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beso_compliance_geometry01.m
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beso_compliance_geometry01.m
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%% cleaning
clc; clear; close all
%% coordinate matrix
nx = 81; % nodes number in x
ny = 51; % nodes number in y
Lx = 80/1000; % length in x [mm]
Ly = 50/1000; % length in y [mm]
Lz = 1/1000; % length in z [mm]
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
%% boundary conditions matrix
bc = zeros(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
%% load matrix
if mod(ey,2) == 0
k = nx*(ny+1)/2;
else
k = nx*(ny)/2;
end
load = [k 2 -1000];
%% material matrix
%E rho nu
material = [100e9 7000 0.3]; % [Pa] [kg/m�] []
%% solver
nload = size(load,1); %load quantity
nnodes = size(coord,1); % number of nodes
alldof = 1:nnodes*2; % degrees of freedom
nel = size(inci,1); % number of elements
F = zeros(2*nnodes,1); %load matrix pre-location
kg = zeros(2*nnodes); % global stiffness matrix pre-location
for i = 1:nload % load vector
if load(i,2) == 1
F(2*load(i,1)-1) = load(i,3);
else
F(2*load(i,1)) = load(i,3);
end
end
for i = 1 : nel % global stiffness matrix
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
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 = Lz*E/(1-nu^2)*[ k(1) k(2) k(3) k(4) k(5) k(6) k(7) k(8)
k(2) k(1) k(8) k(7) k(6) k(5) k(4) k(3)
k(3) k(8) k(1) k(6) k(7) k(4) k(5) k(2)
k(4) k(7) k(6) k(1) k(8) k(3) k(2) k(5)
k(5) k(6) k(7) k(8) k(1) k(2) k(3) k(4)
k(6) k(5) k(4) k(3) k(2) k(1) k(8) k(7)
k(7) k(4) k(5) k(2) k(3) k(8) k(1) k(6)
k(8) k(3) k(2) k(5) k(4) k(7) k(6) k(1)];
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
end
freedof = alldof;
ke_min = 1e-3*ke;
for k = 1 : size(bc,1) % degrees of freedom
freedof(2*bc(k,1)-(2-bc(k,2))) = 0;
end
F_aux = sparse(F(logical(freedof),1)); % column & rows elimination
u = zeros(size(F_aux,1),1);
%% BESO parameters
Vf = 0.5;
Er = 1/100;
Ar_max = 5/100;
r_min = 3/1000;
tau = 0.01/100;
node_set = zeros(size(inci,1), 100);
node_set_aux = zeros(size(inci,1), size(coord,1));
distance = zeros(size(inci,1), 100);
for i = 1 : size(inci,1)
x_nodes = coord(inci(i,2:5),2); % nodal coordinates in x
y_nodes = coord(inci(i,2:5),3); % nodal coordinates in y
x_c = mean(x_nodes,1); % coordinate of the center of the element in x
y_c = mean(y_nodes,1); % coordinate of the center of the element in y
k = 1;
for j = 1 : size(coord,1)
if (coord(j,2)-x_c)^2 + (coord(j,3)-y_c)^2 <= r_min^2
node_set(i,k) = coord(j,1);
node_set_aux(i,coord(j,1)) = 1;
k = k + 1;
end
end
for j = 1 : length(nonzeros(node_set(i,:)))
distance(i,node_set(i,j)) = r_min - sqrt((coord(node_set(i,j),2)-x_c)^2 + (coord(node_set(i,j),3)-y_c)^2);
end
end
figure()
patch('Faces',inci(:,2:5),'Vertices',coord(:,2:3),'FaceColor','blue')
axis equal
hold on
% chr = int2str(coord(:,1));
% text(coord(:,2),coord(:,3),chr)
%% BESO compliance optimization
presence = ones(nel,1); % elements presence
iter = 1 ; % iteration counter
nel_1 = nel; % volume fraction * n of full elements each iteraction
nel_f = nel * Vf; % n of elements at the end of the optimization
nel_add_max = nel * Ar_max; % max n of elements added each iteraction
complaince = zeros(10,1); % complaince value
historical = zeros(nel,2); % history of the sensivity number vector
elements = ones(nel,2); % presence of elements each two iteractions
complaince_hist = zeros(100,1);
conv = 0;
kk = 0;
tic
while conv == 0
if nel_1 > nel_f
nel_0 = nel_1;
nel_1 = ceil(nel_1*(1-Er));
end
kg_aux = sparse(kg(logical(freedof),logical(freedof))); % column & rows elimination
u(logical(freedof),1) = kg_aux\F_aux; % displacement vector
alpha_elemental = zeros(nel,1); % sensivity number vector
alpha_n = zeros(nnodes,1); % sensitivity nodal number
C = 0; % complaince value
for i = 1 : nel % alpha_elemental / alpha_nodal
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
nodes = [no1,no2,no3,no4]; % set of nodes
loc = [no1*2-1 no1*2 no2*2-1 no2*2 no3*2-1 no3*2 no4*2-1 no4*2]; % localization vector
u_aux = u(loc,1); % elementar displacement vector
if presence(i) == 1
alpha_elemental(i,1) = 0.5*u_aux'*ke*u_aux; % sensivity vector
C = C + alpha_elemental(i,1);
else
alpha_elemental(i,1) = 0.5*u_aux'*ke_min*u_aux; % sensivity vector
C = C + alpha_elemental(i,1);
end
for j = 1:4
x = coord(nodes(j),2); % nodal coordinate in x
y = coord(nodes(j),3); % nodal coordinate in y
if x == 0 || x == Lx
if y == 0 || y == Ly
M = 1;
else
M = 2;
end
elseif y == 0 || y == Ly
M = 2;
else
M = 4;
end
w = 1 / M;
alpha_n(nodes(j)) = alpha_n(nodes(j)) + w * alpha_elemental(i);
end
end
complaince_hist(iter,1) = C;
if iter <= 10
complaince(iter,1) = C;
else
iter_2 = mod(iter,10);
if iter_2 == 0
iter_2 = 10;
end
complaince(iter_2,1) = C;
error = abs(sum(complaince(6:10)) - sum(complaince(1:5)))/sum(complaince(6:10));
end
alpha_i = zeros(nel,1);
for i = 1 : nel
w_rij = nonzeros(distance(i,:));
alpha_j = alpha_n(logical(node_set_aux(i,:)));
alpha_i(i,1) = w_rij' * alpha_j/sum(w_rij);
end % alpha_i
historical(:,1) = historical(:,2);
historical(:,2) = alpha_i;
alpha_mean(:,1) = 1:nel;
alpha_mean(:,2) = mean(historical,2);
alpha_aux_th = sortrows(alpha_mean,-2);
th = round(alpha_aux_th(nel_1,2),10);
remove_elements = alpha_mean( (presence == 1) & alpha_mean(:,2)<= th);
nel_del = length(remove_elements);
add_elements = alpha_mean((presence == 0) & alpha_mean(:,2) > th);
nel_add = length(add_elements);
if nel_add > nel_add_max
nel_add = nel_add_max;
add_elements = alpha_mean(alpha_mean(presence == 0, 2) > th,:);
add_elements = sortrows(add_elements, -2);
add_elements = add_elements(1:nel_add);
nel_del = nel_add + sum(presence)*Er;
remove_elements = alpha_mean(alpha_mean(presence == 1, 2) <= th,:);
remove_elements = sortrows(remove_elements, 2);
remove_elements = remove_elements(1:nel_del);
end
presence(remove_elements(:,1)) = 0;
presence(add_elements(:,1)) = 1;
for j = 1:nel_del
i = remove_elements(j,1);
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
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 + ke_min; % global stiffness matrix assemble
end
inci_aux = inci(remove_elements(:,1),:);
patch('Faces',inci_aux(:,2:5),'Vertices',coord(:,2:3),'FaceColor','white')
pause(1e-6);
for j = 1:nel_add
i = add_elements(j,1);
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
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_min + ke; % global stiffness matrix assemble
end
inci_aux = inci(add_elements(:,1),:);
patch('Faces',inci_aux(:,2:5),'Vertices',coord(:,2:3),'FaceColor','blue')
pause(1e-6);
if nel_1 <= nel_f && error < tau
conv = 1;
end
if mod(iter, 15) == 0
kk = kk + 1;
filename = sprintf('%d_%d_%d', ex, ey, kk);
saveas(gcf,filename,'png')
end
iter = iter + 1;
end
toc
iter = 1:iter;
complaince_hist = complaince_hist(iter);
complaince_hist = nonzeros(complaince_hist);
iter = iter(1:length(complaince_hist));
figure('Name','Mean Complaince','NumberTitle','off');
plot(iter,complaince_hist)
ylabel('Mean complaince (Nmm)')
xlabel('Iteration')