A Brief Review on Polygonal/Polyhedral Finite Element Methods

Perumal, Logah

doi:10.1155/2018/5792372

Cited by 28 publications

(8 citation statements)

References 181 publications

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“…However, we have to keep in mind that other polygonal/polyhedral element formulations based on, e.g., the Voronoi cell finite element method (VCFEM) [25,29], the virtual element method (VEM) [30,31] and the scaled boundary finite element method (SBFEM) [32][33][34], are also possible. An extensive description of these and other related approaches can be found in the review article by Perumal [35]. The main challenges of the polytopal FEM can be seen in the lack of commercial and free mesh generation software, construction of appropriate (high-order) interpolants and in the need for accurate numerical integration schemes.…”

Section: Polytopal Finite Element Methodsmentioning

confidence: 99%

Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants

et al. 2020

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Polygonal finite elements offer an increased freedom in terms of mesh generation at the price of more complex, often rational, shape functions. Thus, the numerical integration of rational interpolants over polygonal domains is one of the challenges that needs to be solved. If, additionally, strong discontinuities are present in the integrand, e.g., when employing fictitious domain methods, special integration procedures must be developed. Therefore, we propose to extend the conventional quadtree-decomposition-based integration approach by image compression techniques. In this context, our focus is on unfitted polygonal elements using Wachspress shape functions. In order to assess the performance of the novel integration scheme, we investigate the integration error and the compression rate being related to the reduction in integration points. To this end, the area and the stiffness matrix of a single element are computed using different formulations of the shape functions, i.e., global and local, and partitioning schemes. Finally, the performance of the proposed integration scheme is evaluated by investigating two problems of linear elasticity.

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Section: Polytopal Finite Element Methodsmentioning

confidence: 99%

Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants

et al. 2020

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“…2D Mixed meshes composed of triangles and quadrilaterals have also been used, but are not so common as the previous ones [32]. Other numerical methods such as the Voronoi Cell Finite Element Method (VCFEM) [33] and Polygonal Finite Element method (PFEM) [34] use the constrained Voronoi diagram as the polygonal mesh, where the Voronoi cells are the mesh elements [35][36][37].…”

Section: Related Workmentioning

confidence: 99%

POLYLLA: Polygonal meshing algorithm based on terminal-edge regions

Salinas¹,

Hitschfeld²,

Ortiz-Bernardin³

et al. 2022

Preprint

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This paper presents an algorithm to generate a new kind of polygonal mesh obtained from triangulations. Each polygon is built from a terminal-edge region surrounded by edges that are not the longestedge of any of the two triangles that share them. The algorithm is termed Polylla and is divided into three phases. The first phase consists of labeling each edge of the input triangulation according to its size; the second phase builds polygons (simple or not) from terminaledges regions using the label system; and the third phase transforms each non simple polygon into simple ones. The final mesh contains polygons with convex and non convex shape. Since Voronoi based meshes are currently the most used polygonal meshes, we compare some geometric properties of our meshes against constrained Voronoi meshes. Several experiments were run to compare the shape and size of polygons, the number of final mesh points and polygons. For the same input, Polylla meshes contain less polygons than Voronoi meshes and

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“…FEM traditionally uses tetrahedral or hexahedral meshes (in 3D). However, FEM has recently been successfully generalized to meshes of convex polyhedrons, which has brought about significant advantages [17][18][19][20]. Their applications also extend to computer graphics [21].…”

Section: Introductionmentioning

confidence: 99%

GTMesh: A Highly Efficient C++ Template Library for Numerical Schemes on General Topology Meshes

Jakubec,

Strachota

2023

Applied Sciences

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This article introduces GTMesh, an open-source C++ library providing data structures and algorithms that facilitate the development of numerical schemes on general polytopal meshes. After discussing the features and limitations of the existing open-source alternatives, we focus on the theoretical description of geometry and the topology of conforming polytopal meshes in an arbitrary-dimensional space, using elements from graph theory. The data structure for mesh representation is explained. The main part of the article focuses on the implementation of data structures and algorithms (computation of measures, centers, normals, cell coloring) by using State-of-the-Art template metaprogramming techniques for maximum performance. The geometrical algorithms are designed to be valid regardless of the dimension of the underlying space. As an integral part of the library, a template implementation of class reflection in C++ has been created, which is sufficiently versatile and suitable for the development of numerical and data I/O algorithms working with generic data types. Finally, the use of GTMesh is demonstrated on a simple example of solving the heat equation by the finite volume method.

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A Brief Review on Polygonal/Polyhedral Finite Element Methods

Cited by 28 publications

References 181 publications

Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants

Enhanced numerical integration scheme based on image compression techniques: Application to rational polygonal interpolants

POLYLLA: Polygonal meshing algorithm based on terminal-edge regions

GTMesh: A Highly Efficient C++ Template Library for Numerical Schemes on General Topology Meshes

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