Goal-oriented h-adaptivity for the Helmholtz equation: error estimates, local indicators and refinement strategies

The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-010-0557-2This paper introduces a new goal-oriented adaptive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: (1) different error representations, (2) assessment of the dispersion error, and (3) different remeshing criteria.Peer ReviewedPostprint (author's final draft

A simple strategy to assess the error in the numerical wave number of the finite element solution of the Helmholtz equation

The standard approach for goal oriented error estimation and adaptivity uses an error representation via an adjoint problem, based on the linear functional output representing the quantity of interest. For the assessment of the error in the approximation of the wave number for the Helmholtz problem (also referred to as dispersion or pollution error), this strategy cannot be applied. This is because there is no linear extractor producing the wave number from the solution of the acoustic problem. Moreover, in this context, the error assessment paradigm is reverted in the sense that the exact value of the wave number, κ, is known (it is part of the problem data) and the effort produced in the error assessment technique aims at obtaining the numerical wave number, κH, as a postprocess of the numerical solution, uH. The strategy introduced in this paper is based on the ideas used in the a priori analysis. A modified equation corresponding to a modified wave number κm is introduced. Then, the value of κm such that the modified problem better accommodates the numerical solution uH is taken as the estimate of the numerical wave number κH. Thus, both global and local versions of the error estimator are proposed. The obtained estimates of the dispersion error match the a priori predicted dispersion error and, in academical examples, the actual values of the error in the wave number.Peer ReviewedPostprint (author’s final draft

Goal-oriented h-adaptivity for the Helmholtz equation: error estimates, local indicators and refinement strategies

This paper introduces a new goal-oriented adap- tive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: 1) different error representations, 2) assessment of the dispersion error, and 3) different remeshing criteria

REDUCING ORDER METHODS APPLIED TO RESERVOIR SIMULATION

Results obtained by numerical simulations techniques are used in the whole productive life of the reservoir, since exploration until enhanced oil recovery. Numerical simulations involves many cells and heterogeneities and are still limited by the computational time and memory. Reducing Order Methods are a solution to these problems. These methods allow the transformation of high-dimensional models into meaningful representations. It reduces the dimension of the matrices used during the simulations, and consequently, the time and effort. One of the methods used to get a reduced model is the Proper Orthogonal Decomposition (POD). In this work, the mathematical model and equations of a considered reservoir are first presented, and in sequence the discrete system obtained by Finite Difference and Finite Volumes methods. Then, the POD procedure will be described and applied to the problem considered. Finally, the size of new matrices and pressures will be evaluated before and after the reduction, as well as the error involved. The results obtained after the reduction agreed with the physical of the problem and, as expected, the number of unknowns reduced significantly

Adaptividade e estimativas de erro orientadas por metas aplicadas a um benchmark test de propagação de onda

O objetivo deste artigo é estudar a eficiência e a robustez de técnicas adaptativas e estimativas de erro orientadas por metas para um benchmark test. As técnicas utilizadas aqui são baseadas em um simples pós-processo das aproximações de elementos finitos. As estimativas de erro orientadas por metas são obtidas por analisar o problema direto e um problema auxiliar, o qual está relacionado com a quantidade de interesse específico. O procedimento proposto é válido para quantidades lineares e não-lineares. Além disso, são discutidas diferentes representações para o erro e é analisada a influência do erro de dispersão. Os resultados numéricos mostram que as estimativas de erro fornecem boas aproximações ao erro real e que a técnica de refino adaptativo proposta conduz a uma redução mais rápida do erro

THE STUDY OF THE INFLUENCE OF BOUNDARY CONDITIONS AND HETEROGENEITY IN THE PERFORMANCE OF THE NUMERICAL UPSCALING METHOD FOR ABSOLUTE PERMEABILITY

In a previous worka numerical upscaling technique for absolute permeability was developed. This method is a non-local technique that uses a cell layer around the upscaling zone to reduce boundary conditions influence. Upscaling zone is the set of cells of interest for upscaling, and the cell layers (or rings) are the adjascent cells in the fine grid. The following is an extension of the method and it studies the use of more than one ring around the upscaling zone and the effect of high heterogeneity areas in upscaling. Intuition says that a greater number of rings should improve the results, since it leads to reducing boundary conditions effect. However, the use of more layers implies in a higher complexity in the upscaling algorithm. In current study, the use of 1 and 2 rings to upscale a permeability grid was considered. Computational time and percentual error were compared for performance analysis. In addition, the method was compared to the harmonic and arithmetic average techniques. Flow simulations were performed usingfinite-differencemethod and incompressible single-phase flow based. The method was applied to SPE’s dataset 1 and some permeability fields generated by numerical probability distribution

Desenvolvimento de uma metodologia integrada para otimização de forma de mecânica de fluidos

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Engenharia Mecânica.Este trabalho propõe um procedimento numérico integrado para problemas que envolvem a otimização de forma aplicada ao escoamento de fluidos. O procedimento é denominado integrado porque reúne diversos módulos distintos para o tratamento do problema, como modelagem geométrica, geração de malhas por elementos finitos, análise não-linear do escoamento, análise de sensibilidade, programação matemática e otimização de forma. A solução eficiente desta classe de problemas de otimização, principalmente os algoritmos de programação matemática, depende fortemente da estratégia utilizada na análise de sensibilidade da resposta do contorno e da determinação efetiva do gradiente da função objetivo e suas restrições, com relação às variáveis de projeto. Por este motivo, neste trabalho, foi aplicado um procedimento para melhorar a qualidade da análise de sensibilidade. As sensibilidades são obtidas analiticamente e o procedimento implementado para o gradiente da função objetivo propõe uma alternativa chamada de método adjunto, que consiste na adição de um termo à função objetivo, o que reduz consideralvemente o custo computacional. O problema de otimização é definido com base no modelo geométrico, a função objetivo a ser considerada é a dissipação viscosa e as variáveis de projeto são as coordenadas dos pontos-chave, os quais descrevem o contorno do domínio, que é representado por segmentos de B-splines cúbicas. A definição paramétrica de curvas em função de um conjunto de pontos-chave e condições de contorno em seus vértices extremos são discutidas detalhadamente. É dada ênfase à interpolação através de segmentos de B-splines, devido à sua simplicidade, eficácia e flexibilidade. A correta definição da geometria do domínio é responsável pelo sucesso do processo de otimização. O problema discretizado de otimização é formulado através do Método do Lagrangeano Aumentado, considerando-se restrições laterais e volumétricas. Os escoamentos aqui tratados são bidimensionais, supondo as hipóteses de incompressibilidade e em regime permanente, modelados pelas equações de Navier-Stokes. Estas equações são discretizadas pelo método dos elementos finitos de Galerkin, via elementos triangulares T7/C3 (sete nós para a velocidade e três para a pressão), resultando em um sistema de equações não-lineares, que é solucionado pelo método de Newton. O problema a ser solucionado neste trabalho consiste na otimização de forma de contornos, visando à redução da dissipação viscosa decorrente do escoamento em torno de um dado corpo ou em um canal

Derivation of shallow water models for steady and transient phenomena, required assumptions and resulting equations

The shallow water models describe water flow in rivers, lakes and shallow seas. They are used to study many physical phenomena of interest, such as, modeling environmental effects, commercial activities on fisheries and coastal wildlife, remediation of contaminated bays and estuaries for the purposes of improving water quality. However, without any simplifications are very difficult to solve. These assumptions, with respect to several physical aspects are the general source of uncertainty within the modeling process. In this work, we get the derivation of shallow water models for steady and transient phenomena, required assumptions and resulting equations.Peer Reviewe

Simple assessment of the numerical wave number in the fe solution of the helmholtz equation

When numerical methods are applied to the computation of stationary waves, it is observed that "numerical waves" are dispersive for high wave numbers. The numerical wave shows a phase velocity which depends on the wave number "k" of the Helmholtz equation. Recent works on goal-oriented error estimation techniques with respect to socalled quantities of interest or output functionals are developing. Thus, taken into account such aspects, the main purpose of this paper is a posteriori error estimation through of a assessment of the numerical wave number in finite element solution fot he simulation of acoustic wave propagation problems adressed by Helmholtz equation. A method to measure the dispersion on classical Galerkin FEM is presented. In this analysis, the phase difference between the exact and numerical solutions is researched. Fundamental results from a priori error estimation for one-dimensional are presented and issues dealing with pollution error at high wave numbers also are discussed

Simple assessment of the numerical wave number in the fe solution of the helmholtz equation

When numerical methods are applied to the computation of stationary waves, it is observed that "numerical waves" are dispersive for high wave numbers. The numerical wave shows a phase velocity which depends on the wave number "k" of the Helmholtz equation. Recent works on goal-oriented error estimation techniques with respect to socalled quantities of interest or output functionals are developing. Thus, taken into account such aspects, the main purpose of this paper is a posteriori error estimation through of a assessment of the numerical wave number in finite element solution fot he simulation of acoustic wave propagation problems adressed by Helmholtz equation. A method to measure the dispersion on classical Galerkin FEM is presented. In this analysis, the phase difference between the exact and numerical solutions is researched. Fundamental results from a priori error estimation for one-dimensional are presented and issues dealing with pollution error at high wave numbers also are discussed