A 3D refinement/derefinement algorithm for solving evolution problems

Plaza, Ángel; Padrón, Miguel A.; Carey, G. F.

doi:10.1016/s0168-9274(99)00060-4

Cited by 25 publications

(14 citation statements)

References 20 publications

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“…The efficiency arises because coarsening operations prevent the exponential increase in the number of nodes that would otherwise occur from refinement operations. To date, derefinement operations have been developed for nested meshes generated from edge bisection [24] and none exist for 8-subtetrahedron subdivision. A local derefinement algorithm in the nested meshes created by the 8-subtetrahedron subdivision has been developed by the authors and implemented also in this work.…”

Section: Mesh Derefinement-meshmentioning

confidence: 99%

“…where d i,jk is defined by (22) The plane equations for the triangles F j , F k , and F l become (23) with α, β, γ = j, k, l cyclic, respectively. 4-vectors r j , r k , and r l for intersection points with respect to F j , F k , and F l can be derived from (9) and (23), and the resulting interpolation coefficients t j , t k , and t l are given by (24) with α, β, γ = j, k, l cyclic. At the end, any points inside C will be switched into P, Q, R, or Sbased representation using (18)-(21) for finite element assembly related two leaf tetrahedrons separately from M and M′.…”

Section: Intersections In C 4a -Containermentioning

confidence: 99%

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Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation

Lee

Joshi

Sevick‐Muraca

2008

Journal of Computational Physics

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A variety of biomedical imaging techniques such as optical and fluorescence tomography, electrical impedance tomography, and ultrasound imaging can be cast as inverse problems, wherein image reconstruction involves the estimation of spatially distributed parameter(s) of the PDE system describing the physics of the imaging process. Finite element discretization of imaged domain with tetrahedral elements is a popular way of solving the forward and inverse imaging problems on complicated geometries. A dual-adaptive mesh-based approach wherein, one mesh is used for solving the forward imaging problem and the other mesh used for iteratively estimating the unknown distributed parameter, can result in high resolution image reconstruction at minimum computation effort, if both the meshes are allowed to adapt independently. Till date, no efficient method has been reported to identify and resolve intersection between tetrahedrons in independently refined or coarsened dual meshes. Herein, we report a fast and robust algorithm to identify and resolve intersection of tetrahedrons within nested dual meshes generated by 8-similar subtetrahedron subdivision scheme. The algorithm exploits finite element weight functions and gives rise to a set of weight functions on each vertex of disjoint tetrahedron pieces that completely cover up the intersection region of two tetrahedrons. The procedure enables fully adaptive tetrahedral finite elements by supporting independent refinement and coarsening of each individual mesh while preserving fast identification and resolution of intersection. The computational efficiency of the algorithm is demonstrated by diffuse photon density wave solutions obtained from a single-and a dual-mesh, and by reconstructing a fluorescent inclusion in simulated phantom from boundary frequency domain fluorescence measurements.

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Section: Mesh Derefinement-meshmentioning

confidence: 99%

Section: Intersections In C 4a -Containermentioning

confidence: 99%

Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation

Lee

Joshi

Sevick‐Muraca

2008

Journal of Computational Physics

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“…is certainly member of set (14), and since at least φ (j+1) i is deactivated, the refinement set of φ (j) i is guaranteed not to be complete at the end of the unrefinement step. Hence, activating φ (j) i cannot introduce a linear dependency: the linear independence requirement is preserved.…”

Section: Unrefinementmentioning

confidence: 99%

“…A number of approaches are being used to resolve this issue (see Reference [4] for a good introduction): (i) the unknowns of the incompatibly placed nodes are constrained with respect to other nodes so that compatibility of the resulting approximation is ensured even though the mesh remains incompatible; (ii) incompatibility is treated with Lagrangian multipliers or penalty methods; (iii) compatibility of the mesh is achieved by splitting additional elements until the mesh becomes globally compatible by construction. A number of specialized mesh refinement schemes have been proposed for a variety of practically important cases, for triangular and quadrilateral meshes in two dimensions [17,3,15,16], tetrahedral [19,9,10,14,13,1], or hexahedral meshes in three dimensions [8]. A critical review of the existing adaptive algorithms based on mesh refinement leads to the conclusion that they tend to be quite complex (constraint methods, splitting of neighboring elements), or lead to undesirable algorithmic features (Lagrange multipliers, penalty methods).…”

Section: Introductionmentioning

confidence: 99%

Natural hierarchical refinement for finite element methods

Krysl

Grinspun

Schröder

2003

Numerical Meth Engineering

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key words: Finite element, mesh refinement, hierarchical, adaptive approximation, subdivision surface, subdivision element method SUMMARY Current formulations of adaptive finite element mesh refinement seem simple enough, but their implementations prove to be a formidable task. We offer an alternative point of departure which yields equivalent adapted approximation spaces wherever the traditional mesh refinement is applicable, but our method proves to be significantly simpler to implement. At the same time it is much more powerful in that it is general (no special tricks are required for different types of finite elements), and applicable for some newer approximations where traditional mesh refinement concepts are not of much help, for instance on subdivision surfaces.

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“…Local grid refinement techniques have been developed in two dimensions for triangle and quadrilaterals ( [2,8,[19][20][21]27] and in three dimensions for tetrahedrals [1,12,13,17,18,31] and for hexahedrals [11,14,15]. Barry et al [3] applied libraries developed by the SUMAA3d project to solve plasticity problems on a parallel platform.…”

Section: Introductionmentioning

confidence: 99%

Adaptive meshing in a mixed regime hydrologic simulation model

2010

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Problems in hydrology frequently have moving fronts and dynamic driving mechanisms such as wells. Since the location of important features changes during a simulation, accurate modeling requires uniformly fine resolution or the ability to change resolution during the simulation. We will describe an algorithm for refinement and unrefinement of tetrahedral/triangular meshes that has been implemented in the adaptive hydrology (ADH) code. The codes including the refinement/unrefinement algorithms are implemented in parallel to accommodate problems with large run time and memory requirements. In this paper, we describe the parallel, adaptive grid algorithm used in ADH and show the resulting grids from some example problems.

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A 3D refinement/derefinement algorithm for solving evolution problems

Cited by 25 publications

References 20 publications

Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation

Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation

Natural hierarchical refinement for finite element methods

Adaptive meshing in a mixed regime hydrologic simulation model

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