[xi/4 - 1/4,- xi/4 - 1/4,xi/4 + 1/4,1/4 - xi/4];
J=[dNxi;dNeta]*posxy;
detJ=det(J);
iJ=J^-1;
p = 1:2:8;
Bs(1,p) = iJ(1,1)*dNxi + iJ(1,2)*dNeta;
Bs(2,p+1) = iJ(2,1)*dNxi + iJ(2,2)*dNeta;
Bs(3,p) = Bs(2,p+1);
Bs(3,p+1) = Bs(1,p);
ke = ke + 0.1*Bs'*D*Bs*det(J)*wi*wj; % Elementar stiffness matrix
end
end
loc = [no1*2-1 no1*2 no2*2-1 no2*2 no3*2-1 no3*2 no4*2-1 no4*2]; % localization vector
kg(loc,loc) = kg(loc,loc) - ke; % global stiffness matrix assemble
ke = 1e-3*ke;
kg(loc,loc) = kg(loc,loc) + ke; % global stiffness matrix assemble
me = 0.1*rho*Area/36*[4 0 2 0 1 0 2 0;
0 4 0 2 0 1 0 2;
2 0 4 0 2 0 1 0;
0 2 0 4 0 2 0 1;
1 0 2 0 4 0 2 0;
0 1 0 2 0 4 0 2;
2 0 1 0 2 0 4 0;
0 2 0 1 0 2 0 4];
mg(loc,loc) = mg(loc,loc) - me; % global stiffness matrix assemble
me = 1e-3*me;
mg(loc,loc) = mg(loc,loc) + me; % global stiffness matrix assemble
end
if count == 50
break
end
end
figure()
hold on
count = 1:count;
frequency_1 = frequency(count,1);
plot(count,frequency_1,'r');
frequency_2 = frequency(count,2);
plot(count,frequency_2,'b');
frequency_3 = frequency(count,3);
plot(count,frequency_3,'k');
frequency_4 = frequency(count,4);
plot(count,frequency_4,'g');
legend('\Omega_{1}','\Omega_{2}','\Omega_{3}','\Omega_{4}')