利用拓扑优化ESO算法进行悬臂梁的优化代码示例

以下是一个基于拓扑优化ESO算法的悬臂梁优化代码示例，使用MATLAB语言实现。

%% Parameters
nelx = 120; % number of elements in x direction
nely = 40; % number of elements in y direction
volfrac = 0.5; % volume fraction constraint
penal = 3; % penalization factor
rmin = 2; % filter radius
ft = 1; % filter type (1 = mean value, 2 = density)
maxit = 100; % maximum number of iterations

%% Initialization
x = volfrac * ones(nely, nelx); % design variable (density)
xPhys = x; % physical variable (material property)
loop = 0; % iteration counter
change = 1; % change in design variable
m = nely * nelx; % number of elements
n = (nelx + 1) * (nely + 1); % number of nodes

% FE analysis
[KE] = elementstiffness(); % element stiffness matrix
[coord, edof] = mesh(nelx, nely); % mesh generation
K = stiffness(coord, edof, KE, xPhys); % stiffness matrix
U = zeros(n, 1); % displacement vector
F = force(nelx, nely); % load vector
[U, ~] = solve(K, F, U, []);
compliance = F' * U; % objective function value

%% Optimization loop
while change > 0.01 && loop < maxit
    loop = loop + 1;
    
    % Sensitivity analysis
    [dcdx] = sensitivtyanalysis(K, U, xPhys, KE, coord, edof);
    
    % Filter design variables
    [x] = filterdesignvariables(x, ft, rmin, nely, nelx);
    
    % Optimality criteria update
    [x, change] = optimalitycriteria(x, dcdx, volfrac, penal, nely, nelx);
    
    % Update physical variables
    xPhys = x;
    
    % FE analysis
    K = stiffness(coord, edof, KE, xPhys); % stiffness matrix
    [U, ~] = solve(K, F, U, []);
    compliance = F' * U; % objective function value
    
    % Display iteration information
    fprintf('It.:%5i Obj.:%11.4f Vol.:%7.3f ch.:%7.3f\n',...
        loop, compliance, mean(x(:)), change);
    
    % Plot intermediate results
    colormap(gray); imagesc(-xPhys); axis equal; axis off; drawnow;
end

%% Post-processing
colormap(gray); imagesc(-xPhys); axis equal; axis off; drawnow;
title(['Final compliance: ', num2str(compliance)]);

其中，`elementstiffness()`函数用于计算单元刚度矩阵，`mesh()`函数用于生成初始网格，`stiffness()`函数用于计算刚度矩阵，`force()`函数用于生成载荷向量，`solve()`函数用于求解位移向量和反力向量，`sensitivtyanalysis()`函数用于计算灵敏度，`filterdesignvariables()`函数用于对设计变量进行滤波，`optimalitycriteria()`函数用于进行优化更新。