Open Access

ARTICLE

Application of Polygonal Finite Elements to Two-Dimensional Mechanical and Electro-Mechanically Coupled Problems

K. Jayabal¹, A. Menzel^1,2

1 Institute of Mechanics, TU Dortmund, Germany
2 Division of Solid Mechanics, Lund University, Sweden

Computer Modeling in Engineering & Sciences 2011, 73(2), 183-208. https://doi.org/10.3970/cmes.2011.073.183

Abstract

Naturally evolving Voronoi discretisation of two-dimensional plane domains renders representative microstructures that turn out to be useful for the modelling and simulation of polycrystalline materials. Hybrid finite element approaches are employed on such polygonal discretisations to solve, for instance, mechanical and electromechanical problems within a finite element context. In view of solving mechanical problems, varying order of polynomial functions are suggested in the literature to sufficiently approximate stresses within the polygonal finite elements. These are, in addition to the order of the approximation functions for the displacements, characterised by the number of edges in the polygonal elements. It appears, as demonstrated in this work, that the naturally evolving Voronoi discretisations exhibit a specific property when combined with a hybrid polygonal finite element approach. This property allows the choice of stress approximating functions in polygonal finite elements to be based only on the order of the displacement approximating functions regardless of the number of edges in the element. Such a relation also appears to hold in coupled electromechanical problems between the approximating functions for the electric displacements and the electric potential. The realisation of such a property is demonstrated through several standard numerical examples and also with an application on a representative piezoceramic microstructure.

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