Goal-oriented space-time adaptivity for transient dynamics using a modal description of the adjoint solution

Verdugo, Francesc; Mariné, Núria Parés; Dı́ez, Pedro

doi:10.1007/s00466-014-0988-2

Cited by 9 publications

(11 citation statements)

References 30 publications

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“…Figure 18 are the same because the problem is symmetric. Figure 19 shows the adapted space-time mesh after 4 iterations and the relative error in the quantity of interest when we employ the symmetric operator (41). We conclude that with estimator…”

Section: Diffusion Problemmentioning

confidence: 81%

“…The aforementioned methods are implicit in time so that is the reason why most authors employ implicit methods in time for goal-oriented adaptivity. In [10,35,36], goal-oriented adaptive strategies are explained for wave propagation phenomena; in [31,[37][38][39] for parabolic problems; in [40,41] the error representation is derived for structural transient dynamics; and finally, in [42,43] the goal-oriented approach is extended to nonlinear problems. Recently, in [44], the authors expressed some multi-step implicit-explicit (IMEX) schemes as Galerkin methods in time for PDEs by using specific quadrature rules for time integration.…”

Section: Introductionmentioning

confidence: 99%

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Explicit-in-time goal-oriented adaptivity

Muñoz‐Matute

Calo

Pardo

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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Goal-oriented adaptivity is a powerful tool to accurately approximate physically relevant solution features for partial differential equations. In time dependent problems, we seek to represent the error in the quantity of interest as an integral over the whole space-time domain. A full space-time variational formulation allows such representation. Most authors employ implicit time marching schemes to perform goal-oriented adaptivity as it is known that they can be reinterpreted as Galerkin methods. In this work, we consider variational forms for explicit methods in time. We derive an appropriate error representation and propose a goal-oriented adaptive algorithm in space. For that, we derive the forward Euler method in time employing a discontinuous-in-time Petrov-Galerkin formulation. In terms of time domain adaptivity, we impose the Courant-Friedrichs-Lewy condition to ensure the stability of the method. We provide some numerical results in 1D space + time for the diffusion and advection-diffusion equations to show the performance of the proposed explicit-in-time goal-oriented adaptive algorithm.

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Section: Diffusion Problemmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Explicit-in-time goal-oriented adaptivity

Muñoz‐Matute

Calo

Pardo

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…Indeed, we can apply this technique to a wide range of problems, including adaptivity in time domain 61,62 or hp-adaptive algorithms. 4,6 We are also working on extending the proposed approach to other discretizations such as Petrov-Galerkin, Discontinuous Galerkin, and/or some version of the discontinuous Petrov Galerkin method.…”

Section: Discussionmentioning

confidence: 99%

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

Darrigrand

Rodríguez-Rozas

Muga

et al. 2017

Numerical Meth Engineering

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SummaryIn goal-oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element-wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal-oriented adaptivity.While the method can be applied to a variety of problems, we focus here on two-and three-dimensional (2-D and 3-D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p-adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection-dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging-while-drilling problem to illustrate the applicability of the proposed method.

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“…In particular when there is a strong transport phenomena, such as in structural dynamics, cf. Verdugo et al [16], the optimal spatial mesh varies dramatically with time. Hence, there is a great need for arbitrary space-time adaptivity.…”

Section: Introductionmentioning

confidence: 98%

A sequential-adaptive strategy in space–time with application to consolidation of porous media

Larsson

Runesson

2015

Computer Methods in Applied Mechanics and Engineering

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Highlights• A novel approach for space-time adaptive finite element analysis is presented.• Global error control is obtained by the technique of solving a dual problem. • Space and time errors are estimated using a hierarchical decomposition of the dual.• Recursive adaptations of the whole space-time mesh is avoided.• The coupled consolidation problem in geomechanics is considered as an application. AbstractIssues related to space-time adaptivity for a class of nonlinear and time-dependent problems are discussed. The dG(k)-methods are adopted for the time integration, and the a posteriori error control is based on the appropriate dual problem in space-time. One key ingredient is to decouple the error generation in space and time with a hierarchical decomposition of the discrete space of dual solutions. The main idea put forward in the paper is to increase the computational efficiency of the adaptive scheme by avoiding recursive adaptations of the whole time-mesh; rather, the space-mesh and the time-step defining each finite space-time slab are defined in a truly sequential fashion.The proposed adaptive strategy is applied to the coupled consolidation problem in geomechanics involving large deformations. Its performance is investigated with the aid of a numerical example in 2D.

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Goal-oriented space-time adaptivity for transient dynamics using a modal description of the adjoint solution

Cited by 9 publications

References 30 publications

Explicit-in-time goal-oriented adaptivity

Explicit-in-time goal-oriented adaptivity

Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation

A sequential-adaptive strategy in space–time with application to consolidation of porous media

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