Goal-oriented error estimation and adaptivity for fluid–structure interaction using exact linearized adjoints

Zee, van der Kg Kristoffer; Brummelen, van Eh Harald; Akkerman, I.; Borst, de R René

doi:10.1016/j.cma.2010.12.010

Cited by 51 publications

(36 citation statements)

References 60 publications

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“…This can be achieved using a similar Lagrange multiplier construction. Further extensions of this approach include the application to deformable boundary such as that which occurs in fluid-solid interactions problems [36].…”

Section: Discussionmentioning

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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Section: Discussionmentioning

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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“…In [40,41], for example, the presented domain-map and shape-linearization approaches are considered in the context of a fluid-structure-interaction problem.…”

Section: Discussionmentioning

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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“…The term ∂M /∂u may be obtained from (21) as ∂f/∂u. Finally, the term ∂F /∂z accounts for the sensitivity of the fluid problem with respect to variations in the positions of the fluid mesh, both in the FSI interface and its internal nodes due to the ALE formulation.…”

Section: V2 Discrete Adjoint Solution Of Fluid-structure Interactiomentioning

confidence: 99%

“…Given the ability of the adjoint method to efficiently compute sensitivities with respect to thousands of design variables, recent applications have tackled very large multiphysics problems, such as those associated with the shape optimization of full aircraft configurations [16]. Adjoint methods for FSI have also been investigated in other disciplines, such as parameter identification for medical applications [17,18], coupled electrostatic FSI problems [19], goal-oriented error estimation [20,21], steering applications [22], to name a few. However, the complexity of this technique often leads to simplifications in their implementation that have limited the scope of their application.…”

Section: Introductionmentioning

confidence: 99%