Design Sensitivity Analysis in Structural Mechanics.II. Eigenvalue Variations

Haug, Edward J.; Rousselet, Bernard

doi:10.1080/03601218008907358

Cited by 115 publications

(22 citation statements)

References 5 publications

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“…Note that (13) is similar to the standard results of eigenvalue optimization (Haug and Rousselet 1980;Seyranian et al 1994;de Gournay 2006), and that (14) incorporates jump terms that are typical of shape sensitivity with discontinuous coefficients (Bernardi and Pironneau 2003;Pantz 2005). Note also that (13) holds for any eigenvalue (and not only the first one) and in the case of a simple eigenvalue dλ is linear w.r.to θ , hence Fréchet differentiability holds.…”

Section: Statement Of the Main Resultssupporting

confidence: 52%

Shape optimization for the generalized Graetz problem

Gournay

Fehrenbach

Plouraboué

2014

Struct Multidisc Optim

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion equation. The problem boils down to the optimization of generalized eigenvalues on a two phases domain. Shape sensitivity analysis is performed with respect to the evolution of the interface between the fluid and solid phase. In particular physical settings, counterexamples where there is no optimal domains are exhibited. Numerical examples of optimal domains with different physical parameters and constraints are presented. Two different numerical methods (level-set and mesh-morphing) are show-cased and compared.

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Section: Statement Of the Main Resultssupporting

confidence: 52%

Shape optimization for the generalized Graetz problem

Gournay

Fehrenbach

Plouraboué

2014

Struct Multidisc Optim

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“…In the previous relation, we can use either h or −h, therefore the minimality of Ω * gives the desired result (14), (15). Now, let us consider the case of a segment Σ.…”

Section: Optimality Conditionsmentioning

confidence: 99%

“…Let us denote by v = V.n such a concave function. Replacing in (22) and using (14), (15) yields on the segment Σ:…”

Section: Optimality Conditionsmentioning

confidence: 99%

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Henrot

Oudet

2003

Archive for Rational Mechanics and Analysis

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Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditionsAntoine HENROT, Edouard OUDET AbstractIn this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the "stadium", convex hull of two identical tangent disks. We refute this conjecture. Nevertheless, we prove existence of a minimzer. We also study some qualitative properties of the minimizer (regularity, geometric properties).

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“…For prior attempts in this context see Haug and Rousselet [18] and Choi and Haug [7]. Our principal tool is the generalized gradient of Clarke [10].…”

Section: Necessary Conditionsmentioning

confidence: 99%

On the Optimal Design of Columns Against Buckling

Cox¹,

Overton²

1992

SIAM J. Math. Anal.

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-We establish existence, derive necessary conditions, and construct and test an algorithm for the maximization of a column's Euler buckling load under a variety of boundary conditions over a general class of admissible designs. We prove that symmetric clamped-clamped columns possess a positive first eigenfunction and introduce a symmetric rearrangement that does not decrease the column's buckling load. Our necessary conditions, expressed in the language of Clarke's generalized gradient [10] In our numerical treatment of the associated finite dimensional optimization problem we build on the work of Overton [26] in devising an efficient means of extracting an ascent direction from the column's least eigenvalue. Owing to its possible multiplicity this is indeed a nonsmooth problem and again the ideas of Clarke [10] are exploited.

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Design Sensitivity Analysis in Structural Mechanics.II. Eigenvalue Variations

Cited by 115 publications

References 5 publications

Shape optimization for the generalized Graetz problem

Shape optimization for the generalized Graetz problem

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

On the Optimal Design of Columns Against Buckling

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