99行拓扑优化 代码解析

主要的解析来自百度，但是添加了很多新手看不懂的解释，毕竟我是小白……

function top(nelx,nely,volfrac,penal,rmin);

nelx=80; 		% x轴方向上单元个数
nely=20; 		% y轴方向上单元个数
volfrac=0.4; 	 %体积比
penal=3; 		%材料插值的惩罚因子
rmin=2; 		%敏度过滤半径

% INITIALIZE
x(1:nely,1:nelx) = volfrac;  % x是设计变量(单元伪密度)
loop = 0; 					%存放迭代次数的变量
change = 1.;					%每次迭代，目标函数（柔度）的改变值，用来判断何时收敛

% START ITERATION
while change > 0.01 %当两次连续目标函数迭代的差<=0.01时，迭代结束
loop = loop + 1;
xold = x; %把前一次的设计变量付给xold

% FE-ANALYSIS
[U]=FE(nelx,nely,x,penal); %有限元分析，得到位移矢量U

% OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS
[KE] = lk; 	 %单位刚度矩阵
c = 0.;		%用来存放目标函数的变量，这里刚度最大，柔度最小

for ely = 1:nely
for elx = 1:nelx
n1 = (nely+1)*(elx-1)+ely;  %左上角的单元节点
n2 = (nely+1)* elx   +ely;  %右上角的单元节点

%所示单元的自由度，左上，右上，右下，左下
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);

c = c + x(ely,elx)^penal*Ue'*KE*Ue; %计算目标函数的值（柔度）
dc(ely,elx) = -penal*x(ely,elx)^(penal-1)*Ue'*KE*Ue; %目标函数的灵敏度
end
end

% FILTERING OF SENSITIVITIES
[dc]   = check(nelx,nely,rmin,x,dc); %灵敏度过滤，为了边界光顺一点

% DESIGN UPDATE BY THE OPTIMALITY CRITERIA METHOD
[x]    = OC(nelx,nely,x,volfrac,dc);

% PRINT RESULTS 屏幕上显示迭代信息
change = max(max(abs(x-xold))); %计算目标函数的改变量
disp([' It.: ' sprintf('%4i',loop) ' Obj.: ' sprintf('%10.4f',c) ...
' Vol.: ' sprintf('%6.3f',sum(sum(x))/(nelx*nely)) ...
' ch.: ' sprintf('%6.3f',change )])

% PLOT DENSITIES 优化结果的图形显示
colormap(gray); imagesc(-x); axis equal; axis tight; axis off;pause(1e-6);
end

%%%%%%%%%% OPTIMALITY CRITERIA UPDATE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [xnew]=OC(nelx,nely,x,volfrac,dc)

l1 = 0; l2 = 100000; %用于体积约束的拉格朗日乘子
move = 0.2;

while (l2-l1 > 1e-4)
lmid = 0.5*(l2+l1);

%即论文公式的综合
xnew = max(0.001,max(x-move,min(1.,min(x+move,x.*sqrt(-dc./lmid)))));

if sum(sum(xnew)) - volfrac*nelx*nely > 0;%二乘法减半
l1 = lmid;
else
l2 = lmid;
end
end

%%%%%%%%%% MESH-INDEPENDENCY FILTER 敏度过滤技术子程序%%%%%%%%%%%%%%%%%%%
function [dcn]=check(nelx,nely,rmin,x,dc)

dcn=zeros(nely,nelx);

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);

dcn(j,i) = dcn(j,i) + max(0,fac)*x(l,k)*dc(l,k);
end
end

dcn(j,i) = dcn(j,i)/(x(j,i)*sum);
end
end

%%%%%%%%%% FE-ANALYSIS 有限元求解子程序%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [U]=FE(nelx,nely,x,penal) %自定义函数，最后返回[U]

[KE] = lk; %单元刚度矩阵

% sparse 把一个全矩阵转化为一个稀疏矩阵，只存储每一个非零元素的三个值：元素值，元素的行号和列号
%总体刚度矩阵的稀疏矩阵
% *2是因为x，y都有一个数
K = sparse(2*(nelx+1)*(nely+1), 2*(nelx+1)*(nely+1));
%力矩阵的稀疏矩阵
F = sparse(2*(nely+1)*(nelx+1),1);
U = zeros(2*(nely+1)*(nelx+1),1); %零矩阵

for elx = 1:nelx
for ely = 1:nely
%一列列的排序
n1 = (nely+1)*(elx-1)+ely; %左上
n2 = (nely+1)* elx   +ely;  %右上

% 左上，右上，右下，左下 自由度
% 一个点有两个，所以要*2。第一个从1开始，所以*2之后要-1。
edof = [2*n1-1;2*n1; 2*n2-1;2*n2; 2*n2+1;2*n2+2; 2*n1+1;2*n1+2];

%将单元刚度矩阵组装成总的刚度矩阵
K(edof,edof) = K(edof,edof) + x(ely,elx)^penal*KE;
end
end

% DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM)
F(2,1) = -1; % 应用了一个在左上角的垂直单元力。
%按着图上来的，最左边和右下角已经固定
fixeddofs   = union([1:2:2*(nely+1)],[2*(nelx+1)*(nely+1)]); %固定结点
alldofs     = [1:2*(nely+1)*(nelx+1)]; 						%所有结点

% setdiff 因无约束自由度与固定自由度的不同来找到无约束自由度
freedofs    = setdiff(alldofs,fixeddofs); %不受约束的自由度

% SOLVING
U(freedofs,:) = K(freedofs,freedofs) \ F(freedofs,:);
U(fixeddofs,:)= 0; % 矩阵A的第r行：A（r，：）

%%%%%%%%%% ELEMENT STIFFNESS MATRIX 单元刚度矩阵的子程序%%%%%%%%%%%%%%%%%%%%
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];

%u1,v1,  u2,v2,   u3,v3,   u4,v4
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)];

附录 99行代码

1 %%%% A99 LINE TOPOLOGY OPTIMIZATION CODE BY OLE

SIGMUND,OCTOBER 1999 %%%

2function top(nelx,nely,volfrac,penal,rmin);

3 %INITIALIZE

4x(1:nely,1:nelx) = volfrac;

5 loop =0;

6 change= 1.;

7 %START ITERATION

8 whilechange > 0.01

9 loop =loop + 1;

10 xold= x;

11 %FE-ANALYSIS

12[U]=FE(nelx,nely,x,penal);

13 %OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS

14 [KE]= lk;

15 c =0.;

16 forely = 1:nely

17 forelx = 1:nelx

18 n1 =(nely+1)*(elx-1)+ely;

19 n2 =(nely+1)* elx +ely;

20 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);

21 c = c+ x(ely,elx)^penal*Ue’*KE*Ue;

22 dc(ely,elx)= -penal*x(ely,elx)^(penal-1)*

Ue’*KE*Ue;

23 end

24 end

25 %FILTERING OF SENSITIVITIES

26 [dc]= check(nelx,nely,rmin,x,dc);

27 %DESIGN UPDATE BY THE OPTIMALITY CRITERIA METHOD

28 [x] =OC(nelx,nely,x,volfrac,dc);

29 %PRINT RESULTS

30change = max(max(abs(x-xold)));

31disp([’ It.: ’ sprintf(’%4i’,loop) ’ Obj.: ’

sprintf(’%10.4f’,c)...

32 ’Vol.: ’ sprintf(’%6.3f’,sum(sum(x))/

(nelx*nely))...

33 ’ch.: ’ sprintf(’%6.3f’,change )])

34 %PLOT DENSITIES

35colormap(gray); imagesc(-x); axis equal; axis

tight;axis off;pause(1e-6);

36 end

37%%%%%%%%%% OPTIMALITY CRITERIA UPDATE %%%%%%%%%

38function [xnew]=OC(nelx,nely,x,volfrac,dc)

39 l1 =0; l2 = 100000; move = 0.2;

40 while(l2-l1 > 1e-4)

41 lmid= 0.5*(l2+l1);

42 xnew= max(0.001,max(x-move,min(1.,min(x+move,x.

*sqrt(-dc./lmid)))));

43 ifsum(sum(xnew)) - volfrac*nelx*nely > 0;

44 l1 =lmid;

45 else

46 l2 =lmid;

47 end

48 end

49%%%%%%%%%% MESH-INDEPENDENCY FILTER %%%%%%%%%%%

50function [dcn]=check(nelx,nely,rmin,x,dc)

51 dcn=zeros(nely,nelx);

52 for i= 1:nelx

53 for j= 1:nely

54sum=0.0;

55 for k= max(i-round(rmin),1):

min(i+round(rmin),nelx)

56 for l= max(j-round(rmin),1):

min(j+round(rmin),nely)

57 fac =rmin-sqrt((i-k)^2+(j-l)^2);

58 sum =sum+max(0,fac);

59dcn(j,i) = dcn(j,i) + max(0,fac)*x(l,k)

*dc(l,k);

60 end

61 end

62dcn(j,i) = dcn(j,i)/(x(j,i)*sum);

63 end

64 end

65%%%%%%%%%% FE-ANALYSIS %%%%%%%%%%%%

66function [U]=FE(nelx,nely,x,penal)

67 [KE]= lk;

68 K =sparse(2*(nelx+1)*(nely+1), 2*(nelx+1)*

(nely+1));

69 F =sparse(2*(nely+1)*(nelx+1),1); U =

sparse(2*(nely+1)*(nelx+1),1);

70 forely = 1:nely

71 forelx = 1:nelx

72 n1 =(nely+1)*(elx-1)+ely;

73 n2 =(nely+1)* elx +ely;

74 edof= [2*n1-1; 2*n1; 2*n2-1; 2*n2; 2*n2+1;

2*n2+2;2*n1+1;2*n1+2];

75 K(edof,edof)= K(edof,edof) +

x(ely,elx)^penal*KE;

76 end

77 end

78 %DEFINE LOADSAND SUPPORTS(HALF MBB-BEAM)

79F(2,1) = -1;

80fixeddofs = union([1:2:2*(nely+1)],

[2*(nelx+1)*(nely+1)]);

81alldofs = [1:2*(nely+1)*(nelx+1)];

82freedofs = setdiff(alldofs,fixeddofs);

83 %SOLVING

84U(freedofs,:) = K(freedofs,freedofs) \

F(freedofs,:);

85U(fixeddofs,:)= 0;

86%%%%%%%%%% ELEMENT STIFFNESS MATRIX %%%%%%%

87function [KE]=lk

88 E =1.;

89 nu =0.3;

90 k=[1/2-nu/6 1/8+nu/8 -1/4-nu/12 -1/8+3*nu/8 ...

91 -1/4+nu/12-1/8-nu/8 nu/6 1/8-3*nu/8];

92 KE =E/(1-nu^2)*

[ k(1)k(2) k(3) k(4) k(5) k(6) k(7) k(8)

93 k(2)k(1) k(8) k(7) k(6) k(5) k(4) k(3)

94 k(3)k(8) k(1) k(6) k(7) k(4) k(5) k(2)

95 k(4)k(7) k(6) k(1) k(8) k(3) k(2) k(5)

96 k(5)k(6) k(7) k(8) k(1) k(2) k(3) k(4)

97 k(6)k(5) k(4) k(3) k(2) k(1) k(8) k(7)

98 k(7)k(4) k(5) k(2) k(3) k(8) k(1) k(6)

99 k(8)k(3) k(2) k(5) k(4) k(7) k(6) k(1)];