propSym: a tool to establish relationships between property constants for material property tensors of any order

The properties of crystalline materials can be described mathematically by tensors whose components are generally known as property constants. Tabulations of these constants in terms of the independent components are well known for common material properties (e.g. elasticity, piezoelectricity etc.) aptly described by tensors of lower rank (e.g. ranks 2–4). General relationships between constants of higher rank are often unknown and sometimes reported incorrectly. A computer program is developed here to calculate the property constant relationships of a property of any order, represented by a tensor of any rank and point group. Tensors up to rank 12, e.g. the tensor of sixth-order elastic constants c 6 ijklmnpqrs , can be calculated on a standard computer, while ranks higher than 12 are best handled on a supercomputer. Output is provided in either full index form or a reduced index form, e.g. the Voigt index notation common to elasticity. As higher-order tensors are often associated with nonlinear material responses, the program provides an accessible means to investigate the important constants involved in nonlinear material modeling. The routine has been used to discover several incorrect relationships reported in the literature.

Terminology and notation

Notation and sign conventions adopted for the rest of the book are explained. The book employs index notation, but not abstract index notation. The metric signature for GR is taken as (-1,1,1,1). Terminology such as “local inertial frame” and “Rieman normal coordinates” is explained.

On the Unification of Schemes and Software for Wavelets on the Interval

We revisit the construction of wavelets on the interval with various degrees of polynomial exactness, and explain how existing schemes for orthogonal- and Spline wavelets can be extended to compactly supported delay-normalized wavelets. The contribution differs substantially from previous ones in how results are stated and deduced: linear algebra notation is exploited more heavily, and the use of sums and complicated index notation is reduced. This extended use of linear algebra eases translation to software, and a general open source implementation, which uses the deductions in this paper as a reference, has been developed. Key features of this implementation is its flexibility w.r.t. the length of the input, as well as its generality regarding the wavelet transform.

Multidimensional Arrays, Indices and Kronecker Products

Much of the algebra that is associated with the Kronecker product of matrices has been rendered in the conventional notation of matrix algebra, which conceals the essential structures of the objects of the analysis. This makes it difficult to establish even the most salient of the results. The problems can be greatly alleviated by adopting an orderly index notation that reveals these structures. This claim is demonstrated by considering a problem that several authors have already addressed without producing a widely accepted solution.

Notation

This chapter explains the notation for the basic construction of the correction. It employs the Einstein summation convention, according to which there is an implied summation when a pair of indices is repeated, and the conventions of abstract index notation, so that upper indices and lower indices distinguish contravariant and covariant tensors. It also presents the notation concerning multi-indices which will later prove helpful for expressing higher order derivatives of a composition. In this notation, a K-tuple of multi-indices is said to form an ordered K-partition of a multi-index if there is a partition whereby the subsets are pairwise disjoint and are ordered by their largest elements.