Free surfaces: shape sensitivity analysis and numerical methods

Kärkkäinen, Kari; Tiihonen, Timo

doi:10.1002/(sici)1097-0207(19990320)44:8<1079::aid-nme543>3.0.co;2-i

Cited by 22 publications

(21 citation statements)

References 17 publications

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“…[26][27][28]. The use of shape calculus appears to provide a convenient rigorous setting compared to formal asymptotic developments as in [2,16].…”

Section: Discussionmentioning

confidence: 99%

“…It is therefore more related to Newton-based iteration algorithms in continuous settings as presented in [3,16], for example. In particular, we mention the works of Kärkkäinen [26] and Kärkkäinen and Tiihonen [27,28] who appropriately apply the techniques of shape differential calculus, though in a formal sense.…”

Section: Van Der Zee Van Brummelen and De Borstmentioning

confidence: 99%

“…The first is by means of a Lagrange multiplier that enforces the constraint u = h on Γ. This is the approach taken by Kärkkäinen [26] and Kärkkäinen and Tiihonen [27,28]. The downsides of this approach are the additional difficulty and dual interpretation of the Lagrange multiplier and the inability to perform the shape linearization without unnecessary smoothness assumptions.…”

Section: Very Weak Form Of the Free-boundary Problemmentioning

confidence: 99%

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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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“…[26][27][28]. The use of shape calculus appears to provide a convenient rigorous setting compared to formal asymptotic developments as in [2,16].…”

Section: Discussionmentioning

confidence: 99%

Section: Van Der Zee Van Brummelen and De Borstmentioning

confidence: 99%

Section: Very Weak Form Of the Free-boundary Problemmentioning

confidence: 99%

See 1 more Smart Citation

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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“…The shape-linearization approach has been investigated for Bernoulli-type free-boundary problems in [14,25].…”

Section: Der Zee Van Brummelen and De Borstmentioning

confidence: 99%

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

Zee¹,

Brummelen²,

Borst³

2010

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

Abstract. In free-boundary problems, the accuracy of a goal quantity of interest depends on both the accuracy of the approximate solution and the accuracy of the domain approximation. We develop duality-based a posteriori error estimates for functional outputs of solutions of free-boundary problems that include both sources of error. The derivation of an appropriate dual problem (linearized adjoint) is, however, nonobvious for free-boundary problems. To derive an appropriate dual problem, we present the domain-map linearization approach. In this approach, the free-boundary problem is first transformed into an equivalent problem on a fixed reference domain after which the dual problem is obtained by linearization with respect to the domain map. We show for a Bernoulli-type free-boundary problem that this dual problem corresponds to a Poisson problem with a nonlocal Robin-type boundary condition. Furthermore, we present numerical experiments that demonstrate the effectivity of the dual-based error estimate and its usefulness in goal-oriented adaptive mesh refinement. 1. Introduction. Free-boundary problems arise in various applications such as free-surface flow, fluid-structure interaction, and Stefan problems; see [15,18]. The numerical simulation of free-boundary problems is a challenging endeavor, as it requires the simultaneous solution of both the unknown function and its domain of definition and these two solution components can display distinct length (and/or time) scales. In many free-boundary problems, practical interest is restricted to a prescribed response quantity in the form of a goal functional of the solution rather than full norm resolution. However, the accuracy of the goal quantity depends on both the accuracy of the approximate solution and the accuracy of the domain approximation. In general, this dependence is nonobvious, and heuristic approaches, such as a priori mesh refinement in the vicinity of the free boundary [14,42], lead to inefficient approximations of the goal quantity.Finite-element techniques employing goal-oriented adaptive strategies can offer a significant efficiency improvement in such simulations. Starting with a coarse discretization, only those refinements are made which benefit substantially to the accuracy of the goal functional, in contrast to global norm-oriented adaptive strategies which make refinements which benefit the accuracy of the solution in the full

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“…First, a variational formulation may be considered and the corresponding cost function minimized [16,20,23]; this requires the calculations of shape gradients. Second, a fixed point type approach can be set up where a sequence of elliptic problems are solved in a sequence of converging domains, those domains being obtained through some updating rule at each iteration [5,8,21]. The method studied in this paper falls in the latter category.…”

Section: Introductionmentioning

confidence: 99%