Voronoi cell finite elements

Ghosh, Somnath; Mallett, R. L.

doi:10.1016/0045-7949(94)90435-9

Cited by 65 publications

(24 citation statements)

References 15 publications

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“…The following section briefly discusses the polygonal finite element approach used in this work. For detailed background information the reader is referred to [29,32].…”

Section: Polygonal Finite Element Methodsmentioning

confidence: 99%

“…As seen in Eq. (29), within the element (B e ) solely stresses and electric displacements need to be determined-apart from S-whereas displacements and the electric potential have to be specified only along the element boundary (∂B e ). The latter (u and φ) also must satisfy the continuity conditions.…”

Section: Mixed Formulationmentioning

confidence: 99%

“…One way to proceed further from this point is to subdiscretise the individual irregular polygons by means of several standard quadrilateral or triangular elements-with, for instance, the displacement field and the electric potential as degrees of freedom. Alternatively, one may apply Voronoi-cell-based finite elements [29,30]. Such hybrid formulations, as proposed even earlier in [31], allow to avoid further subdiscretisation of a crystal grain structure, the geometry of which is here assumed to be represented by a polygon.…”

Section: Introductionmentioning

confidence: 98%

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Micromechanical modelling of switching phenomena in polycrystalline piezoceramics: application of a polygonal finite element approach

et al. 2011

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A micromechanically motivated model is proposed to capture nonlinear effects and switching phenomena present in ferroelectric polycrystalline materials. The changing remnant state of the ferroelectric crystal is accounted for by means of so-called back fields-such as back stressesto resist or assist further switching processes in the crystal depending on the local loading history. To model intergranular effects present in ferroelectric polycrystals, the computational model elaborated is embedded into a mixed polygonal finite element approach, whereby an individual ferroelectric grain is represented by one single irregular polygonal finite element. This computationally efficient coupled simulation framework is shown to reproduce the specific characteristics of the responses of ferroelectric polycrystals under complex electromechanical loading conditions in good agreement with experimental observations.

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“…The following section briefly discusses the polygonal finite element approach used in this work. For detailed background information the reader is referred to [29,32].…”

Section: Polygonal Finite Element Methodsmentioning

confidence: 99%

Section: Mixed Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Micromechanical modelling of switching phenomena in polycrystalline piezoceramics: application of a polygonal finite element approach

et al. 2011

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“…Usually, the FEM is employed as a means to analyze grain stresses in polycrystals by using different kinds of elements, [1,4,9,[29][30][31][32][33][34]. In the FE treatment, one often imposes two extreme boundary conditions of (i) no grain boundary sliding, [4,9], and (ii) free grain boundary sliding, [9], to account for the intergranular interaction.…”

Section: Self-adaptive Methodsmentioning

confidence: 99%

Computational assessment of material structure weaknesses in polycrystalline materials

2004

Archive of Applied Mechanics

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By incorporating local grain orientation, grain geometry and macroscopic elastic properties, a numerical procedure has been developed for computational prediction of mesoscopic stress and strain distributions in simulated polycrystalline material samples. The numerical procedure is developed on the basis of the concept of grain-average fields, Kröner-Kneer model, Waldvogel-Rodin algorithm and a self-adaptive method. Repeated computer tests were performed to investigate mesoscopic stress variation in the samples, and find coherent interrelations of material structure weaknesses (MSWs) with local microstructure of the samples. It was found that the stronger the single crystal elastic anisotropy, the stronger the inhomogeneity of mesoscopic stress distribution. Not only the elastic anisotropy, but also the grain geometry, may produce significant local stress disturbances. It has been found that the defined 'orientation-geometry factor' and 'correlation parameter' are two adequate physical quantities which account for synergetic interactions due to grain-orientation geometry-induced anisotropy. By using the two quantities, MSWs can be well correlated with local microstructure. Computer tests also show that 250-400 conjoining grains are necessary to homogenize the mesoscopic stress distribution in the considered materials.

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“…The optimum spatial adjacency of the interpolation scheme in conjunction with properties such as partition of unity and linear completeness render n-n interpolation to be a promising choice in a Galerkin method for the solution of PDEs. For a few applications of Voronoi polygons in continuum mechanics, see Cruz and Patera (1982), Ghosh and Mallett (1994), and Cannmo, Runesson, and Ristinmaa (1995). Braun and Sambridge (1995) introduced natural neighbor interpolation for the solution of partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Dispersive properties of the natural element method

Bueche¹,

Sukumar²,

Moran

2000

Computational Mechanics

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The Natural Element Method (NEM) is a meshfree numerical method for the solution of partial differential equations. In the natural element method, natural neighbor coordinates, which are based on the Voronoi tesselation of a set of nodes, are used to construct the interpolant. The performance of NEM in two-dimensional linear elastodynamics is investigated. A standard Galerkin formulation is used to obtain the weak form and a centraldifference time integration scheme is chosen for time history analyses. Two different applications are considered: vibration of a cantilever beam and dispersion analysis of the wave equations. The NEM results are compared to ®nite element and analytical solutions. Excellent dispersive properties of NEM are observed and good agreement with analytical solutions is obtained.

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Voronoi cell finite elements

Cited by 65 publications

References 15 publications

Micromechanical modelling of switching phenomena in polycrystalline piezoceramics: application of a polygonal finite element approach

Micromechanical modelling of switching phenomena in polycrystalline piezoceramics: application of a polygonal finite element approach

Computational assessment of material structure weaknesses in polycrystalline materials

Dispersive properties of the natural element method

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