We consider a class of piezoelectric materials with defects, voids, and/or elastic dielectric or piezoelectric inclusions. We develop computationally highly efficient as well as mathematically highly accurate methods for the Direct Numerical Simulation (DNS) of micro/meso mechanics of such materials, for the purposes of: 1. determining the meso/macro physical properties of such materials, and 2. studying the mechanics of damage initiation at the micro-level in such materials. In this paper, we develop what we label as "Trefftz-Lekhnitskii Grains (TLGs)", each of which can model a single grain of piezoelectric materials with voids. These TLGs are of arbitrary geometrical shapes, to mimic the natural shape of each micro-grain of the material. The TLGs are based on expressing the mechanical and electrical fields in the interior of each grain in terms of the Trefftz solution functions derived from Lekhnitskii formulation for piezoelectric materials. The potential functions are written in terms of Laurent series which can describe interior or exterior domains where negative exponents are used only in the latter case. The boundary conditions at the outer boundaries of each TLG can be enforced using a boundary variational principle, collocation or least squares method, while the boundary conditions at the inner (void/inclusion) boundary can be enforced using collocation/least squares, or by using the special solution set which satisfy the traction-free, charge-free boundary conditions at the void periphery. These various methods of enforcing the boundary conditions generate different grains which are denoted as TLG-BVPs, TLG-C, TLG-Cs, TLG-LS, TLG-LSs (where BVP refers to "boundary variational principle", C refers to " collocation", LS refers to "Least Squares", and s refers to "special solution set"). Several examples of the DNS of micro/meso mechanics of porous piezoelectric materials are presented, not only to determine the macro physical properties of such materials, but also to study the mechanisms for damage precursors in such intelligent materials.
- Voronoi cells