A new goal-oriented formulation of the finite element method

Kenan Kergrene; Serge Prudhomme; Ludovic Chamoin; Marc Laforest

doi:10.1016/j.cma.2017.09.018

A new goal-oriented formulation of the finite element method

Kenan Kergrene, Serge Prudhomme, Ludovic Chamoin, Marc Laforest

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

In this paper, we introduce, analyze, and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary-value problem. The objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal-oriented adaptivity, which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement. In the current approach, we first solve the adjoint problem, then use the adjoint information as a minimization constraint for the primal problem. As a result, the constrained finite element solution is enhanced with respect to the quantities of interest, while maintaining near-optimality in energy norm. We describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples.

Original language	English (US)
Pages (from-to)	256-276
Number of pages	21
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	327
DOIs	https://doi.org/10.1016/j.cma.2017.09.018
State	Published - Dec 2017
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2021-04-06
Acknowledged KAUST grant number(s): CRG3, OCRF-2014-CRG
Acknowledgements: SP is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. He also acknowledges the support by KAUST under Award Number OCRF-2014-CRG3-2281. Moreover, the authors gratefully acknowledge Jonathan Vacher for fruitful discussions on the subject.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

General Physics and Astronomy
Mechanics of Materials
Mechanical Engineering
Computational Mechanics
Computer Science Applications

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10.1016/j.cma.2017.09.018

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title = "A new goal-oriented formulation of the finite element method",

abstract = "In this paper, we introduce, analyze, and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary-value problem. The objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal-oriented adaptivity, which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement. In the current approach, we first solve the adjoint problem, then use the adjoint information as a minimization constraint for the primal problem. As a result, the constrained finite element solution is enhanced with respect to the quantities of interest, while maintaining near-optimality in energy norm. We describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples.",

author = "Kenan Kergrene and Serge Prudhomme and Ludovic Chamoin and Marc Laforest",

note = "KAUST Repository Item: Exported on 2021-04-06 Acknowledged KAUST grant number(s): CRG3, OCRF-2014-CRG Acknowledgements: SP is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. He also acknowledges the support by KAUST under Award Number OCRF-2014-CRG3-2281. Moreover, the authors gratefully acknowledge Jonathan Vacher for fruitful discussions on the subject. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

year = "2017",

month = dec,

doi = "10.1016/j.cma.2017.09.018",

language = "English (US)",

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AU - Prudhomme, Serge

AU - Chamoin, Ludovic

AU - Laforest, Marc

N1 - KAUST Repository Item: Exported on 2021-04-06 Acknowledged KAUST grant number(s): CRG3, OCRF-2014-CRG Acknowledgements: SP is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. He also acknowledges the support by KAUST under Award Number OCRF-2014-CRG3-2281. Moreover, the authors gratefully acknowledge Jonathan Vacher for fruitful discussions on the subject. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2017/12

Y1 - 2017/12

N2 - In this paper, we introduce, analyze, and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary-value problem. The objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal-oriented adaptivity, which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement. In the current approach, we first solve the adjoint problem, then use the adjoint information as a minimization constraint for the primal problem. As a result, the constrained finite element solution is enhanced with respect to the quantities of interest, while maintaining near-optimality in energy norm. We describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples.

AB - In this paper, we introduce, analyze, and numerically illustrate a method for taking into account quantities of interest during the finite element treatment of a boundary-value problem. The objective is to derive a method whose computational cost is of the same order as that of the classical approach for goal-oriented adaptivity, which involves the solution of the primal problem and of an adjoint problem used to weigh the residual and provide indicators for mesh refinement. In the current approach, we first solve the adjoint problem, then use the adjoint information as a minimization constraint for the primal problem. As a result, the constrained finite element solution is enhanced with respect to the quantities of interest, while maintaining near-optimality in energy norm. We describe the formulation in the case of a problem defined by a symmetric continuous coercive bilinear form and demonstrate the efficiency of the new approach on several numerical examples.

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DO - 10.1016/j.cma.2017.09.018

M3 - Article

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SP - 256

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JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

ER -

A new goal-oriented formulation of the finite element method

Abstract

Bibliographical note

ASJC Scopus subject areas

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