# Optimization of Electro-Mechanical Smart Structures

## Contents

Lehrstuhl für Angewandte Mathematik III
Uni Erlangen

Tel: 09131 85-25225
Fax: 09131 85-25228

Applied Mechatronics
Alpen-Adria Universität Klagenfurt, Austria

phone: +(43) 463 2700 3562
phone: +(43) 463 2700 3565 (secr.)
fax: +(43) 463 2700 3698

homepage: http://am.uni-klu.ac.at

Lehrstuhl für Angewandte Mathematik II
Uni Erlangen

Tel: 09131 8527509
Fax: 09131 85-28126

## DFG funded assistants

Fabian Schury (Uni Erlangen)
Homepage: http://www.am.uni-erlangen.de/~schury

## Overview

Our objective is to perform topology optimization of PDE-based problems.

Model setup without PML

We use applied mathematics and sophisticated engineering methods to cope with the challenging goal of numerically realizing a robust, validated and efficient optimization strategy for a problem that originally arose in engineering practice. Furthermore, we investigate the analytical structure of the underlying partial differential equations concerning existence, uniqueness and wellposedness and the analysis of mathematical concepts like topology and shape gradients for coupled PDE problems.

## Problem setting

Discretization of the piezoelectric-mechanical-acoustic system. Note the different meshing for air and structure via non-matching grids.

PDEs:

$\rho_{ m} \ddot{u} - \mathcal{B}^T \left( [c^E] \mathcal{B} u + [e ]^T \nabla \phi \right) = 0\ \ \ \textrm{in}\qquad \Omega_{\text{piezo}}$

$\mathcal{B}^{T} \left( [e] \mathcal{B} u - [\varepsilon^S] \nabla \phi \right) = 0\ \ \ \textrm{in}\qquad \Omega_{\text{piezo}}$

$\rho_{ m} \ddot u - \mathcal{B}^T [c] \mathcal{B} u = 0\ \ \ \textrm{in} \qquad\Omega_{\text{plate}}$

$\frac{1}{c^2}\, \ddot p_{ a} - \Delta p_{ a} = 0 \ \ \ \textrm{in}\qquad \Omega_{\text{air}}$

$\frac{1}{c^2}\, \ddot p_{ a} - {\mathcal A}^2 \, p_{ a} = 0 \ \ \ \textrm{in}\qquad \Omega_{\text{PML}}$

Interface conditions:

$n \cdot \ddot u = - \frac{1}{\rho_{ a}} \frac{\partial p_{ a}}{\partial n} \qquad \textrm{ on } \qquad \Gamma_\textrm{iface}\times (0,T)$

$\sigma_{ n} = - n \ p_a \qquad \textrm{ on } \qquad \Gamma_\textrm{iface}\times (0,T)$

## Methods

As the analysis of the piezoelectric topology gradient was performed concurrently to the first project phase only, expending efforts in the extension of the SIMP method now gives the unique situation of having both kinds of topology optimization simultaneously implemented in the multiphysics simulation package CFS++ of PI M. Kaltenbacher. Thus comparison and combination of the methods based on realistic large scale 2D and 3D models can be performed on an elaborated numerical base.

### SIMP

Optimal topology for piezoelectric loudspeaker at 800 Hz (SIMP)

For the variable density approach (often denoted by the SIMP method (Solid Isotropic Material with Penalization), an artificial density function $0 < \rho \leq 1$ is introduced in order to represent the material distribution within the optimization domain. Via an interpolation function μ(ρ) the material is locally modified by piecewise constant ρ such that full material is represented for ρ close to one and void material for ρ close to a lower bound. Meanwhile, this approach has been adapted to several physical models, mainly elasticity but also piezoelectricity, acoustics and many more.

During the course of this project, an important phenomenon was observed which has not, to our knowledge, been published before: For our models and objective function, we were able to achieve topology optimization of piezoelectric materials without a volume constraint (no appearance of unphysical intermediate material for most cases). Additionally, no penalizing interpolation functions (which change the actual problem) are necessary and no filtering of the objective gradients has to be performed. Without a volume constraint it is furthermore possible to analyze the optimization for different target frequencies.

### Topology and Shape Gradient

3D elasticity problem with topology gradient

While the SIMP-method provided us with a well established method in order to achieve topology optimization for the relaxed problem as described above, the proposed method involving topological gradients and shape sensitivities guarantees a 0-1-design. However, as was established during the first part of the funding period, the analytical framework for establishing the topological gradient of piezoelectric material is far more involved than the one known for 3D elasticity. Moreover, the existing results on shape sensitivities for 3D elasticity did not directly generalize to the system under consideration. It is for this reason that the project leaders decided to first extend the SIMP method to the model in order to be able to keep up with the experiments. It was anticipated that the results obtained by the SIMP-method would serve as a validation platform.

Levelset with topology gradient

In order to proceed along the two lines indicated, we decided to first pursue, in addition to the SIMP-approach and according to the working plan, the topology-shape-optimization approach first for 2D and 3D elasticity in order to gain numerical expertise for full piezoelectric-mechanical-acoustic coupling.

Trying to mimic existing codes for topology optimization for 3D elasticity using FEM would have left us with the problem of properly extending the FEM to piezoelectric-mechanical-acoustic coupling envisaged in this project. As the software package CFS++ (M. Kaltenbacher) focuses exactly on the latter system and is well established and validated, it is mandatory that a solid 3D topology optimization based on topological gradients and shape sensitivities was the avenue along which we should proceed.

## Selected Publications

During the course of the project, several publications have been written by the project members, partly in cooperation with other researchers.

• with D. Braess: Efficient 3D Finite-Element-Formulation for Thin Mechanical and Piezo-electrical Structures, appeared in Int. J. Num. Meth. Eng.
• with B. Kaltenbacher: Topology Optimization of Piezoelectric Layers Using the SIMP Method, Preprint SPP1253-03-01, submitted to J. Struct, Multidisc. Optim.
• Topology Optimization of a Piezoelectric-Mechanical Actuator With Single- and Multiple-Frequency Excitation, Preprint SPP1253-03-02, accepted at Int. J. Appl. Electromagn. Mech.
• with G. Perla-Menzala, A. Novotny and J. Sokolowski: Wellposedness of a Dynamic Wave-Propagation Problem in a Coupled Piezoelectric-Elastic-Acoustic Structure, in preparation
• with G. Perla-Menzala, A. Novotny and J. Sokolowski: Shape Sensitivities for Layered Piezoelectric Materials, in preparation
• with M. Ertl and M. Meiler: Physical Modeling and Numerical Computation of Magnetostriction, accepted for Comp. Math. Electr. Electron. Eng.