Second-order shape sensitivity analysis for nonlinear problems

Taroco, E.; Buscaglia, Gustavo C.; Feljóo, R. A.

doi:10.1007/bf01278496

Cited by 12 publications

(38 citation statements)

References 12 publications

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“…where Σ ε is the Eshelby energy-momentum tensor (see, for instance, [5,26]) given in this particular case by…”

Section: Shape Sensitivity Analysismentioning

confidence: 99%

“…However, several approaches to compute the topological derivative may be found in the literature. In particular, we proposed an alternative method based on classical shape sensitivity analysis (see [1,13,14,22,25,26,27] and references therein). This approach, called Topological-Shape Sensitivity Method, was already applied in the following two-dimensional problems:…”

Section: Introductionmentioning

confidence: 99%

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Topological sensitivity analysis for three-dimensional linear elasticity problem

Novotny

Feijóo

Taroco

et al. 2007

Computer Methods in Applied Mechanics and Engineering

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In this work we use the Topological-Shape Sensitivity Method to obtain the topological derivative for three-dimensional linear elasticity problems, adopting the total potential energy as the cost function and the equilibrium equation as the constraint. This method, based on classical shape sensitivity analysis, leads to a systematic procedure to compute the topological derivative. In particular, firstly we present the mechanical model, later we perform the shape derivative of the corresponding cost function and, finally, we compute the final expression for the topological derivative using the Topological-Shape Sensitivity Method and results from classical asymptotic analysis around spherical cavities.

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“…where Σ ε is the Eshelby energy-momentum tensor (see, for instance, [5,26]) given in this particular case by…”

Section: Shape Sensitivity Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Topological sensitivity analysis for three-dimensional linear elasticity problem

Novotny

Feijóo

Taroco

et al. 2007

Computer Methods in Applied Mechanics and Engineering

Self Cite

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“…For simplicity we shall adopt as cost function the potential energy associated to the state equation (the total potential energy of the bar under analysis in the case of the kinematic formulation and the total complementary energy in the case of the static formulation). In fact, besides having a clear physical meaning, these cost functions simplify the calculation, Taroco et al (1998). We also assume that during the shape change of the transverse cross-section of the bar, the external actions, the torque (M t ) or the twist angle ( a) in each of the two formulations, remain unchanged.…”

Section: First Order Shape Sensitivitymentioning

confidence: 99%

Shape sensitivity analysis and the energy momentum tensor for the kinematic and static models of torsion

Taroco

Feijóo

2006

International Journal of Solids and Structures

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The aim of this paper is to bridge shape sensitivity analysis and configurational mechanics by means of a widespread use of the shape derivative concept. This technique will be applied as a systematic procedure to obtain the EshelbyÕs energy momentum tensor associated to the problem under consideration. In order to highlight special features of this procedure and without loss of generality, we focus our attention in the application of shape sensitivity analysis to the problem of twisted straight bars within the framework of linear elasticity.Kinematic and static variational formulations as well as the direct method of sensitivity analysis are used to perform shape derivatives of both models. Integral expressions of first and second order shape derivatives of the total potential energy and the complementary potential energy with respect to an arbitrary transverse cross-section shape change, are achieved. These integral expressions put in evidence the relationship between shape sensitivity analysis and the first and second order EshelbyÕs energy momentum tensors. Also, the null divergence property of these tensors is easily proved by comparing, in each case, the domain and boundary integral shape derivative arrived at. Finally, an example with a known exact solution, corresponding to an elastic bar with elliptical transverse cross-section submitted to twist, is presented in order to illustrate the usefulness of these tensors to compute the corresponding shape derivatives.

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“…In Equation (19), the partial derivatives with respect to the design variables and grid coordinates can be interchanged, yielding…”

Section: The Continuous Av Approach-first-order Sensitivitiesmentioning

confidence: 99%

Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach

Papadimitriou

Giannakoglou

2011

Numerical Methods in Fluids

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SUMMARYIn this paper, the so-called 'continuous adjoint-direct approach' is used within the truncated Newton algorithm for the optimization of aerodynamic shapes, using the Euler equations. It is known that the direct differentiation (DD) of the flow equations with respect to the design variables, followed by the adjoint approach, is the best way to compute the exact matrix, for use along with the Newton optimization method. In contrast to this, in this paper, the adjoint approach followed by the DD of both the flow and adjoint equations (i.e. the other way round) is proved to be the most efficient way to compute the product of the Hessian matrix with any vector required by the truncated Newton algorithm, in which the Newton equations are solved iteratively by means of the conjugate gradient (CG) method. Using numerical experiments, it is demonstrated that just a few CG steps per Newton iteration are enough. Considering that the cost of solving either the adjoint or the DD equations is approximately equal to that of solving the flow equations, the cost per Newton iteration scales linearly with the (small) number of CG steps, rather than the (much higher, in large-scale problems) number of design variables. By doing so, the curse of dimensionality is alleviated, as shown in a number of applications related to the inverse design of ducts or cascade airfoils for inviscid flows.

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Second-order shape sensitivity analysis for nonlinear problems

Cited by 12 publications

References 12 publications

Topological sensitivity analysis for three-dimensional linear elasticity problem

Topological sensitivity analysis for three-dimensional linear elasticity problem

Shape sensitivity analysis and the energy momentum tensor for the kinematic and static models of torsion

Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach

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