If the finite element mesh size becomes large, the traditional direct solver (line 72) used to address the finite element analysis is suffered by longer solving time and some other issues. However, iterative solver can solve large-scale problems efficiently. In this tutorial, you will learn how to replace direct solver by iterative solver in order to handle large-scale topology optimization problem.To this end, line 72 is replaced by a built-in $latex textsc{Matlab}$ function $latex texttt{pcg}$, called preconditioned conjugate gradients method, as shown in the following

[matlab firstline="72" highlight="74"]

tolit = 1e-8;

maxit = 8000;

M = diag(diag(K(freedofs,freedofs)));

U(freedofs,:) = pcg(K(freedofs,freedofs),F(freedofs,:),tolit,maxit,M);

[/matlab]

Direct solver is a special case by setting the preconditioner (line 74) to

[matlab light="true"]M = inv(K);[/matlab]

Of course you can use other preconditioner such as incomplete choleksy by replace the code snippets above with the following

[matlab firstline="72"]

L = ichol(K(freedofs,freedofs));

U(freedofs,:) = pcg(K(freedofs,freedofs),F(freedofs,:),1e-8,1000,L,L');

[/matlab]

The following are some large-scale problem we solved using top3d and iterative solver