Development of T-Trefftz four-node quadrilateral and Voronoi Cell Finite Elements for macro-& micromechanical modeling of solids

L. Dong, S. N. Atluri

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


In this paper, we explore three different ways of developing T-Trefftz finite elements of quadrilateral as well as polygonal shapes. In all of these three approaches, in addition to assuming an inter-element compatible displacement field along the element boundary, an interior displacement field for each element is independently assumed as a linear combination of T-Trefftz trial functions. In addition, a characteristic length is defined for each element to scale the T-Trefftz modes, in order to avoid solving systems of ill-conditioned equations. The differences between these three approaches are that, the compatibility between the independently assumed fields at the boundary and in the interior, are enforced alternatively, using a two-field boundary variational principle, collocation, and the least squares method. The corresponding four-node quadrilateral elements with/without drilling degrees of freedom are developed, for modeling macrostructures of solids. These three approaches are also used to derive T-Trefftz Voronoi Cell Finite Elements (VCFEM), for micromechanical analysis of heterogeneous materials. Several two dimensional macro-& micromechanical problems are solved using these elements. Computational results demonstrate that the elements derived using the collocation method are very simple, accurate and computationally efficient. Because the elements derived by this approach are also not plagued by LBB conditions, which are almost impossible to be satisfied a priori, we consider this class of elements to be useful for engineering applications in micromechanical modeling of heterogeneous materials.

Original languageEnglish
Pages (from-to)69-118
Number of pages50
JournalCMES - Computer Modeling in Engineering and Sciences
Issue number1
StatePublished - 2011


  • Collocation
  • Drilling degree of freedom
  • Finite element
  • LBB conditions
  • Least squares
  • T-Trefftz
  • Variational principle


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