✅作者简介热爱科研的Matlab仿真开发者擅长毕业设计辅导、数学建模、数据处理、算法改进、程序设计科研仿真。完整代码获取 定制创新 论文复现私信个人信条做科研博学之、审问之、慎思之、明辨之、笃行之是为博学慎思明辨笃行。 内容介绍结构轻量化对于提高能源效率和减少二氧化碳排放至关重要。此外在许多应用中高导热性是实现高效能量传递的必要条件同时还能增加产品刚度并减轻重量。近年来随着 3D 打印技术的发展人们的注意力转向了对减重有很大帮助的多孔材料。因此这项教学研究旨在介绍一种并发多尺度拓扑优化方法该方法利用多目标优化方法来设计轻质、高导热且刚度良好的多孔结构。本研究采用归一化多目标函数以使导热性和刚度最大化。因此目标准则包括最小化热阻和机械柔度。利用 SIMP固体各向同性材料惩罚模型方法通过伴随法从理论上推导了多尺度灵敏度分析和优化公式以降低计算成本并在 MATLAB 代码中实现。研究了二维案例并得到了合适的帕累托前沿。结果显示宏观和微观尺度设计之间实现了良好的耦合。⛳️ 运行结果 部分代码function [CH,dCH] Homogenization_full(Micro,select_case)%% Initialize input data[nelx,nely]size(Micro);for i1:nelxfor j1:nelyif Micro(i,j).1;x(i,j)2;else;x(i,j)1;end;end;endlx1;ly1;lambda[1 0];mu[1 0];dx lx/nelx; dy ly/nely;nel nelx*nely;Qones(3,3);[keLambda, keMu, feLambda, feMu,~] elementMatVec(dx/2, dy/2,Q,select_case);nodenrs reshape(1:(1nelx)*(1nely),1nely,1nelx);edofVec reshape(2*nodenrs(1:end-1,1:end-1)1,nel,1);edofMat repmat(edofVec,1,8)repmat([0 1 2*nely[2 3 0 1] -2 -1],nel,1);switch(select_case);case(1);case(2)keMu(1:2:end,1:2:end) keMu(1:2:end,1:2:end)keMu(2:2:end, 2:2:end);end%% Impose Periodic Boundary Conditionsnn (nelx1)*(nely1);nnP (nelx)*(nely);nnPArray reshape(1:nnP, nely, nelx);nnPArray(end1,:) nnPArray(1,:);nnPArray(:,end1) nnPArray(:,1);dofVector zeros(2*nn, 1);dofVector(1:2:end) 2*nnPArray(:)-1;dofVector(2:2:end) 2*nnPArray(:);edofMat dofVector(edofMat);ndof 2*nnP;%% Assemble Stiffness MatrixiK kron(edofMat,ones(8,1));jK kron(edofMat,ones(1,8));lambda lambda(1)*(x1) lambda(2)*(x2);mu mu(1)*(x1) mu(2)*(x2);for i1 nely;for j1;lambda(i,j)lambda(i,j);mu(i,j)mu(i,j);end;endsK keLambda(:)*lambda(:). keMu(:)*mu(:).;K sparse(iK(:), jK(:), sK(:), ndof, ndof);%% Load Vectors and SolutionsF feLambda(:)*lambda(:).feMu(:)*mu(:).;iF repmat(edofMat,3,1);jF [ones(8,nel); 2*ones(8,nel); 3*ones(8,nel)];F sparse(iF(:), jF(:), sF(:), ndof, 3);% SolveactivedofsedofMat(x1,:);activedofssort(unique(activedofs(:)));switch(select_case);case(1)Xizeros(ndof,3);Xi(activedofs(3:end),:) K(activedofs(3:end),activedofs(3:end))\F(activedofs(3:end),:);case(2)Xi zeros(ndof,2);Xi(activedofs(3:2:end),:) K(activedofs(3:2:end),activedofs(3:2:end))\...[F(activedofs(3:2:end),1) F(activedofs(4:2:end),2)];end%% HomogenizationXi0 zeros(nel, 8, 3);Xi0_e zeros(8, 3);ke keMu keLambda;fe feMu feLambda;Loop0;switch(select_case);case(1)Loop3;Xi0_e([3 5:end],:) ke([3 5:end],[3 5:end])\fe([3 5:end],:);case(2)Loop2;Xi0_e(3:2:end,1:2) keMu(3:2:end,3:2:end)\[feMu(3:2:end,1) feMu(4:2:end,2)]; endXi0(:,:,1) kron(Xi0_e(:,1), ones(nel,1));Xi0(:,:,2) kron(Xi0_e(:,2), ones(nel,1));Xi0(:,:,3) kron(Xi0_e(:,3), ones(nel,1));CH zeros(3);cellVolume lx*ly;for i 1:Loopfor j 1:LoopsumLambda ((Xi0(:,:,i) - Xi(edofMat(i-1)*ndof))*keLambda).*...(Xi0(:,:,j) - Xi(edofMat(j-1)*ndof));sumMu ((Xi0(:,:,i) - Xi(edofMat(i-1)*ndof))*keMu).*...(Xi0(:,:,j) - Xi(edofMat(j-1)*ndof));sumLambda reshape(sum(sumLambda,2), nely, nelx);sumMu reshape(sum(sumMu,2), nely, nelx);qelambda.*sumLambda mu.*sumMu;CH(i,j) 1/cellVolume*sum(sum(qe.*(1e-6Micro.^(3)*(1-1e-6))));dCHq 1/cellVolume*(qe.*(3*Micro.^(3-1)*(1-0.0001)).); dCH{i,j}dCHq;end;end 参考文献[1] Ali M A , Shimoda M .Toward multiphysics multiscale concurrent topology optimization for lightweight structures with high heat conductivity and high stiffness using MATLAB[J].Structural and Multidisciplinary Optimization, 2022, 65(7):1-26.DOI:10.1007/s00158-022-03291-0.更多免费数学建模和仿真教程关注领取