Fast smoothing of mixed volume meshes based on the effective geometric element transformation method

Vartziotis, Dimitris; Wipper, Joachim

doi:10.1016/j.cma.2011.09.008

Cited by 26 publications

(16 citation statements)

References 15 publications

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“…This approach has been presented for tetrahedron in Vartziotis et al (2009), then for hexahedrons in Vartziotis and Wipper (2011) and for mixed elements mesh (tetrahedron, pyramids, prisms and hexahedrons) in Vartziotis and Wipper (2012). This method is called the Geometric Element Transformation Method (GETMe).…”

Section: Improving Robustness By Smoothingmentioning

confidence: 99%

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Robust moving-mesh algorithms for hybrid stretched meshes: Application to moving boundaries problems

Landry

Soulaïmani

Luke

et al. 2016

54th AIAA Aerospace Sciences Meeting

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ACKNOWLEDGEMENTSI would like to express my gratitude:• To my thesis supervisor Azzeddine Soulaïmani Ph.D., Eng. Professor Soulaïmani has given me the chance to explore a complex subject with freedom and trust. I have also gained a lot from his mentoring, reputation, experience and our long discussions for which I am truly grateful;• To Amine Ben Haj Ali Ph.D. for his help on defining the test cases and his experiences in meshing generation;• To Professor Edward Luke Ph.D. who shared with us his mesh-mover algorithm which was crucial to generate great results and study deeper the subject of moving boundaries problems;• To Aziz Madrane Ph.D. for his recommendation on the use of the ADT search algorithm.Also, a special thanks to the École de technologie supérieure for entrusting me the excellence scholarship which allowed me to dedicate my energy completely to this research with the support of this growing institution.I dedicated this work to my dear fiancée Alicia who allowed me to push myself every day and to produce proudly this thesis thanks to her love and her motivation; à ma mère Colette, mon père Georges, mon frère Nicholas et ma soeur Maude pour leur soutien inconditionnel; to all my friends who have help me grow; and to Botas and Rizpita whom entertain me during my redaction. MÉTHODES ROBUSTES DE MOUVEMENT DE MAILLAGES HYBRIDES: APPLICATION AUX PROBLÈMES DE FRONTIÈRES MOBILESJonathan LANDRY RÉSUMÉ L'objectif principal de ce mémoire est de développer un Algorithme de Mouvement de Maillage (AMM) suffisament robuste qui sera capable d'adapter les maillages de la majorité des problèmes de frontières mobiles. Ainsi, une nouvelle méthodologie est implémentée à partir de la meilleure combinaison provenant d'algorithmes connus dans le but de mieux préserver la qualité des maillages initiaux.Dans la majorité des simulations, les AMMs standards sont capables de déplacer le maillage en conservant une bonne qualité de maillage, toutefois quand le mouvement est complexe et/ou qu'il y a une interaction multi-corps ils génèrent des maillages invalides. Alors, la nouvelle approche est constituée de trois algorithmes: la fonction pondérée de la distance inverse (PDI) pour produire le champ de déplacements, les algorithmes de lissages de la Méthode de Transformation Géométrique d'Élément (MéTGÉ) pour améliorer la qualité du maillage résultant et un nouvel algorithme basé sur la MéTGÉ qui est capable de réparer les maillages invalides. Il a été prouvé qu'avec cette méthodologie des problèmes de frontières mobiles très difficiles peuvent être résolus.L'approche proposée a été démontrée efficace pour adapter les maillages de différentes situations aéroélastiques réalistes: une aile symétrique a subi une grande flexion et une grande torsion induites au bout de l'aile; et les dispositifs hypersustentateurs de deux ailes (une rectangulaire et une avec une flèche) ont été déplacés vers différentes positions de vol.Finalement, il a été prouvé qu'avec la méthode proposée, la majorité des maillages pour les problèmes d'IFS peuvent ê...

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Section: Improving Robustness By Smoothingmentioning

confidence: 99%

“…• The complete Geometric Transformation Method (GETMe) smoothing approach described by Vartziotis and Wipper (2012);…”

Section: Chapter 2 Two Approaches For Moving-mesh Algorithmsmentioning

confidence: 99%

Robust moving-mesh algorithms for hybrid stretched meshes: Application to moving boundaries problems

Landry

Soulaïmani

Luke

et al. 2016

54th AIAA Aerospace Sciences Meeting

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“…Furthermore, similar smoothing schemes can also be applied to volumetric meshes. Here, transformations can for example be based on geometric constructions [14] or on the gradient flow of the mean volume [15].…”

Section: Polygon Transformations and Fourier Polygonsmentioning

confidence: 99%

The Fractal Nature of an Approximate Prime Counting Function

Vartziotis

Wipper

2017

Fractal Fract

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“…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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confidence: 99%

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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Fast smoothing of mixed volume meshes based on the effective geometric element transformation method

Cited by 26 publications

References 15 publications

Robust moving-mesh algorithms for hybrid stretched meshes: Application to moving boundaries problems

Robust moving-mesh algorithms for hybrid stretched meshes: Application to moving boundaries problems

The Fractal Nature of an Approximate Prime Counting Function

Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume

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