Four-Dimensional Anisotropic Mesh Adaptation

Caplan, Philip Claude; Haimes, Robert; Darmofal, David L.; Galbraith, Marshall C.

doi:10.1016/j.cad.2020.102915

Cited by 30 publications

(22 citation statements)

References 60 publications

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“…Suitable pentatope mesh refinement procedures have been explored by Neumüller and Steinbach 15 and Grande. 16 Anisotropic four-dimensional mesh adaptation is pioneered by Caplan et al 17,18 and successfully employed in the solution of the advection-diffusion equation. 19 Further, recent application examples of four-dimensional SST meshes from the field of mathematics deal with parabolic evolution problems 20,21 or a broader class of transient PDEs recast as constrained first-order system.…”

Section: F I G U R Ementioning

confidence: 99%

“…16 Anisotropic four-dimensional mesh adaptation is pioneered by Caplan et al 17,18 and successfully employed in the solution of the advection-diffusion equation. 19 Further, recent application examples of four-dimensional SST meshes from the field of mathematics deal with parabolic evolution problems 20,21 or a broader class of transient PDEs recast as constrained first-order system. 22 In the field of computational engineering science, adaptive temporal refinement of pentatope meshes is used for two-phase flow simulations 23 -also combined with complex material laws such as the Carreau-Yasuda-WLF model 24 or the 𝜇(I)-rheology 25 -as well as gas flow simulations in the piston ring-pack of internal combustion engines.…”

Section: F I G U R Ementioning

confidence: 99%

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Four‐dimensional elastically deformed simplex space‐time meshes for domains with time‐variant topology

Danwitz

Antony

Key

et al. 2021

Numerical Methods in Fluids

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Considering the flow through biological or engineered valves as an example, there is a variety of applications in which the topology of a fluid domain changes over time. This topology change is characteristic for the physical behavior, but poses a particular challenge in computer simulations. A way to overcome this challenge is to consider the application‐specific space‐time geometry as a contiguous computational domain. In this work, we obtain a boundary‐conforming discretization of the space‐time domain with four‐dimensional simplex elements (pentatopes). To facilitate the construction of pentatope meshes for complex geometries, the widely used elastic mesh update method is extended to four‐dimensional meshes. In the resulting workflow, the topology change is elegantly included in the pentatope mesh and does not require any additional treatment during the simulation. The potential of simplex space‐time meshes for domains with time‐variant topology is demonstrated in a valve simulation, and a flow simulation inspired by a clamped artery.

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Section: F I G U R Ementioning

confidence: 99%

Section: F I G U R Ementioning

confidence: 99%

Four‐dimensional elastically deformed simplex space‐time meshes for domains with time‐variant topology

Danwitz

Antony

Key

et al. 2021

Numerical Methods in Fluids

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“…Figure 12 shows the quadrature error produced by integrating f poly for different values of m and p. In accordance with expectations, we obtain exact integration whenever p ≥ m, to within machine precision (10 −32 ). 6) 65-pt., (7) 114-pt., (8) 163-pt., (9) Fig. 12: Absolute error in the numerical integration of f poly over the reference cubic pyramid K * .…”

Section: Polynomial Integrationmentioning

confidence: 99%

“…In fact, in most cases the higher degree rules outperform the lower degree rules, as expected. 7-pt., (2) 10-pt., (3) 21-pt., (4) 29-pt., (5) 50-pt., (6) 65-pt., (7) 114-pt., (8) 163-pt., (9) Fig. 15: Absolute value of the error in the numerical integration of f 3 over the domain Ω for mesh parameters M = 1, 2, .…”

Section: Transcendental Integrationmentioning

confidence: 99%

“…A similar approach for generating four-dimensional simplex meshes has been employed by Neumüller and Karabelas in [25]. A detailed survey of four-dimensional structured and unstructured simplicial meshing techniques is provided by Caplan [8] and Frontin et al [18]. Finally, examples of refinement strategies for high-dimensional simplicial meshes have been developed in the work of Brandts et al [6], and Korotov and Křížek [20].…”

Section: Introductionmentioning

confidence: 99%

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Enabling four-dimensional conformal hybrid meshing with cubic pyramids

Petrov¹,

Todorov²,

Walters³

et al. 2021

Preprint

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The main purpose of this article is to develop a novel refinement strategy for four-dimensional hybrid meshes based on cubic pyramids. This optimal refinement strategy subdivides a given cubic pyramid into a conforming set of congruent cubic pyramids and invariant bipentatopes. The theoretical properties of the refinement strategy are rigorously analyzed and evaluated. In addition, a new class of fully symmetric quadrature rules with positive weights are generated for the cubic pyramid. These rules are capable of exactly integrating polynomials with degrees up to 12. Their effectiveness is successfully demonstrated on polynomial and transcendental functions. Broadly speaking, the refinement strategy and quadrature rules in this paper open new avenues for four-dimensional hybrid meshing, and space-time finite element methods.Keywords 4D hybrid meshes • cubic pyramids • bipentatopes • pentatopes • optimal refinement strategy • quadrature

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A unified time discontinuous Galerkin space‐time finite element scheme for non‐Newtonian power law models

Toulopoulos

2023

Numerical Methods in Fluids

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In this work, a stabilized time Discontinuous Galerkin, Space‐Time Finite Element (tDG‐ST‐FE) scheme is presented for discretizing time‐dependent viscous shear‐thinning fluid flow models, which exhibit a usual power‐law stress strain relation. The development of the proposed numerical scheme based mainly on a unified weak space‐time formulation, where simple streamline‐upwind terms have been added in the numerical scheme, for stabilizing the discretization of the associated temporal and convective terms. The original time interval is partitioned into time subintervals, resulting in a subdivision of the space‐time cylinder into space‐time subdomains. Discontinuous Galerkin techniques are applied for the time discretization between the space‐time subdomain interfaces. A stability bound is given for the derived ST‐FE scheme. In the last part numerical examples on benchmark problems are presented for testing the efficiency of the proposed method.

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Four-Dimensional Anisotropic Mesh Adaptation

Cited by 30 publications

References 60 publications

Four‐dimensional elastically deformed simplex space‐time meshes for domains with time‐variant topology

Four‐dimensional elastically deformed simplex space‐time meshes for domains with time‐variant topology

Enabling four-dimensional conformal hybrid meshing with cubic pyramids

A unified time discontinuous Galerkin space‐time finite element scheme for non‐Newtonian power law models

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