Fabian Schury

Design of Auxetic Structures via Mathematical Optimization

The design of auxetic structures is traditionally based on engineering experience, see, for instance, refs. [1,2]. The goal of this article is to demonstrate how an existing auxetic structure can be improved by structural optimization techniques and manufactured by a selective electron beam melting (SEBM) system. Auxetic materials possess negative Poisson’s ratios ν which is in contrast to almost all engineering materials (e. g. metals ν ≈ 0.3). The negative Poisson’s ratio translates into the unusual geometric effect that these materials expand when stretched and contract when compressed. This behavior is of interest for example for fastener systems and fi lters. [ 2 , 3 ] It also results in interesting mechanical behavior like increased penetration resistance for large negative Poisson’s ratios [ 4 ] and in the case of isotropy for higher shear strength relative to the Young’s modulus [ 2 ] of the structure. The auxetic behavior can always be traced back to complex mesoor microscopic geometric mechanisms, e. g. unfolding of inverted elements or rotation of star shaped rigid units relative to each other. [ 5 – 7 ] These geometries can be realized on vastly different scales from specifi cally designed polymers to reactor cores. [ 2 , 8 ]

Topology optimization of a piezoelectric-mechanical actuator with single- and multiple-frequency excitation

We present the topology optimization of an assembly consisting of a piezoelectric layer attached to a plate with support. The optimization domain is the piezoelectric layer. Using the SIMP (Solid Isotropic Material with Penalization) method with forced vibrations by harmonic electrical excitation, we achieve a maximization of the dynamic displacement. We show that the considered objective function can be used under certain boundary conditions to optimize the sound radiation. The vibrational patterns resulting from the optimization are analysed in comparison with the modes from an eigenvalue analysis. Multiple-frequency optimization is achieved by adaptive weighted sums. As a second optimization criterion, a flat frequency response is integrated in the optimization process.

Efficient Two-Scale Optimization of Manufacturable Graded Structures

A new method for the optimal design of inhomogeneous meso-structured components is presented. The method is based on a variant of the free material optimization (FMO) method. Instead of coupling a macroscopic model directly with the meso-scale in the framework of a two-scale approach, information from the meso-scale is incorporated into the FMO model by additional constraints. The FMO result, given in terms of optimized tensors, is interpreted by the inverse homogenization approach. Rather than only computing the meso-structure for each “FMO cell” independently, approximate continuity of the meso-structure is enforced. Thus, the final result is a manufacturable structurally graded material. The feasibility of the whole method is demonstrated by means of an academic example.

The Eshelby Theorem and Application to the Optimization of an Elastic Patch

We present the analysis for finding optimal locations and rotations of anisotropic material inclusions in a matrix material by using the polarization matrix. We compare different types of cost functionals, in particular local ones, and show their respective differences. We use the Eshelby theorem and the representation of stresses based on the link matrix. As an analytical model reduction technique, this allows for efficient numerical computation which is demonstrated for two selected examples.

Optimization of Electro-mechanical Smart Structures

We present topology optimization of piezoelectric loudspeakers using the SIMP method and topology gradient based methods along with analytical and numerical results.