Uma formulação Petrov-Galerkin descontínuo para solução da equação de Helmholtz com minimização do erro de fase
Pollution error is a well known source of inaccuracies in continuous or discontinuous FE approaches to solve the Helmholtz equation. This topic is exaustivelly studied in a large number of papers as well as IHLENBURG e BABUSKA [1], IH- ˇ LENBURG [2] and others references inside there in and others r...
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| Tipo de documento: | tese |
| Estado: | Versão publicada |
| Data de publicação: | 2019 |
| País: | Brasil |
| Recursos: | Universidade Federal do Rio de Janeiro (UFRJ) |
| Repositório: | Repositório Institucional da UFRJ |
| Idioma: | português |
| OAI Identifier: | oai:pantheon.ufrj.br:11422/13684 |
| Acesso em linha: | http://hdl.handle.net/11422/13684 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Galerkin finite element methods Discontinuous Galerkin Petrov-Galerkin Discontinuous (PGD) Helmholtz equation CNPQ::ENGENHARIAS::ENGENHARIA CIVIL |
| Resumo: | Pollution error is a well known source of inaccuracies in continuous or discontinuous FE approaches to solve the Helmholtz equation. This topic is exaustivelly studied in a large number of papers as well as IHLENBURG e BABUSKA [1], IH- ˇ LENBURG [2] and others references inside there in and others references. Robust methodologies for structured square meshes have been developed in recent years. This work seeks to develop a methodology based on Discontinuous PetrovGalerkin formulation (DPG), in order to minimize phase error for structured or unstructured meshes applied for Helmholtz equation in homogeneous media. A Petrov–Galerkin FE formulation is introduced for Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the triangular mesh, a global basis function for the weighting space is obtained, adding to the bilinear C 0 Lagrangian weighting function linear combinations. The optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients α n m of these linear combinations attending to optimal criteria. This is done numerically through a preprocessing technique that is naturally applied to nonuniform and unstructured meshes. In particular, for uniform mesh a quasi optimal interior stencil of the same order of the quasi-stabilized finite element method stencil derived by BABUSKA ˇ et al. [3] is obtained. Numerical results are presented illustrating the great stability and accuracy of this formulation with nonuniform and unstructured meshes. |
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