Numerical minimization of eigenmodes of a membrane with respect to the domain

Oudet, Èdouard

doi:10.1051/cocv:2004011

Cited by 82 publications

(98 citation statements)

References 22 publications

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“…Furthermore, recent numerical work indicates that for eigenvalues of planar domains higher than the fourth, the optimal shape under an area restriction will not have a boundary which may be described in terms of known functions [AF1,O]. Also, the optimal domains obtained in [AF1] for the first fifteen eigenvalues suggest that any underlying structure that one might expect there to exist, such as optimal sets having at least Z 2 symmetry, will most likely be up against some exceptions.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

Bucur

Freitas

2013

Journal of Mathematical Physics

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Abstract. We consider the problem of minimizing the k th Dirichlet eigenvalue of planar domains with fixed perimeter and show that, as k goes to infinity, the optimal domain converges to the ball with the same perimeter. We also consider this problem within restricted classes of domains such as n−polygons and tiling domains, for which we show that the optimal asymptotic domain is that which maximises the area for fixed perimeter within the given family, i.e. the regular n−polygon and the regular hexagon, respectively.

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Section: Introductionmentioning

confidence: 99%

Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

Bucur

Freitas

2013

Journal of Mathematical Physics

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“…In order to avoid the computation of a new mesh at each iteration, we compute an approximation of the solution of (2.4) via a penalization method introduced in [14].…”

Section: General Approach and Level-set Methodsmentioning

confidence: 99%

“…2. Computation of the velocity field by a penalization method introduced in [14] on the fixed triangular mesh deduced from the Cartesian grid. Checking of an exit criterion.…”

Section: A Multi-level Set Methodsmentioning

confidence: 99%

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Local minimizers of functionals with multiple volume constraints

Oudet

Rieger

2008

ESAIM: COCV

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Abstract.We study variational problems with volume constraints, i.e., with level sets of prescribed measure. We introduce a numerical method to approximate local minimizers and illustrate it with some two-dimensional examples. We demonstrate numerically nonexistence results which had been obtained analytically in previous work. Moreover, we show the existence of discontinuous dependence of global minimizers from the data by using a Γ-limit argument and illustrate this with numerical computations. Finally we construct explicitly local and global minimizers for problems with two volume constraints.

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“…The variation of E(µ, f ) with respect to the discrete measure µ h on each element T k (see [14], [20]) is given in the next theorem.…”

Section: Debonding Membranes With Active Fractions Of Glue: Measuresmentioning

confidence: 99%

Analysis and simulation of debonding membranes

Durus¹,

Lux²

2011

Interfaces Free Bound.

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The paper deals with the numerical analysis and simulation of debonding membranes in the framework of quasistatic processes modeled by minimizing movements. Depending on parameters of the model, the "support" of the membrane is a quasi-open set (i.e. the membrane is debonded or glued) or a capacitary measure (i.e. fractions of glue remain active). Time discretization is associated to gradient methods based on shape (measure) derivative and genetic shape (measure) optimization.

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Numerical minimization of eigenmodes of a membrane with respect to the domain

Cited by 82 publications

References 22 publications

Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

Local minimizers of functionals with multiple volume constraints

Analysis and simulation of debonding membranes

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