Paper on Piezoelectric Self-Penalization Published

Our second paper is published (online):

F. Wein, M. Kaltenbacher, B. Kaltenbacher, G. Leugering, E. Baensch and F. Schury; On the effect of self-penalization of piezoelectric composites in topology optimization; Structural and Multidisciplinary Optimization; DOI: 10.1007/s00158-010-0570-2

springerlink

It covers the effect of self-penalization of many piezoelectric topology optimization problems. That means that no penalization, volume constraint and regularization is applied to gain often mesh independent black and white solutions. It's not a method we describe but a phenomenon, so to report when it does not work is part of the job.
I guess we implemented almost all relevant piezoelectric objective functions and to compare them based on the same model is IMHO also of interest. The objective functions are mean transduction, displacement, sound power, electric potential, electric energy, energy conversion and electric power.

We give an explanation based on the piezoelectric model: stiffness, electrostatic and piezoelectric coupling. More density is bad with respect to stiffness but good for the other fields, opposite for less density. That means that there is balance of the effects and when this balance is outside the feasible design set (0,1], there is no greyness.

Self-penalization has been, to my knowledge, first reported by Ole Sigmund and Jakob S. Jensen 2003 with (elastic) wave guiding. It also appears in other, especially non-elastic problems but has never been investigated. Meanwhile I almost belief that probably most non-compliance problems show some form of self-penalization. I hope I can find a general explanation as this might eventually lead to new method in topology optimization. :) I have at least one idea on how to proceed ... Let's see when I find the time and if I have success.

P.S.: It seems that the term "self-penalization" has not been used before in the context of topology optimization (I actually don't want to know what google finds for self-penalization in other contexts :) ) but I learned the expression from Ole Sigmund, so he deserves the credits.

Mean Transduction

In 1999, Silva et al. introduced in

Silva, Nishiwaki, Kikuchi; Design of piezocomposite materials and piezoelectric transducers using topology optimization—Part II; Archives of Computational Methods in Engineering; 1999


the objective function mean transduction. It describes the piezoelectric coupling, and it is quite intuitive to maximize that coupling. The mean transduction was applied quite often by Silva. To my knowledge there are no recent publications, where it is used, but I might be wrong.

I don't use the function but as I started 4 years ago with it, it is still in my mind and somehow I still try to actually understand it properly.

I make currently good progress in writing my thesis and just work on a section about mean transduction. I finally found the link between mean transduction and classical adjoint based sensitivy analysis - which clearly helps in understanding the nature of the mean transduction. Just, I don't know if this is of interest, as I don't know, if the mean transduction is used any more. For sensor applications, maximizing the electric power as done by Cory Rupp is IMHO more practical - and for actor applications the physics might be necessary, e.g. acoustic formulations to prevent acoustic short circuits as I have shown.

This piezo stuff is really heavy (optimization) stuff :)