3D mechanic compliance minimization

The simulations in 3d look much better, but still need some hours to calculate. I hope to speed this up by a factor of 5-10, I will need it for real piezo optimization.

This is a bridge, fixed at the four lower corner points and three load points in the lower plane. I allowed half of the total volume. It is quite symmetric. The optimization is done with the optimiality criteria and the iterations for all this examples are between 15 and 40.


Here the bridge has only a single lower load point.


Here the single load point example but a volume fraction of 10%.


Here Tom (thanks again :)) suggested to optimize the eiffel tower. The top center force in positive z-direction, the lower four corners are fixed.


This is the most challenging example. For this bridge I apply a downwards force on every node on the lower plane. Currently it is still running, so this is a snapshot of iteration 15. The volume fraction is 20% and the penalization is 5 instead of 3. Still I don't think that there will ever be a nice black and white result. The green is for a pseudo density of 0.5. I'm curious if you have any idea to solve this optimization problem. I thought about adaptive penalization (increasingly higher depending on the greyness) - does anyone have experience in this field?


Finally the aqueduct which looks much better in 3d :) Here I use a penalization of 4. With the standard 3 I get nice big holes but I cannot get rid of the greyness. At leas here higher penalization works. The load applied on the upper long central line (a "pipe"). I fix the upper four corners and the front end back point of the feet. Note that there is a bead!

2D mechanic optimizations

From time to time, I calculate 2d and 3d mechanical minimizing compliance problems. This is some kind of leasure while I work on the piezo optimization - so don't take it serious, it's just for fun. On the other side, some optimizations don't really fit to what I expected from publications. So if you also do also thus optimizations, please let me know if you have the same or different results. I can give the parametrization details on request. I use quite fine discretizations and a filter regularization as given in [2] to make the images nicer .:)

The cantilever is in my opinion too thin on the right end which is loaded.


This bridge gets its load from the top. I miss a lower horizontal bar.


Here the bridge has a lower load in the center.


This is a aqueduct, it looks much cooler in 3d. Thanks to Tom for the suggestion for this optimization problem.


If you have any ideas what could interesting to optimize, just let me know.

Using IPOPT as Optimizer

I integrated IPOPT as an external optimizer, again still for the mechanical compliance minimization. I do not use the Hessian of the Lagrangian but let IPTOP to an L-BFGS approximation. The software is really nice - good documentation, open source with a good license (CPL) and a good C++ frontend with cool features.

The results are also good for 2D, the number of iterations is roughly comparable to the optimality criteria and there are about 50% more function evaluations for line search (determine the step length).

Bad news are, that in 3D the performance is much worse - but I still did not do examine what is really going on. Also I had to increase the mechanical load by a few magnitudes - otherwise I had nearly instant convergence with only slight modification of my inital guess for the desing parameters. I'll have to implement the break conditions by myself - what is supported by IPOPT

Scope of current work

Before doing structural optimization of a piezo based loudspeaker, I implement the standard approach in simple compliance minimization in mechanics.

More precisely I do SIMP, following the publications from Sigmund and Bensoe. This includes the simple heuristic optimality criteria update rule. I found the form given in the 99-lines code publication to suit better than the form in the book (see my publications reference list)