Multi-region Trefftz collocation grains (MTCGS) for modeling piezoelectric composite and porous materials in direct and inverse problems

Peter L. Bishay, Abdullah Alotaibi, Satya N. Atluri

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


A simple and efficient method for modeling piezoelectric composite and porous materials to solve direct and inverse 2D problems is presented in this paper. The method is based on discretizing the problem domain into arbitrary polygonal-shaped regions that resemble the physical shapes of grains in piezoelectric polycrystalline materials, and utilizing the Trefftz solution functions derived from the Lekhnitskii formulation for piezoelectric materials, or for elastic dielectric materials, to express the mechanical and electrical fields in the interior of each grain or region. A simple collocation method is used to enforce the continuity of the inter-region primary and secondary fields, as well as the essential and natural boundary conditions. Each region may contain a void, an elastic dielectric inclusion, or a piezoelectric inclusion. The void/inclusion interface conditions are enforced using the collocation method, or using the special solution set which is available only for the case of voids (traction-free, charge-free boundary conditions). The potential functions are written in terms of Laurent series which can describe interior or exterior domains, while the negative exponents are used only in the latter case. Because Lekhnitskii's solution for piezoelectric materials breaks down if there is no coupling between mechanical and electrical variables, the paper presents this solution in a general form that can be used for coupled (piezoelectric) as well as uncoupled (elastic dielectric) materials. Hence, the matrix or the inclusion can be piezoelectric or elastic dielectric to allow modeling of different types of piezoelectric composites. The present method can be used for determining the meso/macro physical properties of these materials as well as for studying the mechanics of damage initiation at the micro level in such materials. The inverse formulation can be used for determining the primary and secondary fields over some unreachable boundaries in piezoelectric composites and devices; this enables direct numerical simulation (DNS) and health monitoring of such composites and devices. Several examples are presented to show the efficiency of the method in modeling different piezoelectric composite and porous materials in different direct and inverse problems.

Original languageEnglish
Pages (from-to)287-312
Number of pages26
JournalJournal of Mechanics of Materials and Structures
Issue number3
StatePublished - 2014


  • Collocation
  • Composites
  • Inclusion
  • Inverse problems
  • Lekhnitskii
  • Piezoelectric
  • Porous
  • Trefftz
  • Void
  • Voronoi cells


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