Topological Sensitivity Analysis Research Articles

A New Strategy for Railway Bogie Frame Designing Combining Structural–Topological Optimization and Sensitivity Analysis

Rolling stock manufacturers are finding innovative structural solutions to improve the quality and reliability of railway vehicles components. Structural optimization processes represent an effective strategy for reducing manufacturing costs, resulting in geometries easier to design and produce. In this framework, the present paper proposes a new methodology to design a railway metro bogie frame, combining structural–topological optimization methods and sensitivity analysis. In addition, manufacturing constraints were included to make the component design suitable for production through sand-casting. A robust sensitivity analysis has highlighted the most critical load conditions acting on the bogie frame. Its effectiveness was verified by carrying out two different structural optimizations based on different loadings. Two equivalent designs were obtained. Computational times were positively reduced by about 57%. The maximum value of stress was reduced about 23%. This new methodology has shown encouraging results to streamline the design process of this complex mechanical system, allowing researchers to also include manufacturing requirements.

On the topological gradient method for an inverse problem resolution

Abstract In this work, we consider the topological gradient method to deal with an inverse problem associated with three-dimensional Stokes equations. The problem consists in detecting unknown point forces acting on fluid from measurements on the boundary of the domain. We present an asymptotic expansion of the considered cost function using the topological sensitivity analysis method. A detection algorithm is then presented using the developed formula. Some numerical tests are presented to show the efficiency of the presented algorithm.

Topological sensitivity analysis for the 3D nonlinear Navier–Stokes equations

This work is devoted to a topological asymptotic expansion for the nonlinear Navier–Stokes operator. We consider the 3D Navier–Stokes equations as a model problem and we derive a topological sensitivity analysis for a design function with respect to the insertion of a small obstacle inside the fluid flow domain. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is examined. The performed mathematical framework can be applied for a large class of design functions and arbitrarily shaped geometric perturbations. The obtained asymptotic formula can serve as a useful tool for solving a variety of topology optimization problems in fluid mechanics.

A Unified Approach to Shape and Topological Sensitivity Analysis of Discretized Optimal Design Problems

We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is represented by a piecewise linear and globally continuous level set function on a fixed finite element mesh and relate perturbations of the level set function to perturbations of the shape or topology of the corresponding design. We illustrate the sensitivity analysis for a problem that is constrained by a reaction–diffusion equation and draw connections between our discrete sensitivities and the well-established continuous concepts of shape and topological derivatives. Finally, we verify our sensitivities and illustrate their application in a level-set-based design optimization algorithm where no distinction between shape and topological updates has to be made.

Topological asymptotic expansion for the full Navier–Stokes equations

This paper is concerned with a topological sensitivity analysis for the two dimensional incompressible Navier–Stokes equations. We derive a topological asymptotic expansion for a shape functional with respect to the creation of a small geometric perturbation inside the fluid flow domain. The geometric perturbation is modeled as a small obstacle. The asymptotic behavior of the perturbed velocity field with respect to the obstacle size is discussed. The obtained results are valid for a large class of shape fonctions and arbitrarily shaped geometric perturbations. The established topological asymptotic expansion provides a useful tool for shape and topology optimization in fluid mechanics.

Singular geometry perturbation based method for shape-topology optimization in unsteady Stokes flow

This paper concerns the topological derivatives and its applications for solving shape and topology optimization problems in fluid mechanics. The fluid flow is governed by the unsteady incompressible Stokes equations in the two dimensional case. We derive a topological sensitivity analysis for this parabolic-type operator. The proposed approach is based on a preliminary estimate describing the variation of the velocity field caused by the presence of a small obstacle inside the fluid flow. We obtain a topological asymptotic expansion for the unsteady Stokes operator valid for a large class of shape functions and an arbitrarily shaped geometric perturbation. Then, the topological gradient is exploited for building an efficient and accurate topology optimization algorithm. Finally, we present some numerical investigations showing the efficiency of the proposed approach.

Approximation of a Wanted Flow via Topological Sensitivity Analysis

We propose an optimization algorithm for the geometric control of fluid flow. the used approach is based on the topological sensitivity analysis method. It consists in studying the variation of a cost function with respect to the insertion of a small obstacle in the domain. Some theoretical and numerical results are presented in 2D and 3D.

Identification of an embedded tumor using the thermography process and the topological sensitivity analysis method

This work is concerned with a geometric inverse problem related to the thermography concept. The aim is the identification of the shape, size and location of a small embedded tumor from measured temperature on the skin surface. The temperature distribution in the biological tissue is governed by the Pennes model equation. Our proposed approach is based on the Kohn–Vogelius formulation and the topological sensitivity analysis method. The ill‐posed geometric inverse problem is reformulated as a topology optimization one. The topological gradient is exploited for locating the zone containing the embedded tumor. The size and shape of the infected zone are approximated using the solution of a scalar parameter estimate problem. The efficiency and accuracy of the proposed approach are justified by some numerical simulations.

Topological Sensitivity Analysis for the Anisotropic Laplace Problem

This paper is concerned with the reconstruction of objects immersed in anisotropic media from boundary measurements. The aim of this paper is to propose an alternative approach based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing an energy-like function. We derive a topological asymptotic expansion for the anisotropic Laplace operator. The unknown object is reconstructed using level-set curve of the topological gradient. We make finally some numerical examples proving the efficiency and accuracy of the proposed algorithm.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part II: recursive computations by the boundary integral equation method

PurposeThe 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/approachThe 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.FindingsAs 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/valueThe 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.

Cavities identification in linear elasticity: energy-gap versus L2-gap cost functionals

We study the geometric inverse problem of cavities identification in linear elasticity equation with partially over-determined boundary data. This work proposes a way to compute the topological derivative based on the topological sensitivity analysis concepts, adopting the energy-gap and the -gap as cost functionals. Then, a comparison of both functionals is presented to determine the best choice for numerical studies. Further, several numerical experiences are shown to explore the efficiency of this method.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

PurposeIn this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.Design/methodology/approachFrom the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.FindingsAs the shape gradient 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/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

Topological sensitivity analysis with respect to a small idealized bolt

PurposeThis 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/approachThe methodology relies on the adjoint method and the variational formulation of the linearized elasticity equations in order to establish the topological sensitivity.FindingsThe numerical results prove the influence of the number and locations of the bolts which strongly influence the final optimized design of the structure.Originality/valueThis paper is the first one to study the topology optimization of bolted systems without a fixed prescribed number of bolts.

Topological sensitivity method for reconstruction of the spatial component in the source term of a time-fractional diffusion equation

In the paper, an inverse source problem for a time-fractional diffusion equation is formulated and proved. The topological sensitivity analysis method is presented. The considered problem is formulated as a topology optimization one. The proposed process leads to a non-iterative reconstruction algorithm can be applied for large class of cost functional. The unknown shape of the term source support is reconstructed using a level-set curve of the topological gradient. Several numerical simulations are presented to show the efficiency and the accuracy of the algorithm.

Bayesian approach to inverse scattering with topological priors

We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite parameter field representing the objects. To construct the prior distribution we use a topological sensitivity analysis. We demonstrate the approach on the Bayesian solution of 2D inverse problems in light and acoustic holography with synthetic data. Statistical information on objects such as their center location, diameter size, orientation, as well as material properties, are extracted by sampling the posterior distribution. Assuming the number of objects known, comparison of the results obtained by Markov Chain Monte Carlo (MCMC) sampling and by sampling a Gaussian distribution found by linearization about the maximum a posteriori estimate show reasonable agreement. The latter procedure has low computational cost, which makes it an interesting tool for uncertainty studies in 3D. However, MCMC sampling provides a more complete picture of the posterior distribution and yields multi-modal posterior distributions for problems with larger measurement noise. When the number of objects is unknown, we devise a stochastic model selection framework.

Geometric inverse problem for the nonstationary Stokes equations using topological sensitivity analysis

We study in this paper a geometric inverse problem in fluid mechanics. The goal is to determine the location of an object in the fluid domain from boundary information. We consider the case of a nonstationary two‐dimensional Stokes flow. Our approach is based on the Kohn–Vogelius concept and the topological gradient method. We develop a new topological asymptotic expansion, which can be used as a basic step for developing accurate detection algorithms.

Kohn-Vogelius formulation and high-order topological asymptotic formula for identifying small obstacles in a fluid medium

Abstract This paper concerns the identification of a small obstacle immersed in a Stokes flow from boundary measurements. The proposed approach is based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. We derive a high order asymptotic formula describing the variation of a Kohn-Vogelius type functional with respect to the insertion of a small obstacle inside the fluid flow domain. The obtained asymptotic formula will serve as very useful tools for developing accurate and robust numerical reconstruction algorithms.

Topological asymptotic analysis for tumor identification problem

This work is concerned with the problem of identifying the shape, size and location of a small embedded tumor from measured temperature on the skin surface. The temperature distribution in the biological tissue is governed by the Pennes model equation. The proposed approach is based on the Kohn–Vogelius formulation and the topological sensitivity analysis method. The ill-posed geometric inverse problem is reformulated as a topology optimization. The temperature field perturbation, caused by the presence of a small anomaly, is analyzed and estimated. A topological asymptotic formula, describing the variation of the considered Kohn–Vogelius type functional with respect to the presence of a small anomaly is derived.

Shape optimization of turbine blade cooling system using topological sensitivity analysis method

The topological sensitivity analysis method gives the variation of a criterion with respect to the creation of a small hole in the domain. In this paper, we use this method to solve an inverse problem related to the turbine blade cooling. The aim is to optimize the hole characteristics created in the blade vane in order to improve the behavior of the cooling system. A topological optimization algorithm is proposed and some numerical results, showing the efficiency of our approach, are presented.

Topological sensitivity analysis based method for solving a geometry reconstruction problem

This paper is concerned with the topological sensitivity analysis for a parabolic type operator. We consider the unsteady-state heat transfer equation as a model problem and we develop the topological asymptotic expansion valid for a large class of cost functions. The obtained theoretical results are based on a simplified mathematical analysis involving the elliptic part of the operator. In the numerical part, we proposed a simple and fast geometry reconstruction algorithm. The proposed algorithm is applied for recovering internal cooling holes in a turbine blade by minimizing various error functions measuring the difference between the current and actual temperature fields.