Polygonal finite elements for three-dimensional Voronoi-cell-based discretisations
DOI:
https://doi.org/10.13052/17797179.2012.702432Keywords:
polygonal finite elements, hybrid finite element method, three-dimensional Voronoi cells, stress approximationAbstract
Hybrid finite element formulations in combination with Voronoi-cell-based discretisation methods can efficiently be used to model the behaviour of polycrystalline materials. Randomly generated three-dimensional Voronoi polygonal elements with varying numbers of surfaces and corners in general better approximate the geometry of polycrystalline microor rather grain-structures than the standard tetrahedral and hexahedral finite elements. In this work, the application of a polygonal finite element formulation to three-dimensional elastomechanical problems is elaborated with special emphasis on the numerical implementation of the method and the construction of the element stiffness matrix. A specific property of Voronoi-based discretisations in combination with a hybrid finite element approach is investigated. The applicability of the framework established is demonstrated by means of representative numerical examples.
Downloads
References
Aurenhammer, F. (1991). Voronoi diagrams – a survey of a fundamental geometric structure. AMC Computing
Surveys, 23, 345–405.
Ghosh, S., & Mallett, R.L. (1994). Voronoi cell finite elements. Computers & Structures, 50, 33–46.
Ghosh, S., & Moorthy, S. (1995). Elastic-plastic analysis of arbitrary heterogeneous materials with the
Voronoi cell finite element method. Computer Methods in Applied Mechanics and Engineering, 121,
–409.
Ghosh, S., & Moorthy, S. (2004). Three dimensional Voronoi cell finite element model for microstructures
with ellipsoidal heterogeneties. Computational Mechanics, 34, 510–531.
Hughes, T.J.R. (1987). The finite element method. Prentice-Hall.
Jayabal, K., & Menzel, A. (2011). Application of polygonal finite elements to two-dimensional mechanical
and electro-mechanically coupled problems. Computer Modeling in Engineering & Sciences, 73,
–208.
Jayabal, K., & Menzel, A. (2012). Voronoi-based three-dimensional polygonal finite elements for electro-
mechanical problems. Computational Materials Science. doi : 10.1016/j.commatsci.2012.02.049.
Jayabal, K., Menzel, A., Arockiarajan, A., & Srinivasan, S.M. (2011). Micromechanical modelling of
switching phenomena in polycrystalline piezoceramics: Application of a polygonal finite element
approach. Computational Mechanics, 48, 421–435.
Mousavi, S.E., & Sukumar, N. (2011). Numerical integration of polynomials and discontinuous functions
on irregular convex polygons and polyhedrons. Computational Mechanics, 47, 535–554.
Pian, T.H.H. (1964). Derivation of element stiffness matrices by assumed stress distribution. American
Institute of Aeronautics and Astronautics, 2, 1333–1336.
Sze, K.Y., & Sheng, N. (2005). Polygonal finite element method for nonlinear constitutive modeling of
polycrystalline ferroelectrics. Finite Elements in Analysis and Design, 42, 107–129.
Timoshenko, S.P., & Goodier, J.N. (1970). Theory of elasticity, 3rd ed. New York: McGraw-Hill.
