On the sensitivities of multiple eigenvalues

Gravesen, Jens; Evgrafov, Anton; Nguyen, Dang-Manh

doi:10.1007/s00158-011-0644-9

Cited by 34 publications

(25 citation statements)

References 5 publications

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“…In such situations, the method proposed by Gravesen et al [30] can be used. If j 1 ¼ j 2 , we can calculate sensitivities of functions j 1 þ j 2 and j 1 j 2 using the following expressions…”

Section: Sensitivity Of Buckling Load Factorsmentioning

confidence: 98%

Topology optimization of continuum structures under buckling constraints

Gao

2015

Computers & Structures

104

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Section: Sensitivity Of Buckling Load Factorsmentioning

confidence: 98%

Topology optimization of continuum structures under buckling constraints

Gao

2015

Computers & Structures

104

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“…In such situations, the sensitivity analysis method proposed by Gravesen et al [2011] can be used. For example, if κ 1 = κ 2 , we can calculate the sensitivities of κ 1 + κ 2 by using the following expression:…”

Section: An Adaptive Continuation Methods For Topology Optimizationmentioning

confidence: 99%

An Adaptive Continuation Method for Topology Optimization of Continuum Structures Considering Buckling Constraints

Gao

2017

Int. J. Appl. Mechanics

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This paper presents an adaptive continuation method for buckling topology optimization of continuum structures using the Solid Isotropic Material with Penalization (SIMP) model. For optimization problems of minimizing structural compliance subject to constraints on material volume and buckling load factors, it has been found that the conflict between the requirements for structural stiffness and stability may have an adverse impact on the performance of existing optimization algorithms. An automatic scheme for adjusting the penalization parameter is introduced to deal with this conflict and thus achieves better designs. Based on an investigation on the effect of the penalization parameter on design evolution during the optimization process, a rule is established to determine the appropriate penalization parameter values. Using this rule, an effective scheme is developed for determining the penalization parameter values such that the buckling constraints are appropriately considered throughout the optimization process. Numerical examples are presented to illustrate the effectiveness of the proposed method.

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“…that is a particularization of what was originally suggested in the work of Amir et al 37 To motivate the use of (17), we can look at Figure 4, showing a plot of the quantity 1 − s T s (k) , where s is the sensitivity computed from an accurate u and s (k) the one computed from an u (k) fulfilling (16) or (17) for a given tolerance. For all x ∈  ad but stationary points, the smaller this quantity is, the closer s (k) will be to the steepest descent direction.…”

Section: And Briggs Et Al 34 For Details and Implementation)mentioning

confidence: 96%

“…First, eigenvalues are likely to become repeated and the corresponding sensitivities nonsmooth during the optimization . This theoretical difficulty has been extensively studied in the last decades, and several methods have been proposed for computing sensitivities of repeated eigenvalues or for working with smooth functions of them . Other approaches overcome this issue by recurring to methods of semidefinite programming or nonconvex optimization .…”

Section: Introductionmentioning

confidence: 99%

“…11,12 This theoretical difficulty has been extensively studied in the last decades, and several methods have been proposed for computing sensitivities of repeated eigenvalues 13,14 or for working with smooth functions of them. [15][16][17] Other approaches overcome this issue by recurring to methods of semidefinite programming or nonconvex optimization. 12,18,19 A second practical limitation is due to the high computational cost required by the solution of a large eigenvalue problem at each optimization step.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalue topology optimization via efficient multilevel solution of the frequency response

Ferrari

Lazarov

Sigmund

2018

Numerical Meth Engineering

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Summary The article presents an efficient solution method for structural topology optimization aimed at maximizing the fundamental frequency of vibration. Nowadays, this is still a challenging problem mainly because of the high computational cost required by spectral analyses. The proposed method relies on replacing the eigenvalue problem with a frequency response one, which can be tuned and efficiently solved by a multilevel procedure. Connections of the method with multigrid eigenvalue solvers are discussed in details. Several applications demonstrating more than 90% savings of the computational time are presented as well.

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On the sensitivities of multiple eigenvalues

Cited by 34 publications

References 5 publications

Topology optimization of continuum structures under buckling constraints

Topology optimization of continuum structures under buckling constraints

An Adaptive Continuation Method for Topology Optimization of Continuum Structures Considering Buckling Constraints

Eigenvalue topology optimization via efficient multilevel solution of the frequency response

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