On the effect of self-penalization of piezoelectric composites in topology optimization

Abstract

We investigate the occurrence of self-penalization in topology optimization problems for piezoceramic-mechanical composites. Our main goal is to give physical interpretations for this phenomenon, i.e., to study the question why for various problems intermediate material values are not optimal in the absence of explicit penalization of the pseudo densities. In order to investigate this effect numerical experiments for several static and/or dynamic actuator and sensor objective functions are performed and their respective results are compared. The objective functions are mean transduction, displacement, sound power, electric potential, electric energy, energy conversion and electric power.

Appendix: Material properties

The applied piezoelectric material is lead zirconate titanate PZT-5A with the following properties (in Voigt notation): mass density 7.75025 kg/m³, damping tanδ = 0.015 at 1,000 Hz, stiffness in GPa $c_{11}^E=c_{22}^E=126$, $c_{13}^E=79.5$, $c_{23}^E=c_{22}^E=84.1$, $c_{44}^E=c_{55}^E=c_{66}^E=23$, coupling in N/C e ₁₅ = e ₂₄ = 17, e ₃₁ = e ₃₂ = − 6.5, e ₃₃ = 23.3, permittivity in 10^− 8 F/m $\varepsilon_{11}^S=\varepsilon_{22}^S=1.51$, $\varepsilon_{33}^S=1.27$.

The supporting aluminum plate has the following isotropic properties: Poisson’s ratio ν = 0.34, Young’s modulus E = 70.7 GPa, mass density 2.7 kg/m³, damping tanδ = 0.03 at 1,000 Hz.

For an accurate simulation model it might be necessary to determine the piezoelectric coupling coefficients by inverse methods as in Rupitsch and Lerch (2009).