Numerical solution of Bernoulli‐type free boundary value problems by variable domain method

Mejak, George

doi:10.1002/nme.1620372407

Cited by 10 publications

(10 citation statements)

References 20 publications

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“…where α α α is the vector of discrete design variables defining the shape of Γ i (Ξ) and u h and v v v h satisfy the discretized coupled system (10) - (12). Here R(θ) is the radius of the free boundary corresponding to α α α (a suitable discretization of g) and R(θ) is the radius of the target free boundary at the angle θ.…”

Section: Discrete Optimization Problemmentioning

confidence: 99%

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Shape optimization of systems governed by Bernoulli free boundary problems

Toivanen¹,

Haslinger²,

Mäkinen³

2008

Computer Methods in Applied Mechanics and Engineering

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In this work we consider shape optimization of systems, which are governed by external Bernoulli free boundary problems. A pseudo-solid approach for solving discrete free boundary problems is introduced. The solution strategy readily allows us to obtain geometrical sensitivities of the system, which can then be used to solve e.g. inverse design problems. Numerical examples show that the location of the free boundary can, to some extent, be controlled by changing the shape of the other component of the boundary.

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Section: Discrete Optimization Problemmentioning

confidence: 99%

“…Usually this means that the dependence of the location of the internal grid points on the locations of the boundary ones must be known. In [12], for example, this is done by constructing a conformal mapping between the computational domain and a simple reference domain. Instead, our solution strategy for the free boundary problem is as follows.…”

mentioning

confidence: 99%

Shape optimization of systems governed by Bernoulli free boundary problems

Toivanen¹,

Haslinger²,

Mäkinen³

2008

Computer Methods in Applied Mechanics and Engineering

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“…Most of these address the free-boundary problem in a discretized setting; see, for instance, [10,30,38]. Our linearization is, however, in the continuous setting where one requires intricate shape differential calculus.…”

Section: Van Der Zee Van Brummelen and De Borstmentioning

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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“…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].…”

Section: Introductionmentioning

confidence: 99%