Topological Sensitivity Research Articles

Structural topology optimization under constraints on instantaneous failure probability

Accurate prediction of stochastic responses of a structure caused by natural hazards or operations of non-structural components is crucial to achieve an effective design. In this regard, it is of great significance to incorporate the impact of uncertainty into topology optimization of structures under constraints on their stochastic responses. Despite recent technological advances, the theoretical framework remains inadequate to overcome computational challenges of incorporating stochastic responses to topology optimization. Thus, this paper presents a theoretical framework that integrates random vibration theories with topology optimization using a discrete representation of stochastic excitations. This paper also discusses the development of parameter sensitivity of dynamic responses in order to enable the use of efficient gradient-based optimization algorithms. The proposed topology optimization framework and sensitivity method enable efficient topology optimization of structures under stochastic excitations, which is successfully demonstrated by numerical examples of structures under stochastic ground motion excitations.

Far field imaging of a dielectric inclusion

A non-iterative topological sensitivity framework for guaranteed far field detection of a dielectric inclusion is presented. The cases of single and multiple measurements of the electric far field scattering amplitude at a fixed frequency are taken into account. The performance of the algorithm is analyzed theoretically in terms of resolution, stability, and signal-to-noise ratio.

Defect Detection from Multi-frequency Limited Data via Topological Sensitivity

In this work we investigate the reconstruction of sound-hard obstacles buried in a bounded material medium by a non-iterative method based on the computation of topological derivatives. The main purpose is to suitably combine multi-frequency data in a fairly demanding measurement configuration: a very reduced number of aligned emitters and receivers are considered. The performance of the algorithm is illustrated for both single and multiple obstacle reconstructions.

Using the comprehensive patent citation network (CPC) to evaluate patent value

Most approaches to patent citation network analysis are based on single-patent direct citation relation, which is an incomplete understanding of the nature of knowledge flow between patent pairs, which are incapable of objectively evaluating patent value. In this paper, four types of patent citation networks (direct citation, indirect citation, coupling and co-citation networks) are combined, filtered and recomposed based on relational algebra. Then, a method based on comprehensive patent citation (CPC) network for patent value evaluation is proposed, and empirical study of optical disk technology related patents has been conducted based on this method. The empirical study was carried out in two steps: observation of network characteristics over the entire process (citation time lag and topological and graphics characteristics), and measurement verification by independent proxies of patent value (patent family and patent duration). Our results show that the CPC network retains the advantages of patent direct citation, and performs better on topological structure, graphics features, centrality distribution, citation lag and sensitivity than a direct citation network; The verified results by the patent family and maintenance show that the proposed method covers more valuable patents than the traditional method.

A staggered approach to structural shape and topology optimization

AbstractThis contribution is concerned with the design and evaluation of a staggered approach to computational shape and topology optimization. The main idea is to interrelate a classical boundary variation scheme based on the results from shape sensitivity analysis and an evolutionary‐type element removal procedure that relies on the topological sensitivity as a rejection criterion. The key ingredient in this perspective is to consider an advancing front algorithm that is gradually removing, from the evolving boundary of the domain, a number of finite elements based on the evaluation of the topological sensitivity.This process is repeated until the minimum topological sensitivity is no longer encountered at the boundary but in the interior of the domain, such that a topological change is to be considered for the current design layout. However, we first aim to establish a more accurate approximation of the true optimal shape of the newly established (zig‐zag representation) of the design boundary. This is achieved by the evaluation of the shape sensitivity and the accompanying boundary variations obtained by a basic descent algorithm. Only then, we create a hole in the interior of the domain by removing all cells that are adjacent to the nodal point that exhibits the minimum topological sensitivity and resume the advancing front algorithm to alter the newly established design boundary. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the elastic-wave imaging and characterization of fractures with specific stiffness

The concept of topological sensitivity (TS) is extended to enable simultaneous 3D reconstruction of fractures with unknown boundary condition and characterization of their interface by way of elastic waves. Interactions between the two surfaces of a fracture, due to e.g. presence of asperities, fluid, or proppant, are described via the Schoenberg’s linear slip model. The proposed TS sensing platform is formulated in the frequency domain, and entails point-wise interrogation of the subsurface volume by infinitesimal fissures endowed with interfacial stiffness. For completeness, the featured elastic polarization tensor – central to the TS formula – is mathematically described in terms of the shear and normal specific stiffness (κs,κn) of a vanishing fracture. Simulations demonstrate that, irrespective of the contact condition between the faces of a hidden fracture, the TS (used as a waveform imaging tool) is capable of reconstructing its geometry and identifying the normal vector to the fracture surface without iterations. On the basis of such geometrical information, it is further shown via asymptotic analysis – assuming “low frequency” elastic-wave illumination, that by certain choices of (κs,κn) characterizing the trial (infinitesimal) fracture, the ratio between the shear and normal specific stiffness along the surface of a nearly-planar (finite) fracture can be qualitatively identified. This, in turn, provides a valuable insight into the interfacial condition of a fracture at virtually no surcharge – beyond the computational effort required for its imaging. The proposed developments are integrated into a computational platform based on a regularized boundary integral equation (BIE) method for 3D elastodynamics, and illustrated via a set of canonical numerical experiments.

Why the high-frequency inverse scattering by topological sensitivity may work.

This study deciphers the topological sensitivity (TS) as a tool for the reconstruction and characterization of impenetrable anomalies in the high-frequency regime. It is assumed that the anomaly is simply connected and convex, and that the measurements of the scattered field are of the far-field type. In this setting, the formula for TS-which quantifies the perturbation of a cost functional due to a point-like impenetrable scatterer-is expressed as a pair of nested surface integrals: one taken over the boundary of a hidden obstacle, and the other over the measurement surface. Using multipole expansion, the latter integral is reduced to a set of antilinear forms featuring Green's function and its gradient. The remaining expression is distilled by evaluating the scattered field on the surface of an obstacle via Kirchhoff approximation, and pursuing an asymptotic expansion of the resulting Fourier integral. In this way, the TS is found to survive upon three asymptotic lynchpins, namely (i) the near-boundary approximation for sampling points close to the 'exposed' surface of an obstacle; (ii) uniform expansions synthesizing the diffraction catastrophes for sampling points near caustic surfaces, lines and points; and (iii) stationary phase approximation. Within the framework of catastrophe theory, it is shown that, in the case of the full source aperture, the TS is asymptotically dominated by the (explicit) near-boundary term-which explains the previously reported reconstruction capabilities of this class of indicator functionals. The analysis further shows that, when the (Dirichlet or Neumann) character of an anomaly is unknown beforehand, the latter can be effectively exposed by assuming point-like Dirichlet perturbation and considering the sign of the leading term inside the reconstruction.

Topological gradient for a fourth order operator used in image analysis

This paper is concerned with the computation of the topological gradient associated to a fourth order Kirchhoff type partial differential equation and to a second order cost function. This computation is motivated by fine structure detection in image analysis. The study of the topological sensitivity is performed both in the cases of a circular inclusion and a crack.

Topological derivative-based topology optimization of structures subject to multiple load-cases

The topological derivative measures the sensitivity of a shape functional with respect to an infinitesimal singular domain perturbation, such as the insertion of holes, inclusions or source-terms. The topological derivative has been successfully applied in obtaining the optimal topology for a large class of physics and engineering problems. In this paper the topological derivative is applied in the context of topology optimization of structures subject to multiple load-cases. In particular, the structural compliance under plane stress or plane strain assumptions is minimized under volume constraint. For the sake of completeness, the topological asymptotic analysis of the total potential energy with respect to the nucleation of a small circular inclusion is developed in all details. Since we are dealing with multiple load-cases, a multi-objective optimization problem is proposed and the topological sensitivity is obtained as a sum of the topological derivatives associated with each load-case. The volume constraint is imposed through the Augmented Lagrangian Method. The obtained result is used to devise a topology optimization algorithm based on the topological derivative together with a level-set domain representation method. Finally, several finite element-based examples of structural optimization are presented.

Inverse acoustic scattering by solid obstacles: topological sensitivity and its preliminary application

A to pological sensitivity approach is developed for the imaging of penetrable solid obstacles embedded in a fluid background medium by way of inverse acoustic scattering. To this end, an asymptotic form for the scattered field caused by a nucleating spherical solid obstacle in the reference fluid medium is derived within the boundary integral equation framework, where the required limiting behaviour of the acceleration field inside the perturbation is established based on solutions of two simplified fluid–solid interaction problems. With this result, the equivalence of acoustic scattering by fluid and solid scatterers in terms of asymptotic behaviour is observed and numerically validated. The direct and adjoint-field topological sensitivity expressions for transient and time-harmonic excitations are obtained accordingly. The utility of the proposed method as a tool for preliminary obstacle reconstruction and bulk modulus characterization is illustrated through numerical examples and an exploratory experimental study. On the basis of adjusted topological sensitivity formulas for fluid reference domains containing pre-existing solid inhomogeneities, an iterative 3D obstacle reconstruction approach is established and its performance with synthetic data is demonstrated.

Design of bi-metallic devices based on the topological derivative concept

In this work the topological derivative concept is applied in the context of topology design of thermo-mechanical devices, where the linear elasticity system (modeled by the Navier equation) is coupled with the steady-state heat conduction problem (modeled by the Laplace equation). The mechanical coupling term comes out from the thermal stress induced by the temperature field. We consider the topology design of bi-metallic devices. The idea is to maximize the displacement in a given direction defined on the boundary of the thermo-elastic body with respect to a bi-metallic material distribution. The topological derivative is obtained by considering the nucleation of a small circular inclusion with different thermal expansion coefficients. A level-set domain representation method is used, together with the derived topological sensitivity formula, to devise a topology design algorithm. Finally, some numerical experiments regarding the conceptual design of thermo-mechanical devices are presented.

A staggered approach to shape and topology optimization using the traction method and an evolutionary-type advancing front algorithm

Abstract We investigate a novel approach for structural shape optimization on the basis of complementary shape and topological sensitivity analysis. As in early approaches to shape optimization, the domain variation is specified by modification of boundary nodal points, hence leading to an updated Lagrangian description for the course of optimization. To overcome the formation of oscillating boundaries in the optimal design trials, we employ the traction method to establish smooth descent directions for shape variation. Therein, an auxiliary elastic problem is solved in which the shape sensitivity constitutes the external loading and the deformation of the auxiliary elastic body is used as a descent direction for shape variation rather than the shape sensitivity itself. We complement this method through an evolutionary-type element removal procedure that is based on the topological sensitivity such that an advancing front algorithm is gradually removing elements from the design boundary of the domain. Once the minimum topological sensitivity is no longer encountered at the design boundary, we create a hole in the domain, again using the topological sensitivity to specify its exact location, and resume the element removal procedure for the newly established design boundary. Since this approach yields only a vague estimate of the true optimal shape of newly established holes, the traction method is again used for shape variation of the extended design boundary.

Topological sensitivity analysis for systems biology.

Mathematical models of natural systems are abstractions of much more complicated processes. Developing informative and realistic models of such systems typically involves suitable statistical inference methods, domain expertise, and a modicum of luck. Except for cases where physical principles provide sufficient guidance, it will also be generally possible to come up with a large number of potential models that are compatible with a given natural system and any finite amount of data generated from experiments on that system. Here we develop a computational framework to systematically evaluate potentially vast sets of candidate differential equation models in light of experimental and prior knowledge about biological systems. This topological sensitivity analysis enables us to evaluate quantitatively the dependence of model inferences and predictions on the assumed model structures. Failure to consider the impact of structural uncertainty introduces biases into the analysis and potentially gives rise to misleading conclusions.

Control of crack propagation by shape-topological optimization

An elastic body weakened by small cracks is considered in the framework of unilateral variational problems in linearized elasticity. The frictionless contact conditions are prescribed on the crack lips in two spatial dimensions, or on the crack faces in three spatial dimensions. The weak solutions of the equilibrium boundary value problem for the elasticity problem are determined by minimization of the energy functional over the cone of admissible displacements. The associated elastic energy functional evaluated for the weak solutions is considered for the purpose of control of crack propagation. The singularities of the elastic displacement field at the crack front are characterized by the shape derivatives of the elastic energy with respect to the crack shape within the Griffith theory. The first order shape derivative of the elastic energy functional with respect to the crack shape, i.e., evaluated for a deformation field supported in an open neighbourhood of one of crack tips, is called the Griffith functional. &nbsp The control of the crack front in the elastic body is performed by the optimum shape design technique. The Griffith functional is minimized with respect to the shape and the location of small inclusions in the body. The inclusions are located far from the crack. In order to minimize the Griffith functional over an admissible family of inclusions, the second order directional, mixed shape-topological derivatives of the elastic energy functional are evaluated. &nbsp The domain decomposition technique [42] is applied to the shape [56] and topological [54,55] sensitivity analysis of variational inequalities. &nbsp The nonlinear crack model in the framework of linear elasticity is considered in two and three spatial dimensions. The boundary value problem for the elastic displacement field takes the form of a variational inequality over the positive cone in a fractional Sobolev space. The variational inequality leads to a problem of metric projection over a polyhedric convex cone, so the concept of conical differentiability applies to shape and topological sensitivity analysis of variational inequalities under consideration.

Multi-constrained topology optimization via the topological sensitivity

The objective of this paper is to introduce and demonstrate a robust method for multi-constrained topology optimization. The method is derived by combining the topological sensitivity with the classic augmented Lagrangian formulation. The primary advantages of the proposed method are: (1) it rests on well-established augmented Lagrangian formulation for constrained optimization, (2) the augmented topological level-set can be derived systematically for an arbitrary set of loads and constraints, and (3) the level-set can be updated efficiently. The method is illustrated through numerical experiments.

Topological sensitivity analysis for the modified Helmholtz equation under an impedance condition on the boundary of a hole

The topological sensitivity analysis consists in providing an asymptotic expansion of a shape functional with respect to emerging of small holes in the interior of the domain occupied by the body. In this paper, such an expansion is obtained for the modified Helmholtz equation with an impedance condition prescribed on the boundary of a hole. The topological derivative is then used for numerical simulations for an inverse problem.

Topological gradient for fourth-order PDE and application to the detection of fine structures in 2D images

In this paper we describe a new approach for the detection of fine structures in an image. This approach is based on the computation of the topological gradient associated with a cost function defined from a regularization of the data (possibly noisy). We get this approximation by solving a fourth-order PDE. The study of the topological sensitivity is made in the cases of both a circular inclusion and a crack. We illustrate our approach by giving two experimental results.

Topology design of plates considering different volume control methods

Purpose – The purpose of this paper is to compare between two methods of volume control in the context of topological derivative-based structural optimization of Kirchhoff plates. Design/methodology/approach – The compliance topology optimization of Kirchhoff plates subjected to volume constraint is considered. In order to impose the volume constraint, two methods are presented. The first one is done by means of a linear penalization method. In this case, the penalty parameter is the coefficient of a linear term used to control the amount of material to be removed. The second approach is based on the Augmented Lagrangian method which has both, linear and quadratic terms. The coefficient of the quadratic part controls the Lagrange multiplier update of the linear part. The associated topological sensitivity is used to devise a structural design algorithm based on the topological derivative and a level-set domain representation method. Finally, some numerical experiments are presented allowing for a comparative analysis between the two methods of volume control from a qualitative point of view. Findings – The linear penalization method does not provide direct control over the required volume fraction. In contrast, through the Augmented Lagrangian method it is possible to specify the final amount of material in the optimized structure. Originality/value – A strictly simple topology design algorithm is devised and used in the context of compliance structural optimization of Kirchhoff plates under volume constraint. The proposed computational framework is quite general and can be applied in different engineering problems.

Piezoresistive device optimization using topological derivative concepts

A piezoresistive sensor is composed of a piezoresistive membrane attached to a flexible plate. The piezoresistive material is anisotropic, and its electrical properties change when subjected to mechanical stresses. In this work, the topology design of a piezoresistive pressure sensor is addressed. More specifically, an optimization technique based on topological sensitivity analysis is proposed in order to obtain the optimized distribution of piezoresistive material over the plate. In most of the works regarding topological sensitivity analysis, isotropic materials are considered. However, the problem of conductivity in an anisotropic non-homogeneous domain has been recently addressed, and a closed form for the topological derivative associated to the energy shape functional has been presented. In this work, on the other hand, a closed form for the topological derivative associated with a multi-objective shape functional related to the steady-state anisotropic current density diffusion problem is presented. To illustrate the applicability of the closed formula and the proposed optimization procedure, numerical examples regarding the conceptual design of piezoresistive sensors, considering distinct optimization parameters and boundary conditions in the conductivity problem, are presented.

2214 境界要素法を用いた二次元動弾性問題におけるトポロジー感度解析について(トポロジー最適化)

We have been investigating topology optimisation methods with the boundary element method (BEM) and the level set method for wave problems. In this paper, we propose a method for topological sensitivity analysis, which is essential to the topology optimisation method, for two-dimensional elastic wave problems with the BEM. The topological derivative is derived in a regorous manner with the help of the Airy stress function.