我正在开发一个有限元软件,可以最大限度地减少机械结构的能量。使用 Octave 及其优化包,我遇到了一个奇怪的问题:当我使用超过 300 个自由度 (DoF) 时,lm_feasible 算法根本不计算。另一种算法 (sqp) 执行计算,但当我使结构复杂化并且不在我的测试用例中时效果不佳。
lm_feasible 算法的 DoF 数量有限制吗?
如果是这样,最大可能有多少 DoF?
给出代码如何工作的概述和一般概念:
[x,y] = geometryGenerator()
U = zeros(lenght(x)*2,1);
U(1:2:end-1) = x;
U(2:2:end) = y;
%Non geometric argument are not optimised, and fixed during calculation
fct =@(U)complexFunctionOfEnergyIWrap(U(1:2:end-1),U(2:2:end), variousMaterialPropertiesAndOtherArgs)
para = optimset("f_equc_idx",contEq,"lb",lb,"ub",ub,"objf_grad",dEne,"objf_hessian",d2Ene,"MaxIter",1000);
[U,eneFinale,cvg,outp] = nonlin_min(fct,U,para)
完整示例:
clear
pkg load optim
function [x,y] = geometryGenerator(r,elts = 100)
teta = linspace(0,pi,elts = 100);
x = r * cos(teta);
y = r * sin(teta);
endfunction
function ene = complexFunctionOfEnergyIWrap (x,y,E,P, X,Y)
ene = 0;
for i = 1:length(x)-1
ene += E*(x(i)/X(i))^4+ E*(y(i)/Y(i))^4- P *(x(i)^2+(x(i+1)^2)-x(i)*x(i+1))*abs(y(i)-y(i+1));
endfor
endfunction
[x,y] = geometryGenerator(5,100)
%Little distance from axis to avoid division by zero
x +=1e-6;
y +=1e-6;
%Saving initial geometry
X = x;
Y = y;
%Vectorisation of the function
%% Initial vector
U = zeros(length(x)*2,1);
U(1:2:end-1) = linspace(min(x),max(x),length(x));
U(2:2:end) = linspace(min(y),max(y),length(y));
%%Constraints
Aeq = zeros(3,length(U));
%%% Blocked bottom
Aeq(1,1) = 1;
Aeq(2,2) = 1;
%%% Sliding top
Aeq(3,end-1) = 1;
%%%Initial condition
beq = zeros(3,1);
beq(1) = U(1);
beq(2) = U(2);
beq(3) = U(end-1);
contEq = @(U) Aeq * U - beq;
%Parameter
Mat = 0.2e9;
pressure = 50;
%% Vectorized function. Non geometric argument are not optimised, and fixed during calculation
fct =@(U)complexFunctionOfEnergyIWrap(U(1:2:end-1),U(2:2:end), Mat, pressure, X, Y)
para = optimset("Algorithm","lm_feasible","f_equc_idx",contEq,"MaxIter",1000);
[U,eneFinale,cvg,outp] = nonlin_min(fct,U,para)
xFinal = U(1:2:end-1);
yFinal = U(2:2:end);
plot(x,y,';Initial geo;',xFinal,yFinal,'--x;Final geo;')
最佳答案
有限元方法通常被表述为最小化问题的最佳标准,这相当于虚拟工作原理(参见 Hughes of Bathe 等书籍)。 Virtual Work 表示一组线性(或非线性)方程,可以更有效地求解(使用 fsolve)。
如果出于某种原因,您必须将问题作为优化问题来解决,那么,如果您正在考虑线性弹性,则应变能是二次的,因此您可以使用 qp Octave 函数。
使用稀疏矩阵也可能会有所帮助。
关于octave - lm_feasible 算法的自由度数有限制吗?如果是这样,限制是多少?,我们在Stack Overflow上找到一个类似的问题: https://stackoverflow.com/questions/56985618/