Optismoothing: An optimization-driven approach to mesh smoothing

Canann, Scott A.; Stephenson, Michael B.; Blacker, Ted D.

doi:10.1016/0168-874x(93)90056-v

Cited by 99 publications

(67 citation statements)

References 9 publications

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“…Após a completa obtenção da malha, podemos aumentar a sua qualidade global através de processos de perturbação nas coordenadas dos vértices [4,6].…”

Section: Técnica De Avanço Da Fronteiraunclassified

Geração Automática de Malhas de Elementos Finitos e a Estrutura de Dados Winged-Edge Modificada

Gois¹,

Piteri²

2002

TEMA - Tendências Em Matemática Aplicada E Computacional

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“…Após a completa obtenção da malha, podemos aumentar a sua qualidade global através de processos de perturbação nas coordenadas dos vértices [4,6].…”

Section: Técnica De Avanço Da Fronteiraunclassified

Geração Automática de Malhas de Elementos Finitos e a Estrutura de Dados Winged-Edge Modificada

Gois¹,

Piteri²

2002

TEMA - Tendências Em Matemática Aplicada E Computacional

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“…Laplacian smoothing is effective for triangular meshes, but for tetrahedral meshes it is less reliable, and often produces poor tetrahedra. We use better smoothing algorithms which are based on numerical optimization [12,42]. These algorithms define a smooth objective function that summarizes the quality of a group of elements, and use a numerical optimization algorithm such that as steepest descent or Newton's method to move a vertex to the optimal location.…”

Section: Tetrahedral Mesh Improvementmentioning

confidence: 99%

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

Madrane¹,

Vaillancourt²

2009

SIAM J. Sci. Comput.

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Abstract.We present an explicit second-order finite volume generalization of the onedimensional (1D) Nessyahu-Tadmor schemes for hyperbolic equations on adaptive unstructured tetrahedral grids. The nonoscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is bypassed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-steps scheme. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in three dimensions (3D), where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods to solve Euler's equations. Numerical experimental tests on classical problems are obtained by our method and by the computational fluid dynamics software Fluent. These tests include results for the 3D Euler system (shock tube problem) and flow around an NACA0012 airfoil. 1. Introduction. The history of schemes on staggered grids can at least be traced back to the famous paper of Courant, Friedrichs, and Lewy in 1928 [13] in which they discovered a scheme on staggered grids for the linear wave equation in one-dimensional (1D). For a special system arising in fluid dynamic problems von Neumann and Richtmyer used staggered grids as well [38]. Four years later, Lax introduced the well-known Lax-Friedrichs scheme and analyzed it [29]. In 1990 Tadmor and Nessyahu [41] picked up the idea to use staggered grids, showed the connection to Godunov's method, and proposed a second-order extension to 1D systems.The main advantage of these schemes is that no information about solutions to local Riemann problems is needed. Using staggered grids one can replace the upwind fluxes by central differences. The price one has to pay is the occurrence of excessive numerical viscosity since the resulting scheme can be interpreted as a Lax-Friedrichs scheme. Therefore, a higher order scheme of monotone upstream-centered schemes for conservation laws (MUSCL)-type in one spatial dimension was proposed in [41]. Later in [4,3,6,7,8,36] central schemes were generalized to multidimensional schemes on unstructured grids. For a Cartesian grid, we refer to [9,22,30,28,31,39,40,45,32] for related work.In the case of staggered unstructured multidimensional grids, there exist only a few convergence results. In [5] convergence of a second-order central scheme on twodimensional (2D) grids has been proven for a linear conservation law. Convergence of the first-order Lax-Friedrichs scheme on the same staggered grids for nonlinear scalar problems has been proven in [19].

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“…Many mathematical techniques have been proposed for improving poor quality meshes. These include methods for improving the mesh even as it is generated (vertex insertion/deletion) [16] and postprocessing methods in which mesh topology is modified (edge or element swapping) [9,6] and/or vertex coordinates are modified (smoothing or optimization of an objective function that measures mesh quality) [3,5,7,10,14,17,4]. The latter set of methods are sometimes referred to as node-movement methods.…”

Section: Background and Introductionmentioning

confidence: 99%

Incorporating the min-max mesh optimization method within the Target-Matrix Paradigm.

Knupp¹,

Day²

2012

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The Target-Matrix Optimization Paradigm (TMOP) is a method for improving mesh quality via node movement [11]. Through the use of appropriate pre-defined quality metrics and Target-matrices, TMOP provides the ability to define mesh quality in terms of element shape, volume, and edge orientation as a function of position within the mesh, thus facilitating high-level goals such as size-adapted meshing. The Paradigm is developed further in this paper by showing how it can be applied to the problem of improving the mesh in locations where the quality is the worst. Improving the worst quality portions of a mesh was investigated by Freitag [8], Amenta [1], and others, who showed that the problem can be formulated as a min-max optimization problem. The Target-paradigm is used to extend the previous work in two important respects. First, by using quality metrics from the Targetparadigm, quality can be user-specified via Target-matrices instead of the quality definition being hard-wired into the quality metric. Second, by making use of the sample point concept in TMOP, one can extend min-max mesh optimization to non-simplicial element types such as quadrilaterals. Numerical experiments were performed to illustrate these extensions and to investigate their utility. A non-gradient solver known as the amoeba was used to numerically compute the optimal solutions. This derivative-free method, in conjunction with the use of barrier metrics from TMOP, enables the solution of the min-max problem without the use of explicit constraints to guarantee that the optimal mesh is non-inverted. The experiments showed that the use of Target-metrics in min-max with triangle (or tetrahedral) element meshes is able to effectively improve the worst quality elements in terms of shape or in terms of size+shape. The experiments also showed that the mesh min-max extension 3 to quadrilateral element meshes can lead to acceptable meshes using barrier metrics from the Target-paradigm, provided that inter-element mesh smoothness is not important. Poor behavior was observed when using non-barrier Target-metrics in min-max with quadrilateral element meshes. For both simplicial and quadrilateral meshes, the min-sum method can be competitive with min-max, however.4

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Optismoothing: An optimization-driven approach to mesh smoothing

Cited by 99 publications

References 9 publications

Geração Automática de Malhas de Elementos Finitos e a Estrutura de Dados Winged-Edge Modificada

Geração Automática de Malhas de Elementos Finitos e a Estrutura de Dados Winged-Edge Modificada

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

Incorporating the min-max mesh optimization method within the Target-Matrix Paradigm.

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