A new adaptive mesh refinement strategy for numerically solving evolutionary PDE's

Burgarelli, Denise; Kischinhevsky, Mauricio; Biezuner, Rodney Josué

doi:10.1016/j.cam.2005.08.013

Cited by 25 publications

(14 citation statements)

References 18 publications

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“…However, the optimal mesh for a given problem cannot be accurately predicted a priori. Various general application spatial adaptivity methods have been developed which increase the accuracy of calculations by using a posteriori errors estimators [2,3,4,5,6]. There are two broad categories of adaptive methods.…”

Section: Introductionmentioning

confidence: 99%

Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes

Goffin

Baker

Buchan

et al. 2013

Journal of Computational Physics

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This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, k eff , is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for k eff with directional dependence. General error estimators are derived for any given functional of the flux and applied to k eff to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The k eff goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.

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Section: Introductionmentioning

confidence: 99%

Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes

Goffin

Baker

Buchan

et al. 2013

Journal of Computational Physics

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“…O ALG é uma estrutura de dados baseada em grafos proposta em [2] como ferramenta para manipulação adaptativa de malhas não uniformes e auxílio na resolução numérica de EDP´s. Em cada passo da execução do algoritmo, uma determinada região da malha sofre um refinamento ou desrefinamento, de acordo com o objetivo do estudo ou comportamento da solução do modelo em questão.…”

Section: Autonomous Leaves Graph (Alg)unclassified

Aplicação de técnicas de análise numérica a um modelo de escoamento de água em áreas amazônicas alagáveis

Leal¹,

Silva²,

Duczmal³

et al. 2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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IntroduçãoA Amazônia é a maior floresta tropical do mundo, cujo território extravasa as fronteiras políticas do Brasil, estendendo-se da Cordilheira dos Andes até o Oceano Atlântico. Apresenta rica biodiversidade e todas as relações no ecossistema procedem com total harmonia. O índice pluviométrico na região, elevado na maior parte do ano, é influenciado por fenômenos como o vapor d'água proveniente do Oceano Atlântico em direção à Cordilheira dos Andes, o processo de evapotranspiração, que é o retorno à atmosfera da umidade não aproveitada pela vegetação, além do próprio degelo decorrente do verão andino. Somados, esses fatores geram ciclos de precipitação sobre a floresta, alimentando as nascentes e os afluentes dos principais rios da bacia amazônica.O aumento do volume de água e a oscilação do regime de chuvas fazem com que os rios transbordem e inundem grandes extensões da floresta. A amplitude de variação do nível da água dos rios pode chegar a 16 metros em algumas áreas. Ecologistas da Amazônia denominam essas variações previsíveis como pulso de inundação.A água que circula nos trechos de floresta alagada transporta sedimentos e material orgânico para outros locais, contribuindo para a redistribuição de nutrientes e energia. Isto constitui um processo fundamental para a floresta, nutrindo a vegetação e disseminando sementes, o que favorece decisivamente a manutenção da cobertura vegetal.Além dos benefícios ecológicos, os alagamentos anuais têm importante papel sócio-econômico. A população ribeirinha, em sua quase totalidade, sobrevive dos processos de extrativismo e exploração da biomassa animal, sobretudo de peixes e crustáceos, que ali se reproduzem e desenvolvem. Além disso, no período de retorno da água ao nível normal, a matéria orgânica diluída na água, proveniente da parte mais elevada na mata, se deposita no solo propiciando a agricultura de subsistência.

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“…The Autonomous Leaves Graph (ALG) was proposed for the Finite Volume Method and a space-filling curve named Modified Hilbert Curve (MHC) was proposed in order to number the mesh control volumes [1]. ALG children nodes become autonomous as their parent node is deleted.…”

Section: Introductionmentioning

confidence: 99%

Autonomous Leaves Graph Applied to the Boundary Layer Problem

Oliveira

Kischinhevsky

2009

Lecture Notes in Computer Science

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Abstract. In physics and fluid mechanics, the boundary layer is the fluid layer in the immediate vicinity of a bounding surface. It is important in many aerodynamic problems. This work presents a numerical simulation of the bidimensional laminar boundary-layer problem considering a steady incompressible flow with no-slip condition on the surface by Autonomous Leaves Graph based on finite volume discretizations. In addition, a Modified Hilbert Curve numbers the control volumes. Initially, the numerical solution of the flat-plate problem is compared to its analytical solution, namely Blasius Solution. Secondly, simulations of the flow along a NACA airfoil shape are presented. Computer experiments show that an adaptive mesh refinement using the Autonomous Leaves Graph with the Modified Hilbert Curve numbering is appropriate for a aerodynamic problem. Finally, results illustrate that the method provides a good trade-off between speed and accuracy.

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A new adaptive mesh refinement strategy for numerically solving evolutionary PDE's

Cited by 25 publications

References 18 publications

Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes

Minimising the error in eigenvalue calculations involving the Boltzmann transport equation using goal-based adaptivity on unstructured meshes

Aplicação de técnicas de análise numérica a um modelo de escoamento de água em áreas amazônicas alagáveis

Autonomous Leaves Graph Applied to the Boundary Layer Problem

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