Goal-Oriented Error Estimation and Adaptivity for Free-Boundary Problems: The Domain-Map Linearization Approach

Zee, Kristoffer G. van der; Brummelen, E. Harald van; Borst, René de

doi:10.1137/080741227

Cited by 18 publications

(29 citation statements)

References 40 publications

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“…Finite differences [127,128] can be applied, but the resulting derivatives can lead to slower convergence of the Newton iterations compared to exact derivatives from shape differentiation [69]. Furthermore, van der Zee et al [205] calculate the shape derivatives by remapping to the reference domain.…”

Section: Monolithic Approachmentioning

confidence: 99%

Partitioned Simulation of Fluid-Structure Interaction

Degroote¹

2013

Arch Computat Methods Eng

123

107

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In this review article, the focus is on partitioned simulation techniques for strongly coupled fluid-structure interaction problems, especially on techniques which use at least one of the solvers as a black box. First, a number of analyses are reviewed to explain why Gauss-Seidel coupling iterations converge slowly or not at all for fluid-structure interaction problems with strong coupling. This provides the theoretical basis for the fast convergence of quasi-Newton and multi-level techniques. Second, several partitioned techniques that couple two black-box solvers are compared with respect to implementation and performance. Furthermore, performance comparisons between partitioned and monolithic techniques are examined. Subsequently, two similar techniques to couple a black-box solver with an accessible solver are analyzed. In addition, several other techniques for fluid-structure interaction simulations are studied and various methods to take into account deforming fluid domains are discussed.

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Section: Monolithic Approachmentioning

confidence: 99%

Partitioned Simulation of Fluid-Structure Interaction

Degroote¹

2013

Arch Computat Methods Eng

123

107

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“…The effect of adjoint approximation has previously been considered by [10,32,37], amongst others. Let the adjoint finite element spacesX 1 ,X 2 , andŶ be such that…”

Section: The Effect Of Adjoint Approximationmentioning

confidence: 99%

Adjoint-Based a Posteriori Error Estimation for Coupled Time-Dependent Systems

Asner¹,

Tavener

Kay

2012

SIAM J. Sci. Comput.

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Abstract. We consider time-dependent parabolic problems coupled across a common interface which we formulate using a Lagrange multiplier construction and solve by applying a monolithic solution technique. We derive an adjoint-based a posteriori error representation for a quantity of interest given by a linear functional of the solution. We establish the accuracy of our error representation formula through numerical experimentation and investigate the effect of error in the adjoint solution. Crucially, the error representation affords a distinction between temporal and spatial errors and can be used as a basis for a blockwise time-space refinement strategy. Numerical tests illustrate the efficacy of the refinement strategy by capturing the distinctive behavior of a localized traveling wave solution. The saddle point systems considered here are equivalent to those arising in the mortar finite element technique for parabolic problems. 1. Introduction. We consider an adjoint-based a posteriori error estimator for parabolic problems that are coupled across a given interface and construct an adaptive space-time finite element scheme. A posteriori error estimation is commonly used in parabolic problems for adaptive mesh refinement and the different approaches are described in a detailed review paper [1], as well as in [4,5,14,20,24,35,39,40,41]. In particular, [14,15,16] develop a technique based on solving an adjoint problem to estimate the error in a quantity of interest given by a functional of the solution as opposed to the error in a norm of the solution.Error estimation for coupled problems solved using operator decomposition techniques was considered in [17,18,9,8]. While operator decomposition is traditionally viewed as easy and inexpensive to implement [19,28], its convergence is rarely guaranteed. In this paper we take the alternative, monolithic approach to solving coupled parabolic problems, which do not suffer from this problem. We introduce a Lagrange multiplier (mortar element) space on the interface between the component domains, resulting in a large-scale complex system, which can nevertheless be solved efficiently with special treatment [33,22]. Moreover, the presence of the Lagrange multiplier variable allows us to derive error estimates for a quantity of interest that is supported along the interface. This is particularly useful in applications where the Lagrange

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“…This is the shape-linearization part of our work on goaloriented error estimation and adaptivity for free-boundary problems; see also [42]. We consider duality-based a posteriori error estimates for functional outputs that include the dependence on both the error in the approximate solution and the error in the domain approximation.In [42], we explained that free-boundary problems elude the standard goaloriented error estimation framework because their typical variational form is noncanonical. In pursuit of a canonical form, we introduced the domain-map linearization approach at a reference domain which in essence reformulates the free-boundary problem to a fixed reference domain.…”

mentioning

confidence: 99%

“…Introduction. This is the shape-linearization part of our work on goaloriented error estimation and adaptivity for free-boundary problems; see also [42]. We consider duality-based a posteriori error estimates for functional outputs that include the dependence on both the error in the approximate solution and the error in the domain approximation.…”

mentioning

confidence: 99%

Goal-Oriented Error Estimation and Adaptivity for Free-Boundary Problems: The Shape-Linearization Approach

Zee¹,

Brummelen²,

Borst³

2010

SIAM J. Sci. Comput.

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Abstract. We develop duality-based a posteriori error estimates for functional outputs of solutions of free-boundary problems via shape-linearization principles. To derive an appropriate dual (linearized adjoint) problem, we linearize the domain dependence of the very weak form and goal functional of interest using techniques from shape calculus. We show for a Bernoulli-type free-boundary problem that the dual problem corresponds to a Poisson problem with a Robin-type boundary condition involving the curvature. Moreover, we derive a generalization of the dual problem for nonsmooth free boundaries which includes a natural extension of the curvature term. The effectivity of the dualbased error estimate and its usefulness in goal-oriented adaptive mesh refinement is demonstrated by numerical experiments.Key words. goal-oriented error estimation, a posteriori error estimation, Bernoulli freeboundary problem, shape derivative, shape differential calculus, linearized adjoint, adaptive mesh refinement AMS subject classifications. 35R35, 49M29, 54C56, 58C20, 65N15, 65N50 DOI. 10.1137/080741239 1. Introduction. This is the shape-linearization part of our work on goaloriented error estimation and adaptivity for free-boundary problems; see also [42]. We consider duality-based a posteriori error estimates for functional outputs that include the dependence on both the error in the approximate solution and the error in the domain approximation.In [42], we explained that free-boundary problems elude the standard goaloriented error estimation framework because their typical variational form is noncanonical. In pursuit of a canonical form, we introduced the domain-map linearization approach at a reference domain which in essence reformulates the free-boundary problem to a fixed reference domain. Accordingly, the dual (linearized adjoint) problem is obtained by linearizing the transformed problem with respect to the domain map. This approach is straightforward. However, the dual problem contains nonstandard and nonlocal interior and boundary terms, which is inconvenient from an implementation point of view. Moreover, there is some arbitrariness in the dual problem due to the heuristic extension of boundary perturbations into the domain. A similar arbitrariness appears in shape optimization in the so-called material derivative approach [23,36]. An elegant alternative in shape optimization is the shape derivative whose variational formulation consists only of standard interior and boundary terms.

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Goal-Oriented Error Estimation and Adaptivity for Free-Boundary Problems: The Domain-Map Linearization Approach

Cited by 18 publications

References 40 publications

Partitioned Simulation of Fluid-Structure Interaction

Partitioned Simulation of Fluid-Structure Interaction

Adjoint-Based a Posteriori Error Estimation for Coupled Time-Dependent Systems

Goal-Oriented Error Estimation and Adaptivity for Free-Boundary Problems: The Shape-Linearization Approach

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