A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model

Abstract

This paper presents a new topology optimization approach based on the so-called Moving Morphable Components (MMC) solution framework. The proposed method improves several weaknesses of the previous approach (e.g., Guo et al. in J Appl Mech 81:081009, 2014a) in the sense that it can not only allow for components with variable thicknesses but also enhance the numerical solution efficiency substantially. This is achieved by constructing the topological description functions of the components appropriately, and utilizing the ersatz material model through projecting the topological description functions of the components. Numerical examples demonstrate the effectiveness of the proposed approach. In order to help readers understand the essential features of this approach, a 188 line Matlab implementation of this approach is also provided.

Appendix: Matlab code (corresponding to the short beam problem)

function MMC188(DW,DH,nelx,nely,x_int,y_int,ini_val,volfrac)

% FEM data initialization

M = [nely + 1, nelx + 1];

EW = DW / nelx; % length of element

EH = DH / nely; % width of element

[x,y] = meshgrid(EW * [0 : nelx], EH * [0 : nely]);

LSgrid.x = x(:);

LSgrid.y = y(:); % coordinate of nodes

% Material properties

h = 1; %thickness

E = 1;

nu = 0.3;

% Component geometry initialization

x0 = x_int/2:x_int:DW; % x-coordinates of the centers of components

y0 = y_int/2:y_int:DH; % y-coordinates of the centers of components

xn = length(x0); % number of component groups in x direction

yn = length(y0); % number of component groups in y direction

x0 = kron(x0,ones(1,2*yn));

y0 = repmat(kron(y0,ones(1,2)),1,xn);

N = length(x0); % total number of components in the design domain

L = repmat(ini_val(1),1,N); % vector of the half length of each component

t1 = repmat(ini_val(2),1,N); % vector of the half width of component at point A

t2 = repmat(ini_val(3),1,N); % vector of the half width of component at point B

t3 = repmat(ini_val(4),1,N); % vector of the half width of component at point C

st = repmat([ini_val(5) -ini_val(5)],1,N/2); % vector of the sine value of the inclined angle of each component

variable = [x0;y0;L;t1;t2;t3;st];

%Parameter of MMA

xy00 = variable(:);

xval = xy00;

xold1 = xy00;

xold2 = xy00;

%Limits of variable:[x0; y0; L; t1; t2; t3; st];

xmin = [0; 0; 0.01; 0.01; 0.01; 0.03; −1.0];

xmin = repmat(xmin,N,1);

xmax = [DW; DH; 2.0; 0.2; 0.2; 0.2; 1.0];

xmax = repmat(xmax,N,1);

low = xmin;

upp = xmax;

m = 1; %number of constraint

Var_num = 7; % number of design variables for each component

nn = Var_num*N;

c = 1000*ones(m,1);

d = zeros(m,1);

a0 = 1;

a = zeros(m,1);

%Define loads and supports(Short beam)

fixeddofs = 1:2*(nely + 1);

alldofs = 1:2*(nely + 1)*(nelx + 1);

freedofs = setdiff(alldofs,fixeddofs);

loaddof = 2*(nely + 1)*nelx + nely + 2;

F = sparse(loaddof,1,-1,2*(nely + 1)*(nelx + 1),1);

%Preparation FE analysis

nodenrs = reshape(1:(1 + nelx)*(1 + nely),1 + nely,1 + nelx);

edofVec = reshape(2*nodenrs(1:end-1,1:end-1)-1,nelx*nely,1);

edofMat = repmat(edofVec,1,8) + repmat([0 1 2*nely + [2 3 4 5] 2 3],nelx*nely,1);

iK = kron(edofMat,ones(8,1))';

jK = kron(edofMat,ones(1,8))';

EleNodesID = edofMat(:,2:2:8)./2;

iEner = EleNodesID';

[KE] = BasicKe(E,nu, EW, EH,h); % stiffness matrix k^s is formed

%Initialize iteration

p = 6;

alpha = 1e-3; % parameter alpha in the Heaviside function

epsilon = 4*min(EW,EH); % regularization parameter epsilon in the Heaviside function

Phi = cell(N,1);

Loop = 1;

change = 1;

maxiter = 1000; % the maximum number of iterations

while change > 0.001 && Loop < maxiter

%Forming Phi^s

for i = 1:N

Phi{i} = tPhi(xy00(Var_num*i-Var_num + 1:Var_num*i),LSgrid.x,LSgrid.y,p);

end

%Union of components

tempPhi_max = Phi{1};

for i = 2:N

tempPhi_max = max(tempPhi_max,Phi{i});

end

Phi_max = reshape(tempPhi_max,nely + 1,nelx + 1);

%Plot components

contourf(reshape(x, M), reshape(y, M),Phi_max,[0,0]);

axis equal;axis([0 DW 0 DH]);pause(1e-6);

% Calculating the finite difference quotient of H

H = Heaviside(Phi_max,alpha,nelx,nely,epsilon);

diffH = cell(N,1);

for j = 1:N

for ii = 1:Var_num

xy001 = xy00;

xy001(ii + (j-1)*Var_num) = xy00(ii + (j-1)*Var_num) + max(2*min(EW,EH),0.005);

tmpPhiD1 = tPhi(xy001(Var_num*j-Var_num + 1:Var_num*j),LSgrid.x,LSgrid.y,p);

tempPhi_max1 = tmpPhiD1;

for ik = 1:j-1

tempPhi_max1 = max(tempPhi_max1,Phi{ik});

end

for ik = j + 1:N

tempPhi_max1 = max(tempPhi_max1,Phi{ik});

end

xy002 = xy00;

xy002(ii + (j-1)*Var_num) = xy00(ii + (j-1)*Var_num)-max(2*min(EW,EH),0.005);

tmpPhiD2 = tPhi(xy002(Var_num*j-Var_num + 1:Var_num*j),LSgrid.x,LSgrid.y,p);

tempPhi_max2 = tmpPhiD2;

for ik = 1:j-1

tempPhi_max2 = max(tempPhi_max2,Phi{ik});

end

for ik = j + 1:N

tempPhi_max2 = max(tempPhi_max2,Phi{ik});

end

HD1 = Heaviside(tempPhi_max1,alpha,nelx,nely,epsilon);

HD2 = Heaviside(tempPhi_max2,alpha,nelx,nely,epsilon);

diffH{j}(:,ii) = (HD1-HD2)/(2*(max(2*min(EW,EH),0.005)));

end

end

%FEA

denk = sum(H(EleNodesID).^2, 2) / 4;

den = sum(H(EleNodesID), 2) / 4;

A1 = sum(den)*EW*EH;

U = zeros(2*(nely + 1)*(nelx + 1),1);

sK = KE(:)*denk(:)';

K = sparse(iK(:),jK(:),sK(:)); K = (K + K')/2;

U(freedofs,:) = K(freedofs,freedofs)\F(freedofs,:);

%Energy of element

energy = sum((U(edofMat)*KE).*U(edofMat),2);

sEner = ones(4,1)*energy'/4;

energy_nod = sparse(iEner(:),1,sEner(:));

Comp = F'*U;

% Sensitivities

df0dx = zeros(Var_num*N,1);

dfdx = zeros(Var_num*N,1);

for k = 1:N

df0dx(Var_num*k-Var_num + 1:Var_num*k,1) = 2*energy_nod'.*H*diffH{k};

dfdx(Var_num*k-Var_num + 1:Var_num*k,1) = sum(diffH{k})/4;

end

%MMA optimization

f0val = Comp;

df0dx = −df0dx/max(abs(df0dx));

fval = A1/(DW*DH)-volfrac;

dfdx = dfdx/max(abs(dfdx));

[xmma,ymma,zmma,lam,xsi,eta,mu,zet,ss,low,upp] = …

mmasub(m,nn,Loop,xval,xmin,xmax,xold1,xold2, …

f0val,df0dx,fval,dfdx,low,upp,a0,a,c,d);

xold2 = xold1;

xold1 = xval;

change = max(abs(xval-xmma));

xval = xmma;

xy00 = round(xval*1e4)/1e4;

disp([' It.: ' sprintf('%4i\t',Loop) ' Obj.: ' sprintf('%6.3f\t',f0val) ' Vol.: ' …

sprintf('%6.4f\t',fval) 'ch.:' sprintf('%6.4f\t',change)]);

Loop = Loop + 1;

end

end

%Forming Phi_i for each component

function [tmpPhi] = tPhi(xy,LSgridx,LSgridy,p)

st = xy(7);

ct = sqrt(abs(1-st*st));

x1 = ct*(LSgridx - xy(1)) + st*(LSgridy - xy(2));

y1 = −st*(LSgridx - xy(1)) + ct*(LSgridy - xy(2));

bb = (xy(5) + xy(4)-2*xy(6))/2/xy(3)^2*x1.^2 + (xy(5)-xy(4))/2*x1/xy(3) + xy(6);

tmpPhi = −((x1).^p/xy(3)^p + (y1).^p./bb.^p-1);

end

%Heaviside function

function H = Heaviside(phi,alpha,nelx,nely,epsilon)

num_all = [1:(nelx + 1)*(nely + 1)]';

num1 = find(phi > epsilon);

H(num1) = 1;

num2 = find(phi < −epsilon);

H(num2) = alpha;

num3 = setdiff(num_all,[num1;num2]);

H(num3) = 3*(1-alpha)/4*(phi(num3)/epsilon-phi(num3).^3/(3*(epsilon)^3)) + (1 + alpha)/2;

end

%Element stiffness matrix

function [KE] = BasicKe(E,nu, a, b,h)

k = [−1/6/a/b*(nu*a^2-2*b^2-a^2), 1/8*nu + 1/8, −1/12/a/b*(nu*a^2 + 4*b^2-a^2), 3/8*nu-1/8, …

1/12/a/b*(nu*a^2-2*b^2-a^2),-1/8*nu-1/8, 1/6/a/b*(nu*a^2 + b^2-a^2), −3/8*nu + 1/8];

KE = E*h/(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

% ~ ~ ~ ~ A Moving Morphable Components (MMC) based topology optimization code

% ~ ~ ~ ~ by Xu Guo Weisheng Zhang and Jie Yuan

% ~ ~ ~ ~ Department of Engineering Mechanics, State Key Laboratory of Structural Analysis

% ~ ~ ~ ~ for Industrial Equipment, Dalian University of Technology

% ~ ~ ~ ~ Please send your suggestions and comments to guoxu@dlut.edu.cn