Topological Derivatives of Shape Functionals. Part III: Second-Order Method and Applications

AbstractThe multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

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Adjoint-based methods to compute higher-order topological derivatives with an application to elasticity

Engineering Computations ◽

10.1108/ec-07-2021-0407 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Phillip Baumann ◽

Kevin Sturm

Keyword(s):

Linear Elasticity ◽

Problem Formulation ◽

Short Review ◽

Higher Order ◽

Second Order ◽

Adjoint Variable ◽

Content Type ◽

Lagrangian Framework ◽

Topological Derivatives ◽

Derivatives Of

PurposeThe goal of this paper is to give a comprehensive and short review on how to compute the first- and second-order topological derivatives and potentially higher-order topological derivatives for partial differential equation (PDE) constrained shape functionals.Design/methodology/approachThe authors employ the adjoint and averaged adjoint variable within the Lagrangian framework and compare three different adjoint-based methods to compute higher-order topological derivatives. To illustrate the methodology proposed in this paper, the authors then apply the methods to a linear elasticity model.FindingsThe authors compute the first- and second-order topological derivatives of the linear elasticity model for various shape functionals in dimension two and three using Amstutz' method, the averaged adjoint method and Delfour's method.Originality/valueIn contrast to other contributions regarding this subject, the authors not only compute the first- and second-order topological derivatives, but additionally give some insight on various methods and compare their applicability and efficiency with respect to the underlying problem formulation.

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Topological Derivatives of Shape Functionals. Part II: First-Order Method and Applications

Journal of Optimization Theory and Applications ◽

10.1007/s10957-018-1419-x ◽

2018 ◽

Vol 180 (3) ◽

pp. 683-710 ◽

Cited By ~ 4

Author(s):

Antonio André Novotny ◽

Jan Sokołowski ◽

Antoni Żochowski

Keyword(s):

Order Method ◽

First Order ◽

Topological Derivatives ◽

First Order Method ◽

Derivatives Of

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Routing Optimization with Efficient Second Order Distributed Approach Using Congestion Control Rules

International Journal of Advanced Research in Computer Science and Software Engineering ◽

10.23956/ijarcsse/v7i7/0208 ◽

2017 ◽

Vol 7 (7) ◽

pp. 363

Author(s):

Jaya Pratha Sebastiyar ◽

Martin Sahayaraj Joseph

Keyword(s):

Congestion Control ◽

Second Order ◽

Back Pressure ◽

Order Method ◽

Delay Performance ◽

Routing Optimization ◽

Distributed Approach ◽

Control Rules ◽

Primal Dual ◽

Queue Stability

Distributed joint congestion control and routing optimization has received a significant amount of attention recently. To date, however, most of the existing schemes follow a key idea called the back-pressure algorithm. Despite having many salient features, the first-order sub gradient nature of the back-pressure based schemes results in slow convergence and poor delay performance. To overcome these limitations, the present study was made as first attempt at developing a second-order joint congestion control and routing optimization framework that offers utility-optimality, queue-stability, fast convergence, and low delay. Contributions in this project are three-fold. The present study propose a new second-order joint congestion control and routing framework based on a primal-dual interior-point approach and established utility-optimality and queue-stability of the proposed second-order method. The results of present study showed that how to implement the proposed second-order method in a distributed fashion.

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