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1 – 10 of 348Phillip Baumann and Kevin Sturm
The 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…
Abstract
Purpose
The 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/approach
The 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.
Findings
The 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/value
In 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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Frédérique Le Louër and María-Luisa Rapún
The purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic…
Abstract
Purpose
The purpose of this paper is to revisit the recursive computation of closed-form expressions for the topological derivative of shape functionals in the context of time-harmonic acoustic waves scattering by sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions).
Design/methodology/approach
The elliptic boundary value problems in the singularly perturbed domains are equivalently reduced to couples of boundary integral equations with unknown densities given by boundary traces. In the case of circular or spherical holes, the spectral Fourier and Mie series expansions of the potential operators are used to derive the first-order term in the asymptotic expansion of the boundary traces for the solution to the two- and three-dimensional perturbed problems.
Findings
As the shape gradients of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.
Originality/value
The authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function in the iterated numerical solution of any shape optimization or imaging problem relying on time-harmonic acoustic waves propagation. When coupled with converging Gauss−Newton iterations for the search of optimal boundary parametrizations, it generates fully automatic algorithms.
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Lucas Fernandez and Ravi Prakash
The purpose of this paper is to present topological derivatives-based reconstruction algorithms to solve an inverse scattering problem for penetrable obstacles.
Abstract
Purpose
The purpose of this paper is to present topological derivatives-based reconstruction algorithms to solve an inverse scattering problem for penetrable obstacles.
Design/methodology/approach
The method consists in rewriting the inverse reconstruction problem as a topology optimization problem and then to use the concept of topological derivatives to seek a higher-order asymptotic expansion for the topologically perturbed cost functional. Such expansion is truncated and then minimized with respect to the parameters under consideration, which leads to noniterative second-order reconstruction algorithms.
Findings
In this paper, the authors develop two different classes of noniterative second-order reconstruction algorithms that are able to accurately recover the unknown penetrable obstacles from partial measurements of a field generated by incident waves.
Originality/value
The current paper is a pioneer work in developing a reconstruction method entirely based on topological derivatives for solving an inverse scattering problem with penetrable obstacles. Both algorithms proposed here are able to return the number, location and size of multiple hidden and unknown obstacles in just one step. In summary, the main features of these algorithms lie in the fact that they are noniterative and thus, very robust with respect to noisy data as well as independent of initial guesses.
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Lalaina Rakotondrainibe, Grégoire Allaire and Patrick Orval
This paper is devoted to the theoretical and numerical study of a new topological sensitivity concerning the insertion of a small bolt connecting two parts in a mechanical…
Abstract
Purpose
This paper is devoted to the theoretical and numerical study of a new topological sensitivity concerning the insertion of a small bolt connecting two parts in a mechanical structure. First, an idealized model of bolt is proposed which relies on a non-local interaction between the two ends of the bolt (head and threads) and possibly featuring a pre-stressed state. Second, a formula for the topological sensitivity of such an idealized bolt is rigorously derived for a large class of objective functions. Third, numerical tests are performed in 2D and 3D to assess the efficiency of the bolt topological sensitivity in the case of no pre-stress. In particular, the placement of bolts (acting then as springs) is coupled to the further optimization of their location and to the shape and topology of the structure for volume minimization under compliance constraint.
Design/methodology/approach
The methodology relies on the adjoint method and the variational formulation of the linearized elasticity equations in order to establish the topological sensitivity.
Findings
The numerical results prove the influence of the number and locations of the bolts which strongly influence the final optimized design of the structure.
Originality/value
This paper is the first one to study the topology optimization of bolted systems without a fixed prescribed number of bolts.
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L.C. Ruspini, E. Dari, C. Padra, G.H. Paissan and N.N. Salva
The purpose of this paper is to present applications of the topological optimization method dealing with fluid dynamic problems in two- and three dimensions. The main goal is to…
Abstract
Purpose
The purpose of this paper is to present applications of the topological optimization method dealing with fluid dynamic problems in two- and three dimensions. The main goal is to develop a tool package able to optimize topology in realistic devices (e.g. inlet manifolds) considering the non-linear terms on Navier–Stokes equations.
Design/methodology/approach
Using an in-house Fortran code, a Galerkin stabilized finite element is implemented method to solve the three equation systems necessary for the topological optimization method: the direct problem, adjoint problem and topological derivative. The authors address the non-linearity in the equations using an iterative method. Different techniques to create holes into a two-dimensional discrete domain are analyzed.
Findings
One technique to create holes produces more accurate and robust results. The authors present several examples of applications in two- and three-dimensional components, which highlight the potential of this method in the optimization of fluid components.
Research limitations/implications
The authors contribute to the methodology and design in engineering.
Practical implications
Engineering fluid flow systems are used in many different industrial applications, e.g. oil flow in pipes; air flow around an airplane wing; sailing submarines; blood flow in synthetic arteries; and thermal and fissure spreading problems. The aim of this work is to create an effective design tool for obtaining efficient engineering structures and devices.
Originality/value
The authors contribute by creating an application of the method to design a tridimensional realistic device, which can be essayed experimentally. Particularly, the authors apply the design tool to an inlet manifold.
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Meisam Takalloozadeh and Gil Ho Yoon
Body forces are always applied to structures in the form of the weight of materials. In some cases, they can be neglected in comparison with other applied forces. Nevertheless…
Abstract
Purpose
Body forces are always applied to structures in the form of the weight of materials. In some cases, they can be neglected in comparison with other applied forces. Nevertheless, there is a wide range of structures in civil and mechanical engineering in which weight or other types of body forces are the main portions of the applied loads. The optimal topology of these structures is investigated in this study.
Design/methodology/approach
Topology optimization plays an increasingly important role in structural design. In this study, the topological derivative under body forces is used in a level-set-based topology optimization method. Instability during the optimization process is addressed, and a heuristic solution is proposed to overcome the challenge. Moreover, body forces in combination with thermal loading are investigated in this study.
Findings
Body forces are design-dependent loads that usually add complexity to the optimization process. Some problems have already been addressed in density-based topology optimization methods. In the present study, the body forces in a topological level-set approach are investigated. This paper finds that the used topological derivative is a flat field that causes some instabilities in the optimization process. The main novelty of this study is a technique used to overcome this challenge by using a weighted combination.
Originality/value
There is a lack of studies on level-set approaches that account for design-dependent body forces and the proposed method helps to understand the challenges posed in such methods. A powerful level-set-based approach is used for this purpose. Several examples are provided to illustrate the efficiency of this method. Moreover, the results show the effect of body forces and thermal loading on the optimal layout of the structures.
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Àlex Ferrer and Sebastián Miguel Giusti
The purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.
Abstract
Purpose
The purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.
Design/methodology/approach
The optimal topology is obtained through a relaxed formulation of the problem by replacing the characteristic function with a continuous design variable, so-called density variable. The constitutive tensor is then parametrized with the density variable through an analytical interpolation scheme that is based on the topological derivative concept. The intermediate values that may appear in the optimal topologies are removed by penalizing the perimeter functional.
Findings
The optimization process benefits from the intermediate values that provide the proposed method reaching to solutions that the topological derivative had not been able to find before. In addition, the presented theory opens the path to propose a new framework of research where the topological derivative uses classical optimization algorithms.
Originality/value
The proposed methodology allows us to use the topological derivative concept for solving the inverse homogenization problem and to fulfil the optimality conditions of the problem with the use of classical optimization algorithms. The authors solved several material design examples through a projected gradient algorithm to show the advantages of the proposed method.
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Guilherme Barros, João Filho, Luiz Nunes and Marcel Xavier
The purpose of this paper is to experimentally validate the crack growth control based on the topological derivative of the famous Rice's integral.
Abstract
Purpose
The purpose of this paper is to experimentally validate the crack growth control based on the topological derivative of the famous Rice's integral.
Design/methodology/approach
Single edge notch tensile specimens with two configurations were tested. Displacement fields near notch were experimentally obtained using the digital image correlation method. These displacements were used to verify the minimization of the associated shape functional, which is defined in terms of the Rice's integral, when a set of controls (holes) positioned according to the topological derivative information, is inserted. Based on the Griffth's energy criterion, this minimization represents an improvement in the fracture toughness of cracked bodies.
Findings
The experimental tests confirmed that a decrease around 27% in the value of the associated shape functional can be obtained following this approach. Therefore, the results allow us to conclude that the predictive methodology for crack growth control based on the topological derivative is feasible.
Originality/value
This is the first work concerning experimental validation of crack growth control based on the topological derivative method.
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Marcel Xavier and Nicolas Van Goethem
In the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.
Abstract
Purpose
In the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.
Design/methodology/approach
The Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.
Findings
Numerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.
Originality/value
This is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.
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Rolando Yera, Luisina Forzani, Carlos Gustavo Méndez and Alfredo E. Huespe
This work presents a topology optimization methodology for designing microarchitectures of phononic crystals. The objective is to get microstructures having, as a consequence of…
Abstract
Purpose
This work presents a topology optimization methodology for designing microarchitectures of phononic crystals. The objective is to get microstructures having, as a consequence of wave propagation phenomena in these media, bandgaps between two specified bands. An additional target is to enlarge the range of frequencies of these bandgaps.
Design/methodology/approach
The resulting optimization problem is solved employing an augmented Lagrangian technique based on the proximal point methods. The main primal variable of the Lagrangian function is the characteristic function determining the spatial geometrical arrangement of different phases within the unit cell of the phononic crystal. This characteristic function is defined in terms of a level-set function. Descent directions of the Lagrangian function are evaluated by using the topological derivatives of the eigenvalues obtained through the dispersion relation of the phononic crystal.
Findings
The description of the optimization algorithm is emphasized, and its intrinsic properties to attain adequate phononic crystal topologies are discussed. Particular attention is addressed to validate the analytical expressions of the topological derivative. Application examples for several cases are presented, and the numerical performance of the optimization algorithm for attaining the corresponding solutions is discussed.
Originality/value
The original contribution results in the description and numerical assessment of a topology optimization algorithm using the joint concepts of the level-set function and topological derivative to design phononic crystals.
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