Thermoelectric Material Tensor Derived from the Onsager–de Groot–Callen Model

The Onsager–de Groot–Callen model describes thermoelectricity in the framework of the

PDE based determination of piezoelectric material tensors

The exact numerical simulation of piezoelectric transducers requires the knowledge of all material tensors that occur in the piezoelectric constitutive relations. To account for mechanical, dielectric and piezoelectric losses, the material parameters are assumed to be complex. The issue of material tensor identification is formulated as an inverse problem: As input measured impedance values for different frequency points are used, the searched-for output is the complete set of material parameters. Hence, the forward operator F mapping from the set of parameters to the set of measurements, involves solutions of the system of partial differential equations arising from application of Newton's and Gauss' law to the piezoelectric constitutive relations. This, via two or three dimensional finite element discretisation, leads to an indefinite system of equations for solving the forward problem. Well-posedness of the infinite dimensional forward problem is proven and efficient solution strategies for its discretized version are presented. Since unique solvability of the inverse problem may hardly be verified, the system of equations we have to solve for recovering the material tensor entries can be rank deficient and therefore requires application of appropriate regularisation strategies. Consequently, inversion of the (nonlinear) parameter-to-measurement map F is performed using regularised versions of Newton's method. Numerical results for different piezoelectric specimens conclude this paper.

Topology Optimization Algorithm for Plates and Shells Subjected to Plastic Deformations

A topology optimization method for plates and shells subjected to plastic deformations is presented. The algorithms is based on the generalized layout optimization method invented by Bendsϕe and Kikuchi (1988), where an admissible design domain is assumed to be composed of microstructures with periodic cavities. The sizes of the cavities and the rotational angles of the microstructures are design variables which are optimized so as to minimize the applied work. The macroscopic material tensor for the porous material is numerically calculated by the homogenization method for the sensitivity analysis. In this paper, the method is applied to two-dimensional elasto-plastic problems. A database of the material tensor and its interpolation technique are presented. The algorithm is expanded into thin shells subjected to finite deformations. Several numerical examples are shown to demonstrate the effectiveness of these algorithms.

Constitutive Relations for Moving Plasmas

A covariant form of Ohm’s Law for bianisotropic plasmas is set up connecting the four-dimensional current density with the field tensor through a material tensor of order three. This tensor is represented by two four-dimensional material tensors of order two, which are closely related to the usual threedimensional conductivity tensors; its symmetry properties are investigated and relations between its components and those of the three-dimensional material tensors are established. In addition a covariant constitutive equation for a plasma is formulated using the polarization model, where the four-dimensional current density is substituted by a polarization tensor. Thereby the plasma properties - like the dielectric and magnetic properties of a medium - are expressed by a material tensor of order four, whose representation is generalized for bianisotropic media