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422 lines
15 KiB
422 lines
15 KiB
% Q: the elastic tensor
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% dQ: the derivatives of elastic tensor for microstructures
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function [Q,dQ,xPhys,xold] = inverseHomog3d_sym8_opx8()
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% 优化八分之一结构
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E = 0.1; nu = 0.15;
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% E = 0.5; nu = 0.35;
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% E = 0.25; nu = 0.35;
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D_expect = E/(1+nu)/(1-2*nu)* [ 1-nu nu nu 0 0 0;
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nu 1-nu nu 0 0 0;
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nu nu 1-nu 0 0 0;
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0 0 0 (1-2*nu)/2 0 0;
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0 0 0 0 (1-2*nu)/2 0;
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0 0 0 0 0 (1-2*nu)/2;];
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disp(D_expect)
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nelxyz = 40;
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% 整个结构的尺寸
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nelx = nelxyz;
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nely = nelxyz;
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nelz = nelxyz;
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nele = nelx*nely*nelz;
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volfrac = 0.5;
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rmin = 2;
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E0 = 1;
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Emin = 1e-9;
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nu = 0.3;
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penal = 5;
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nelx_sym8 = nelx/2;
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nely_sym8 = nely/2;
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nelz_sym8 = nelz/2;
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nele_sym8 = nelx_sym8*nely_sym8*nelz_sym8;
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% the initial definitions of the PUC
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D0 = E0/(1+nu)/(1-2*nu)* [ 1-nu nu nu 0 0 0;
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nu 1-nu nu 0 0 0;
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nu nu 1-nu 0 0 0;
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0 0 0 (1-2*nu)/2 0 0;
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0 0 0 0 (1-2*nu)/2 0;
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0 0 0 0 0 (1-2*nu)/2;];
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disp(D0)
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% dx = lx/nelx; dy = ly/nely; dz = lz/nelz;
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% KE = elementMatVec3D(1.0/2, 1.0/2, 1.0/2, D0);
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KE = elementMatVec3D(1.0, 1.0, 1.0, D0);
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% [KE] = lk_H8(E0,nu);
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Num_node = (1+nely)*(1+nelx)*(1+nelz);
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nodenrs = reshape(1:Num_node,1+nely,1+nelx,1+nelz);
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edofVec = reshape(3*nodenrs(1:end-1,1:end-1,1:end-1)+1,nelx*nely*nelz,1);
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edofMat = repmat(edofVec,1,24)+repmat([0 1 2 3*nely+[3 4 5 0 1 2] -3 -2 -1 3*(nelx+1)*(nely+1)+[0 1 2 3*nely+[3 4 5 0 1 2] -3 -2 -1]], nele, 1);
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iK = reshape(kron(edofMat,ones(24,1))',24*24*nele,1);
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jK = reshape(kron(edofMat,ones(1,24))',24*24*nele,1);
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[H,Hs] = filter(nelx_sym8,nely_sym8,nelz_sym8,rmin);
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% periodic boundary conditions
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n1 = [nodenrs(end, [1 end], 1) nodenrs(1, [end 1], 1) nodenrs(end, [1 end], end) nodenrs(1, [end 1], end)];
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d1 = reshape([3*n1-2; 3*n1-1; 3*n1],3*numel(n1),1);
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n3 = [reshape(squeeze(nodenrs(end,1,2:end-1)),1,numel(squeeze(nodenrs(end,1,2:end-1))))...
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reshape(squeeze(nodenrs(1, 1, 2:end-1)),1,numel(squeeze(nodenrs(1, 1, 2:end-1))))...
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reshape(squeeze(nodenrs(end,2:end-1,1)),1,numel(squeeze(nodenrs(end,2:end-1,1))))...
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reshape(squeeze(nodenrs(1, 2:end-1, 1)),1,numel(squeeze(nodenrs(1, 2:end-1, 1))))...
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reshape(squeeze(nodenrs(2:end-1, 1, 1)),1,numel(squeeze(nodenrs(2:end-1, 1, 1))))...
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reshape(squeeze(nodenrs(2:end-1,1,end)),1,numel(squeeze(nodenrs(2:end-1,1,end))))...
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reshape(squeeze(nodenrs(2:end-1, 2:end-1, 1)),1,numel(squeeze(nodenrs(2:end-1, 2:end-1, 1))))...
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reshape(squeeze(nodenrs(2:end-1, 1, 2:end-1)),1,numel(squeeze(nodenrs(2:end-1, 1, 2:end-1))))...
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reshape(squeeze(nodenrs(end,2:end-1,2:end-1)),1,numel(squeeze(nodenrs(end,2:end-1,2:end-1))))];
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d3 = reshape([3*n3-2; 3*n3-1; 3*n3],3*numel(n3),1);
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n4 = [reshape(squeeze(nodenrs(1, end, 2:end-1)),1,numel(squeeze(nodenrs(1, end, 2:end-1))))...
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reshape(squeeze(nodenrs(end,end,2:end-1)),1,numel(squeeze(nodenrs(end,end,2:end-1))))...
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reshape(squeeze(nodenrs(1, 2:end-1, end)),1,numel(squeeze(nodenrs(1, 2:end-1, end))))...
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reshape(squeeze(nodenrs(end,2:end-1,end)),1,numel(squeeze(nodenrs(end,2:end-1,end))))...
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reshape(squeeze(nodenrs(2:end-1,end,end)),1,numel(squeeze(nodenrs(2:end-1,end,end))))...
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reshape(squeeze(nodenrs(2:end-1, end, 1)),1,numel(squeeze(nodenrs(2:end-1, end, 1))))...
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reshape(squeeze(nodenrs(2:end-1,2:end-1,end)),1,numel(squeeze(nodenrs(2:end-1,2:end-1,end))))...
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reshape(squeeze(nodenrs(2:end-1,end,2:end-1)),1,numel(squeeze(nodenrs(2:end-1,end,2:end-1))))...
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reshape(squeeze(nodenrs(1, 2:end-1, 2:end-1)),1,numel(squeeze(nodenrs(1, 2:end-1, 2:end-1))))];
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d4 = reshape([3*n4-2; 3*n4-1; 3*n4],3*numel(n4),1);
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n2 = setdiff(nodenrs(:),[n1(:);n3(:);n4(:)]);
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d2 = reshape([3*n2-2; 3*n2- 1; 3*n2],3*numel(n2),1);
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% the imposing of six linearly independent unit test strains
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e = eye(6); ufixed = zeros(24,6);
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vert_cor = [0, nelx, nelx, 0, 0, nelx, nelx, 0;
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0, 0, nely, nely, 0, 0, nely, nely;
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0, 0, 0, 0, nelz, nelz, nelz, nelz];
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for i = 1:6
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epsilon = [ e(i,1), e(i,4)/2, e(i,6)/2;
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e(i,4)/2, e(i,2), e(i,5)/2;
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e(i,6)/2, e(i,5)/2, e(i,3)];
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ufixed(:,i) = reshape(epsilon*vert_cor,24,1);
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end
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% 3D boundary constraint equations
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wfixed = [repmat(ufixed(7:9,:),numel(squeeze(nodenrs(end,1,2:end-1))),1);
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repmat(ufixed( 4:6,:)-ufixed(10:12,:),numel(squeeze(nodenrs(1, 1, 2:end-1))),1);
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repmat(ufixed(22:24,:),numel(squeeze(nodenrs(end,2:end-1,1))),1);
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repmat(ufixed(13:15,:)-ufixed(10:12,:),numel(squeeze(nodenrs(1, 2:end-1, 1))),1);
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repmat(ufixed(16:18,:),numel(squeeze(nodenrs(2:end-1, 1, 1))),1);
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repmat(ufixed( 4:6,:)-ufixed(13:15,:),numel(squeeze(nodenrs(2:end-1,1,end))),1);
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repmat(ufixed(13:15,:),numel(squeeze(nodenrs(2:end-1, 2:end-1,1))),1);
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repmat(ufixed(4:6,:),numel(squeeze(nodenrs(2:end-1, 1, 2:end- 1))),1);
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repmat(ufixed(10:12,:),numel(squeeze(nodenrs(end,2:end-1,2:end-1))),1)];
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% intial
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qe = cell(6,6);
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Q = zeros(6,6);
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dQ = cell(6,6);
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dQ_sym8 = cell(6,6);
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% 1/8
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x_sym8 = ones(nely_sym8,nelx_sym8,nelz_sym8).*volfrac;
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x_sym8(nely_sym8/2:nely_sym8/2+1,nelx_sym8/2:nelx_sym8/2+1,nelz_sym8 /2:nelz_sym8/2+1) = Emin;
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x = expandX_3Dsym8(x_sym8);
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xPhys = x;
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% beta = 1;
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% xTilde = x;
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% xPhys = 1-exp(-beta*xTilde)+xTilde*exp(-beta);
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loopbeta = 0; loop = 0; maxloop = 14;
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change = 1;
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OBJ = zeros(maxloop,1);
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VOL = zeros(maxloop,1);
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BWR = zeros(maxloop,1);
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% figure;
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while loop < maxloop && change > 0.01
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loop = loop+1; loopbeta = loopbeta+1;
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% FE-ANALYSIS
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sK = reshape(KE(:)*(Emin+xPhys(:)'.^penal*(1-Emin)),24*24*nele,1);
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K = sparse(iK,jK,sK); K = (K+K')/2;
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Kr = [K(d2,d2), K(d2,d3)+K(d2,d4);
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K(d3,d2)+K(d4,d2),K(d3,d3)+K(d4,d3)+K(d3,d4)+K(d4,d4)];
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U(d1,:) = ufixed;
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U([d2;d3],:) = Kr\(-[K(d2,d1);K(d3,d1)+K(d4,d1)]*ufixed-[K(d2,d4);K(d3,d4)+K(d4,d4)]*wfixed);
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% for i_load=1:6
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% U([d2;d3],i_load) = pcg(Kr, (-[K(d2,d1);K(d3,d1)+K(d4,d1)]*ufixed(:,i_load)-[K(d2,d4);K(d3,d4)+K(d4,d4)]*wfixed(:,i_load)),1e-3,200);
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% end
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U(d4,:) = U(d3,:)+wfixed;
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% homog
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cellVolume = nele;
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for i = 1:6
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for j = 1:6
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U1 = U(:,i);
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U2 = U(:,j);
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qe{i,j} = reshape(sum((U1(edofMat)*KE).*U2(edofMat),2),nely,nelx,nelz);
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Q(i,j) = 1/cellVolume*sum(sum(sum((Emin+xPhys.^penal*(1-Emin)).*qe{i,j})));
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dQ{i,j} = 1/cellVolume*(penal*(1-Emin)*xPhys.^(penal-1).*qe{i,j});
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% compute dQ_sym8
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dQx = dQ{i,j}; % (nely, nelx, nelz);
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dQx_z1 = dQx(:,:,1:nelz_sym8) + flip(dQx(:,:,nelz_sym8+1:end),3);
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dQx_z1_y1 = dQx_z1(1:nely_sym8,:,:) + flipud(dQx_z1(nely_sym8+1:end,:,:));
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dQx_z1_y1_x1 = dQx_z1_y1(:,1:nelx_sym8,:) + fliplr(dQx_z1_y1(:,nelx_sym8+1:end,:));
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dQ_sym8{i,j} = dQx_z1_y1_x1;
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end
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end
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disp(Q)
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% size(dQ_sym8{1,1})
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dQ_sym8_ele = zeros(nelxyz/2,nelxyz/2,nelxyz/2,6,6);
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for i=1:6
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for j=1:6
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dQ_sym8_ele(:,:,:,i,j) = dQ_sym8{i,j};
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end
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end
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c = sum((Q-D_expect).^2,'all');
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dc = zeros(nelxyz/2,nelxyz/2,nelxyz/2);
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for ix = 1:nelxyz/2
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for iy = 1:nelxyz/2
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for iz = 1:nelxyz/2
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dc(iy,ix,iz) = sum((Q-D_expect).'.*squeeze(dQ_sym8_ele(iy,ix,iz,:,:)).*2,'all');
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end
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end
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end
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% % max shear modulus
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% c = -(Q(4,4)+Q(5,5)+Q(6,6));
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% dc = -(dQ_sym8{4,4}+dQ_sym8{5,5}+dQ_sym8{6,6});
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disp('sensitivity:')
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disp([min(dc,[],'all'), max(dc,[],'all')])
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%
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% c = -(Q(1,1)+Q(1,2)+Q(1,3)+Q(2,1)+Q(2,2)+Q(2,3)+Q(3,1)+Q(3,2)+Q(3,3));
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% dc = -(dQ_sym8{1,1}+dQ_sym8{1,2}+dQ_sym8{1,3}+dQ_sym8{2,1}+dQ_sym8{2,2}+dQ_sym8{2,3}...
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% +dQ_sym8{3,1}+dQ_sym8{3,2}+dQ_sym8{3,3});
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%
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% c = -(Q(1,1)+Q(2,2)+Q(3,3));
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% dc = -(dQ_sym8{1,1}+dQ_sym8{2,2}+dQ_sym8{3,3}); % 优化出实心的
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% c = -(Q(1,2)+Q(1,3)+Q(2,1)+Q(2,3)+Q(3,1)+Q(3,2));
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% dc = -(dQ_sym8{1,2}+dQ_sym8{1,3}+dQ_sym8{2,1}+dQ_sym8{2,3}+dQ_sym8{3,1}+dQ_sym8{3,2});
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% c = -(Q(6,6)); % 平板，不连续
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% dc = -(dQ_sym8{6,6});
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% c = -(Q(4,4)+Q(5,5));
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% dc = -(dQ_sym8{4,4}+dQ_sym8{5,5});
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dv = ones(nely_sym8,nelx_sym8,nelz_sym8).*8;
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% FILTERING AND MODIFICATION OF SENSITIVITIES
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dc(:) = H*(dc(:)./Hs);
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dv(:) = H*(dv(:)./Hs);
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% OPTIMALITY CRITERIA UPDATE
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l1 = 0; l2 = 1e9; move = 0.2;move=0.1;
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while (l2-l1)/(l1+l2) > 1e-3
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lmid = 0.5*(l2+l1);
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% xnew_sym8 = max(Emin,max(x_sym8-move,min(1,min(x_sym8+move,x_sym8.*sqrt(-dc./dv/lmid)))));
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xnew_sym8 = max(Emin,max(x_sym8-move,min(1,min(x_sym8+move,x_sym8.*(-dc./dv/lmid)))));
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xPhys_sym8(:) = (H*xnew_sym8(:))./Hs;
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if sum(xPhys_sym8(:)) > volfrac*nele_sym8, l1 = lmid; else l2 = lmid; end
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end
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change = max(abs(xnew_sym8(:)-x_sym8(:)));
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x_sym8 = xnew_sym8;
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x = expandX_3Dsym8(x_sym8);
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xold = xPhys;
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% xPhys = x;
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xPhys = expandX_3Dsym8(xPhys_sym8);
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% PRINT RESULTS
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fprintf(' It.:%5i Obj.:%11.4f Vol.:%7.3f ch.:%7.3f\n',loop,c,mean(xPhys_sym8(:)),change);
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% PLOT DENSITIES
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if mod(loop,30)==1 && loop>1
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% figure(1)
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% clf; display_3D(xPhys_sym8);pause(1e-6);
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figure(2)
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clf;
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display_3D(xPhys);pause(1e-6);
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end
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OBJ(loop) = c;
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VOL(loop) = mean(xPhys(:));
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BWR(loop) = max(xPhys,[],'all')-min(xPhys,[],'all');
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plot_convergence_all(loop,maxloop,OBJ,VOL,BWR);
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end
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figure()
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clf; display_3D(xPhys);
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%%
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% plot: 让微观结构看的更清楚一点，从里到外密度递增
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center_x = nelx/2+0.5;
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center_y = nely/2+0.5;
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center_z = nelz/2+0.5;
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len_diag = sqrt(nelx^2+nely^2+nelz^2)/2;
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x_plot = zeros(nely,nelx,nelz);
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for i=1:nelx
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for j=1:nely
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for k=1:nelz
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dist_x = sqrt((i-center_x)^2+(j-center_y)^2+(k-center_z)^2);
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x_plot(j,i,k) = xPhys(j,i,k)*dist_x/len_diag;
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end
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end
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end
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figure(6)
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clf;
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display_3D(0.9*x_plot);pause(1e-6);
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end
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% function plot_convergence_all(loop,maxloop,sumCompliance,volfraction)
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% figure(3)
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% % figure()
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% % loop=75;maxloop=loop;
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% set(gcf,'position',[100 100 1200 400])
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% subplot(2,1,1)
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% plotC=sumCompliance.';
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% % plot([1:loop].',plotC(1:loop))
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% semilogy([1:loop].',plotC(1:loop))
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% xlim([0 maxloop])
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% subplot(2,1,2)
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% plotV=volfraction.';
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% plot([1:loop].',plotV(1:loop))
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% xlim([0 maxloop])
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% pause(1e-6);
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% end
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% === DISPLAY 3D TOPOLOGY (ISO-VIEW) ===
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function display_3D(rho)
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[nely,nelx,nelz] = size(rho);
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hx = 1; hy = 1; hz = 1; % User-defined unit element size
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face = [1 2 3 4; 2 6 7 3; 4 3 7 8; 1 5 8 4; 1 2 6 5; 5 6 7 8];
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set(gcf,'Name','ISO display','NumberTitle','off');
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for k = 1:nelz
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z = (k-1)*hz;
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for i = 1:nelx
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x = (i-1)*hx;
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for j = 1:nely
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y = nely*hy - (j-1)*hy;
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if (rho(j,i,k) > 0.5) % User-defined display density threshold
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vert = [x y z; x y-hx z; x+hx y-hx z; x+hx y z; x y z+hx;x y-hx z+hx; x+hx y-hx z+hx;x+hx y z+hx];
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vert(:,[2 3]) = vert(:,[3 2]); vert(:,2,:) = -vert(:,2,:);
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patch('Faces',face,'Vertices',vert,'FaceColor',[0.2+0.8*(1-rho(j,i,k)),0.2+0.8*(1-rho(j,i,k)),0.2+0.8*(1-rho(j,i,k))]);
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hold on;
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end
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end
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end
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end
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axis equal; axis tight; axis off; box on; view([30,30]); pause(1e-6);
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end
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% PREPARE FILTER
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function [H,Hs] = filter(nelx,nely,nelz,rmin)
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nele = nelx*nely*nelz;
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iH = ones(nele*(2*(ceil(rmin)-1)+1)^2,1);
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jH = ones(size(iH));
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sH = zeros(size(iH));
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k = 0;
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for k1 = 1:nelz
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for i1 = 1:nelx
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for j1 = 1:nely
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e1 = (k1-1)*nelx*nely + (i1-1)*nely+j1;
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for k2 = max(k1-(ceil(rmin)-1),1):min(k1+(ceil(rmin)-1),nelz)
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for i2 = max(i1-(ceil(rmin)-1),1):min(i1+(ceil(rmin)-1),nelx)
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for j2 = max(j1-(ceil(rmin)-1),1):min(j1+(ceil(rmin)-1),nely)
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e2 = (k2-1)*nelx*nely + (i2-1)*nely+j2;
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k = k+1;
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iH(k) = e1;
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jH(k) = e2;
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sH(k) = max(0,rmin-sqrt((i1-i2)^2+(j1-j2)^2+(k1-k2)^2));
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|
end
|
|
end
|
|
end
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|
end
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|
end
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|
end
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H = sparse(iH,jH,sH);
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Hs = sum(H,2);
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|
end
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|
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function Ke = elementMatVec3D(a, b, c, DH)
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GN_x=[-1/sqrt(3),1/sqrt(3)];
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GN_y=GN_x;
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GN_z=GN_x;
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GaussWeigh=[1,1];
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Ke = zeros(24,24); L = zeros(6,9);
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L(1,1) = 1; L(2,5) = 1; L(3,9) = 1;
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L(4,2) = 1; L(4,4) = 1; L(5,6) = 1;
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L(5,8) = 1; L(6,3) = 1; L(6,7) = 1;
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|
for i=1:length(GN_x)
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|
for j=1:length(GN_y)
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|
for k=1:length(GN_z)
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|
x = GN_x(i);y = GN_y(j);z = GN_z(k);
|
|
dNx = 1/8*[-(1-y)*(1-z), (1-y)*(1-z), (1+y)*(1-z), -(1+y)*(1-z), -(1-y)*(1+z), (1-y)*(1+z), (1+y)*(1+z) -(1+y)*(1+z)];
|
|
dNy = 1/8*[-(1-x)*(1-z) -(1+x)*(1-z), (1+x)*(1-z), (1-x)*(1-z), -(1-x)*(1+z), -(1+x)*(1+z), (1+x)*(1+z), (1-x)*(1+z)];
|
|
dNz = 1/8*[-(1-x)*(1-y) -(1+x)*(1-y) -(1+x)*(1+y) -(1-x)*(1+y), (1-x)*(1-y), (1+x)*(1-y), (1+x)*(1+y), (1-x)*(1+y)];
|
|
J = [dNx;dNy;dNz]*[ -a a a -a -a a a -a ;
|
|
-b -b b b -b -b b b;
|
|
-c -c -c -c c c c c]';
|
|
G = [inv(J) zeros(3) zeros(3);zeros(3) inv(J) zeros(3);zeros(3) zeros(3) inv(J)];
|
|
dN(1,1:3:24) = dNx; dN(2,1:3:24) = dNy; dN(3,1:3:24) = dNz;
|
|
dN(4,2:3:24) = dNx; dN(5,2:3:24) = dNy; dN(6,2:3:24) = dNz;
|
|
dN(7,3:3:24) = dNx; dN(8,3:3:24) = dNy; dN(9,3:3:24) = dNz;
|
|
Be = L*G*dN;
|
|
Ke = Ke + GaussWeigh(i)*GaussWeigh(j)*GaussWeigh(k)*det(J)*(Be'*DH*Be);
|
|
end
|
|
end
|
|
end
|
|
end
|
|
|
|
% === GENERATE ELEMENT STIFFNESS MATRIX ===
|
|
function [KE] = lk_H8(E0,nu)
|
|
A = [32 6 -8 6 -6 4 3 -6 -10 3 -3 -3 -4 -8;
|
|
-48 0 0 -24 24 0 0 0 12 -12 0 12 12 12];
|
|
k = 1/144*A'*[1; nu];
|
|
|
|
K1 = [k(1) k(2) k(2) k(3) k(5) k(5);
|
|
k(2) k(1) k(2) k(4) k(6) k(7);
|
|
k(2) k(2) k(1) k(4) k(7) k(6);
|
|
k(3) k(4) k(4) k(1) k(8) k(8);
|
|
k(5) k(6) k(7) k(8) k(1) k(2);
|
|
k(5) k(7) k(6) k(8) k(2) k(1)];
|
|
K2 = [k(9) k(8) k(12) k(6) k(4) k(7);
|
|
k(8) k(9) k(12) k(5) k(3) k(5);
|
|
k(10) k(10) k(13) k(7) k(4) k(6);
|
|
k(6) k(5) k(11) k(9) k(2) k(10);
|
|
k(4) k(3) k(5) k(2) k(9) k(12)
|
|
k(11) k(4) k(6) k(12) k(10) k(13)];
|
|
K3 = [k(6) k(7) k(4) k(9) k(12) k(8);
|
|
k(7) k(6) k(4) k(10) k(13) k(10);
|
|
k(5) k(5) k(3) k(8) k(12) k(9);
|
|
k(9) k(10) k(2) k(6) k(11) k(5);
|
|
k(12) k(13) k(10) k(11) k(6) k(4);
|
|
k(2) k(12) k(9) k(4) k(5) k(3)];
|
|
K4 = [k(14) k(11) k(11) k(13) k(10) k(10);
|
|
k(11) k(14) k(11) k(12) k(9) k(8);
|
|
k(11) k(11) k(14) k(12) k(8) k(9);
|
|
k(13) k(12) k(12) k(14) k(7) k(7);
|
|
k(10) k(9) k(8) k(7) k(14) k(11);
|
|
k(10) k(8) k(9) k(7) k(11) k(14)];
|
|
K5 = [k(1) k(2) k(8) k(3) k(5) k(4);
|
|
k(2) k(1) k(8) k(4) k(6) k(11);
|
|
k(8) k(8) k(1) k(5) k(11) k(6);
|
|
k(3) k(4) k(5) k(1) k(8) k(2);
|
|
k(5) k(6) k(11) k(8) k(1) k(8);
|
|
k(4) k(11) k(6) k(2) k(8) k(1)];
|
|
K6 = [k(14) k(11) k(7) k(13) k(10) k(12);
|
|
k(11) k(14) k(7) k(12) k(9) k(2);
|
|
k(7) k(7) k(14) k(10) k(2) k(9);
|
|
k(13) k(12) k(10) k(14) k(7) k(11);
|
|
k(10) k(9) k(2) k(7) k(14) k(7);
|
|
k(12) k(2) k(9) k(11) k(7) k(14)];
|
|
KE = E0/((nu+1)*(1-2*nu))*...
|
|
[ K1 K2 K3 K4;
|
|
K2' K5 K6 K3';
|
|
K3' K6 K5' K2';
|
|
K4 K3 K2 K1'];
|
|
end
|