Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation

Vartziotis, Dimitris; Wipper, Joachim; Papadrakakis, Manolis

doi:10.1016/j.finel.2012.11.004

Cited by 28 publications

(19 citation statements)

References 29 publications

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“…We see, that the dual element-based and our new smoothing algorithm produce meshes of a comparable mean ratio quality in about the same amount of time. Only Mesh #9 has elements with exceptionally low quality, but for this example the parameters had been manually optimized for the dual-element based transformation as in [44]. Still, the arithmetic means are comparable throughout the examples.…”

Section: Numerical Testsmentioning

confidence: 99%

“…In this paper, regularization refers to the convergence of the iteratively transformed polyhedra towards a regular polyhedron, and we ask the reader not to confuse this terminology with the methods in numerical analysis that allow you to deal with ill-conditioned problems. GETMe smoothing also significantly reduces errors in solutions of the finite element method and improves solution efficiency for meshes [44]. The search for a mathematical proof of its qualitative behavior led us to the discovery of a simple and reasonable quality measure for volume elements optimized by polyhedral generalizations of the tetrahedral GETMe algorithm in [45].…”

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confidence: 99%

“…The tetrahedral GETMe algorithm. In [45] a powerful heuristic method for smoothing tetrahedral meshes was developed and further generalized to other volume elements in [40,41,42,43,44]. Let us review the key concept from [45].…”

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Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume

Vartziotis¹,

Himpel²

2014

SIAM J. Numer. Anal.

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Abstract. The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted to this sphere is a quality measure for volume elements. We show that, the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra, hexahedra, prisms, pyramids and octahedra, that is, the optimization process converges to regular polyhedra. We show that the (normalized) gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method (GETMe). Furthermore, we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally.Key words. hybrid mesh, smoothing, quality metric, quality measure, polyhedron, optimization, finite element method, GETMe, octahedron, tetrahedron, hexahedron, pyramid, prism AMS subject classifications. 52B70, 58C05, 37C101. Introduction. In the context of the finite element method, mesh quality affects numerical stability as well as solution accuracy of this method [37]. The geometric element transformation method (GETMe) as a smoothing algorithm for tetrahedral meshes is introduced in [45], which was generalized to hybrid meshes in [40,41,42,43,44]. We consider GETMe as a class of smoothing methods based on simple geometric transformations applied iteratively to all elements individually. Several numerical tests have confirmed that the geometric element transformation given in [45] and its generalizations reliably and efficiently regularizes the polyhedron types, which are relevant for the finite element method. In this paper, regularization refers to the convergence of the iteratively transformed polyhedra towards a regular polyhedron, and we ask the reader not to confuse this terminology with the methods in numerical analysis that allow you to deal with ill-conditioned problems. GETMe smoothing also significantly reduces errors in solutions of the finite element method and improves solution efficiency for meshes [44]. The search for a mathematical proof of its qualitative behavior led us to the discovery of a simple and reasonable quality measure for volume elements optimized by polyhedral generalizations of the tetrahedral GETMe algorithm in [45]. Therefore, the purpose of our work is threefold: 1. We introduce the mean volume and turn it into a scaling-invariant quality measure. 2. We consider the gradient flow of the mean volume and discuss its close relationship to the tetrahedral GETMe algorithm presented in [45]. 3. We analyze the new GETMe optimizing and untangling algorithms for volume elements induced by the gradient flow of the mean volume both theoretically and experimentally.

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Section: Numerical Testsmentioning

confidence: 99%

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Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume

Vartziotis¹,

Himpel²

2014

SIAM J. Numer. Anal.

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“…1(a), which seriously affects mesh's application in many fields. For instance, in finite element analysis, the high-quality mesh (equilateral triangle) is in favor of improving the accuracy of the numerical results and the efficiency of the numerical simulation (Vartziotis, et al, 2013). Thus, these meshes need to undergo a further process which is commonly known as remeshing in order to improve the mesh quality (Alliez, et al, 2007), as shown in Fig.…”

Section: Introductionmentioning

confidence: 99%

As-equilateral-as-possible surface remeshing

Jin

Tong

2015

JAMDSM

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Surface remeshing aims to produce a high-quality mesh from a given input mesh. In this paper, we present a practical approach to isotropic surface remeshing of triangular meshes in an as-equilateral-as-possible (AEAP) manner. The proposed approach iterates in a two-step manner. In the first step, we optimize the mesh connectivity to reduce the number of irregular vertices. And in the second step, we improve the elements' shape by locally fitting an equilateral triangle separately and globally stitching them to a new mesh with feature constraints. The proposed approach can be effectively used in preprocessing for finite element computations. Experiments are presented to show that our approach is simple and robust, and the remeshed results generate more well-shaped triangles and preserve the features of the original surface, which are favored in many applications.

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“…A priori grid quality assessment has been based primarily on geometric characteristics of the elements such as ratios of sizes of neighboring elements, as well as on element shape measures, such as angles of the elements [18][19][20]. In the Finite Element method, the quality of a mesh is often given in terms of the element/mesh regularity [21].…”

Section: Introductionmentioning

confidence: 99%

Quality Index and Improvement of the Interfaces of General Hybrid Grids

Fotia

Kallinderis

2014

Procedia Engineering

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Hybrid grids constitute a significant class of meshes for complex-geometry flow simulations. They consist of prisms and/or hexahedra covering viscous regions primarily, while tetrahedra discretize the rest of the domain with pyramids serving as transitional elements. The local changes in element topology form grid interfaces. This is especially so for cases where the number of prisms/hexahedra per layer (stack) is not constant ("chopped" layers). The present work deals with the quality of the interfaces encountered in general hybrid grids with non-constant number of layers. A priori estimation of the quality is based on truncation error terms. Four representative types of interfaces are considered. The effect of grid stretching and skewness was examined for the interfaces. Further, the quality is improved via repositioning of the central point of each of the interface types. The improvement is quantified via application to several hybrid grids.

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Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation

Cited by 28 publications

References 29 publications

Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume

Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume

As-equilateral-as-possible surface remeshing

Quality Index and Improvement of the Interfaces of General Hybrid Grids

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