Reconstructing high-order surfaces for meshing

Jiao, Xiangmin; Wang, Duo

doi:10.1007/s00366-011-0244-8

Cited by 40 publications

(38 citation statements)

References 13 publications

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“…We achieve this by projecting the newly inserted mid‐edge points onto the curved geometry, as illustrated in Figure . The projection can be performed either analytically if the geometry is known or through a high‐order reconstruction of the geometry . Note that if the mesh is too coarse at a concave region, some mesh smoothing may be needed to avoid inverted elements (i.e., elements with a negative Jacobian).…”

Section: Hybrid Multigrid Methodsmentioning

confidence: 99%

“…At a high level, our overall approach may be summarized as follows. Our hierarchical mesh generator starts from a good‐quality coarse unstructured mesh that is a sufficiently accurate representation and allows accurate high‐order reconstruction of the geometry . It iteratively refines the mesh with guaranteed mesh quality (by uniform refinements) and geometric accuracy (by high‐order boundary reconstruction).…”

Section: Introductionmentioning

confidence: 99%

“…From a practical point of view, this hybrid strategy is motivated by our observations that very‐large‐scale meshes often have a few levels of mesh hierarchies. This is because it is increasingly common for high‐resolution meshes to be generated by iteratively refining a moderate‐resolution mesh on parallel computers . Leveraging this inherent mesh hierarchy, HyGA can effectively couple the rigor, accuracy, and runtime‐and‐memory efficiency of GMG at finer resolutions, with the flexibility and simplicity of AMG at coarser resolutions.…”

Section: Introductionmentioning

confidence: 99%

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A hybrid geometric + algebraic multigrid method with semi‐iterative smoothers

Jiao

Missirlis

2014

Numerical Linear Algebra App

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SUMMARYWe propose a multigrid method for solving large‐scale sparse linear systems arising from discretizations of partial differential equations, such as those from finite element and generalized finite difference methods. Our proposed method has the following two characteristics. First, we introduce a hybrid geometric+algebraic multigrid method, or HyGA, to leverage the rigor, accuracy, and efficiency of geometric multigrid (GMG) for hierarchical unstructured meshes, with the flexibility of algebraic multigrid (AMG). Second, we introduce efficient smoothers based on the Chebyshev–Jacobi method for both symmetric and asymmetric matrices. We also introduce a unified derivation of restriction and prolongation operators for weighted‐residual formulations over unstructured hierarchical meshes and apply it to both finite element and generalized finite difference methods. We present numerical results of our method for Poisson equations in both 2‐D and 3‐D and compare our method against the classical GMG and AMG as well as Krylov subspace methods. Copyright © 2014 John Wiley & Sons, Ltd.

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Section: Hybrid Multigrid Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A hybrid geometric + algebraic multigrid method with semi‐iterative smoothers

Jiao

Missirlis

2014

Numerical Linear Algebra App

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“…However, this would compromise the accuracy of the geometry and in turn that of the finite element solver. To address this issue, we are currently exploring integration with geometric models, when available, or to use a polynomial based high-order boundary reconstruction [7,8]. We have also developed a tool on top of this functionality in MOAB which can be used to read in a mesh, generate the hierarchies in parallel for a given number of levels and a sequence of degree of refinements.…”

Section: Refinement Algorithmmentioning

confidence: 98%

Array-based Hierarchical Mesh Generation in Parallel

Ray

Grindeanu

Zhao

et al. 2015

Procedia Engineering

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In this paper, we describe an array-based hierarchical mesh generation capability through uniform refinement of unstructured meshes for efficient solution of PDE's using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial mesh that can be used for a number of purposes such as multi-level methods to generating large meshes. The capability is developed under the parallel mesh framework "Mesh Oriented dAtaBase" a.k.a MOAB [16]. We describe the underlying data structures and algorithms to generate such hierarchies and present numerical results for computational efficiency and mesh quality. We also present results to demonstrate the applicability of the developed capability to a multigrid finite-element solver. c 2015 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of organizing committee of the 24 th International Meshing Roundtable (IMR24).

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“…The first group of methods [57,33,58,59,60] curve the surface mesh boundary and apply local topological operations, such as refinement, edge removal, or edge and face swapping, to adapt the mesh topology to the curved surfaces. Then, edge nodes and inner face nodes are relocated.…”

Section: Related Workmentioning

confidence: 99%

A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

Gargallo-Peiró

Roca

Peraire

et al. 2015

Numerical Meth Engineering

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A framework to validate and generate curved nodal high-order meshes on CAD surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin-shell and 3D finite element analysis with unstructured high-order methods. First, we define a distortion (quality) measure for high-order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high-order), and handles with low-quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high-order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, we curve them to match the geometry, and we optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process.high-order methods; high-order mesh generation; quality measure; mesh optimization; curved elements; CAD ; parameterized surfaces 1

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Reconstructing high-order surfaces for meshing

Cited by 40 publications

References 13 publications

A hybrid geometric + algebraic multigrid method with semi‐iterative smoothers

A hybrid geometric + algebraic multigrid method with semi‐iterative smoothers

Array-based Hierarchical Mesh Generation in Parallel

A distortion measure to validate and generate curved high‐order meshes on CAD surfaces with independence of parameterization

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