基于变密度法的多相拓扑优化MATLAB实现

一、核心代码实现

%% 参数设置
nelx = 60; nely = 30; % 网格尺寸
volfrac = 0.5;        % 目标体积分数
penal = 3;            % SIMP惩罚因子
rmin = 1.5;           % 过滤半径
ft = 1;               % 过滤类型(1=灵敏度,2=密度)

%% 材料属性定义（两相材料）
E1 = 70e9; nu1 = 0.3; % 相1参数(钢)
E2 = 200e9; nu2 = 0.3; % 相2参数(复合材料)
rho_min = 1e-3;       % 最小密度

%% 网格初始化
x = repmat([0.5*ones(nely,1) 0.5*ones(nely,1)],1,nelx);
X = repmat(x',nelx,1); % 初始密度场

%% 有限元参数
KE = lk; % 单元刚度矩阵(需预先定义)
nodes = reshape(1:(1+nelx)*(1+nely),1+nely,1+nelx);
DOF = 2*(1+nely)*(1+nelx);

%% 优化循环
for iter = 1:200
    % 计算刚度矩阵和位移
    [U] = FE(nelx,nely,X,penal,E1,E2,nu1,nu2);
    c = 0;
    dc = zeros(size(X));

    % 灵敏度分析
    for elx = 1:nelx
        for ely = 1:nely
            n1 = (nely+1)*(elx-1)+ely;
            n2 = (nely+1)* elx   +ely;
            Ue = U([2*n1-1;2*n1; 2*n2-1;2*n2; 2*n2+1;2*n2+2; 2*n1+1;2*n1+2],1);
            dc(elx,ely) = -penal*X(elx,ely)^(penal-1)*sum(Ue.^2)*KE(:);
            c = c + X(elx,ely)^penal*Ue'*KE*Ue;
        end
    end

    % 过滤灵敏度
    if ft==1
        dc = check(nelx,nely,rmin,dc);
    else
        X = check(nelx,nely,rmin,X);
    end

    % OC优化准则更新
    l1 = 0; l2 = 1e9; move = 0.2;
    while (l2-l1 > 1e-4)
        lmid = 0.5*(l2+l1);
        xnew = max(0.001,max(x-move,min(1.,min(x+move,x.*sqrt(-dc./lmid)))));

        % 体积约束
        lvol = sum(sum(xnew))/sum(sum(x));
        if lvol > volfrac
            l1 = lmid;
        else
            l2 = lmid;
        end
    end
    x = xnew;

    % 显示迭代信息
    fprintf('Iteration: %d | Compliance: %.3f\n', iter, c);
end

%% 结果可视化
colormap(gray); imagesc(1-x); axis equal; axis tight; hold on;
contour(1-x, [0.5 0.5], 'r', 'LineWidth', 2);
title('Multi-phase Topology Optimization Result');

二、关键函数

1. 单元刚度矩阵计算

function KE = 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(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)];
end

2. 灵敏度过滤

function [xnew] = check(nelx,nely,rmin,x)
    xnew = zeros(nelx,nely);
    for i = 1:nelx
        for j = 1:nely
            sum = 0.0;
            for k = max(i-floor(rmin),1):min(i+floor(rmin),nelx)
                for l = max(j-floor(rmin),1):min(j+floor(rmin),nely)
                    fac = rmin - sqrt((i-k)^2 + (j-l)^2);
                    sum = sum + max(0,fac);
                    xnew(i,j) = xnew(i,j) + max(0,fac)*x(k,l);
                end
            end
            xnew(i,j) = xnew(i,j)/(x(i,j)+1e-3)*sum;
        end
    end
end

三、算法流程解析

四、性能优化

1. 并行计算

   parfor elx = 1:nelx
       for ely = 1:nely
           % 并行计算单元刚度矩阵
       end
   end
   

2. GPU加速

   X_gpu = gpuArray(X);
   K_gpu = gpuArray(KE);
   U_gpu = gpuArray(U);
   

3. 自适应网格

   if iter > 50
       nelx = nelx*2; nely = nely*2; % 后期加密网格
   end
   

参考代码 基于变密度法的简单多相拓扑优化MATLAB程序 www.youwenfan.com/contentalf/81305.html

五、应用场景扩展

1. 轻量化设计

   % 汽车摆臂优化
   nelx = 120; nely = 60;
   volfrac = 0.45;
   penal = 3.5;
   

2. 热-机耦合优化

   % 添加热载荷项
   Q = thermal_load(); % 热载荷向量
   F = [F_mechanical; Q];
   

3. 多尺度优化

   % 宏观-微观联合优化
   macro_x = global_optimization();
   micro_x = local_refinement(macro_x);
   

六、常见问题解决

1. 收敛性问题 调整惩罚因子p（建议2-4） 增加初始密度均匀性

2. 内存不足

   • 使用稀疏矩阵存储刚度矩阵

   K = sparse(DOF,DOF);
   

3. 局部极小值

   • 引入随机扰动

   if rand < 0.01
       x = x + 0.05*randn(size(x));
   end