Topological Derivative Concept Research Articles

Full-waveform reconstruction of micro-seismic events via topological derivative approach

Micro-seismic events, naturally occurring within geological formations and quasi-brittle engineered systems, provide a powerful window into the evolving processes of material degradation and failure. Accurate characterization of these events is critical for a comprehensive understanding of the underpinning fracturing mechanisms and potential implications. In this work, we present an algorithm for the spatial reconstruction and characterization of micro-seismic events in a three-dimensional bounded elastic body (with known geometry and nominal material properties) via combined source location and moment tensor inversion. Assuming availability of the full-waveform “acoustic emission” traces whose spectral content can be exposed via Fourier transform, the inverse solution is based on (i) a time-harmonic (forward) elastodynamic model and (ii) the concept of topological derivative as a framework for robust event reconstruction. On exploiting an equivalence between the elastic wavefield generated by the creation of a new micro-surface and that stemming from a suitable set of dipoles and double-couples (whose strengths are synthesized via the seismic moment tensor), we formulate the inverse problem as that for the real (in-phase) and imaginary (out-of-phase) components of the moment tensor at trial grid locations. In this way the optimal solution is obtained via a combinatorial search over a prescribed grid, inherently allowing for successive refinements of event reconstruction over the region(s) of interest. The analysis is illustrated by numerical experiments highlighting the key features of the inversion scheme including the reconstruction of multiple (i.e. contemporaneous) events, localization of the “off-grid” micro-seismic events, and the ability to handle noisy data. The results in particular highlight the utility of multi-frequency event reconstruction toward reducing the demand on the number of sensing locations.

Efficient and Accurate Separable Models for Discretized Material Optimization: A Continuous Perspective Based on Topological Derivatives

Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman–Morrison–Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two—at a first sight—very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline stage. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.

Topology optimization of a rotating electric machine by the topological derivative

AbstractWe consider the topology optimization of a rotating electric machine in magnetoquasistatic operation in two space dimensions. This amounts to a topology optimization problem subject to a parabolic PDE constraint on a moving domain which we intend to solve by means of the topological derivative concept. For that purpose, we consider a topological perturbation of the materials in the space‐time cylinder along a trajectory given by the rotation of the machine. Using a Lagrangian approach, we derive the topological derivative formula, which depends on the solution of an exterior problem that is bounded in time and unbounded in the space directions. We use this sensitivity information for the design optimization of an electric motor by means of a level set method.

Level set-based topology optimization for the design of labyrinthine acoustic metamaterials

We present a topology optimization method for the design of labyrinthine metamaterials with a large path length for acoustic wave propagation. An objective function maximizes the pressure gradient in an air-filled region at a low frequency, and a level set-based optimization method enables the optimal design of the labyrinthine structure with high computational efficiency owing to the concept of topological derivative. First, we explain the design settings of labyrinthine metamaterials. Next, the objective functions to obtain labyrinthine structures are introduced, and the corresponding topological derivatives are obtained. The optimization problem is formulated in the framework of the level set topology optimization method, and two-dimensional numerical examples are provided to demonstrate the validity of the proposed method. Based on the frequency responses of the effective parameters and dispersion analysis, the optimized designs are characterized by a negative refractive index over a wide frequency interval, which can be controlled by a parameter introduced in the objective function.

Velocity Field Level Set Method Incorporating Topological Derivatives for Topology Optimization

Abstract The velocity field level set method constructs the velocity field by velocity design variables and basis functions, and thus facilitates the use of general optimizers while still retaining the level set-based implicit topological representation. This paper incorporates the topological derivative concept into the velocity field level set method to enable automatic nucleation of interior holes. In each design iteration, a specified volume fraction of new holes is inserted at locations with smaller values of topological derivatives. Thus, the method provides a way to directly change the structural topology during the boundary evolution using the velocity field based on the shape sensitivity. Compared with the original velocity field level set method, the current implementation can further accelerate the topological and shape evolution during the optimization process. More importantly, the capability of hole nucleation eliminates the need of prescribing initial holes and thus alleviates the dependency of the optimized design on the initial design. Several numerical examples in both 2D and 3D design domains are presented to demonstrate the validity and efficiency of the proposed method.

Topological derivative-based topology optimization of incompressible structures using mixed formulations

In this work an algorithm for topological optimization, based on the topological derivative concept, is proposed for both nearly and fully incompressible materials. In order to deal with such materials, a new decomposition of the Polarization tensor is proposed in terms of its deviatoric and volumetric components. Mixed formulations applied in the context of linear elasticity do not only allow to deal with incompressible material behavior but also to obtain a higher accuracy in the computation of stresses. The system is stabilized by means of the Variational Multiscale method based on the decomposition of the unknowns into resolvable and subgrid scales in order to prevent fluctuations. Several numerical examples are presented and discussed to assess the robustness of the proposed formulation and its applicability to Topology Optimization problems for incompressible elastic solids.

Inverse homogenization using the topological derivative

PurposeThe purpose of this study is to solve the inverse homogenization problem, or so-called material design problem, using the topological derivative concept.Design/methodology/approachThe 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.FindingsThe 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/valueThe 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.

Imaging of small penetrable obstacles based on the topological derivative method

PurposeThe purpose of this paper is to present topological derivatives-based reconstruction algorithms to solve an inverse scattering problem for penetrable obstacles.Design/methodology/approachThe 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.FindingsIn 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/valueThe 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.

Topology optimization of connections in mechanical systems

One of the issues for the automotive industry is weight reduction. For this purpose, topology optimization is used for mechanical parts and usually involves a single part. Its connections to other parts are assumed to be fixed. This paper deals with a coupled topology optimization of both the structure of a part and its connections (location and number) to other parts. The present work focuses on two models of connections, namely rigid support and spring that prepares work for bolt connection. Rigid supports are modeled by Dirichlet boundary conditions while bolt-like connections are idealized and simplified as a non-local interaction to be representative enough at a low computational cost. On the other hand, the structure is modeled by the linearized elasticity system and its topology is represented by a level set function. A coupled optimization of the structure and the location of rigid supports is performed to minimize the volume of an engine accessories bracket under a compliance constraint. This coupled topology optimization (shape and connections) provides more satisfactory performance of a part than the one given by classical shape optimization alone. The approach presented in this work is therefore one step closer to the optimization of assembled mechanical systems. Thereafter, the concept of topological derivative is adapted to create an idealized bolt. The main idea is to add a small idealized bolt at the best location and to test the optimality of the solution with this new connection. The topological derivative is tested with a 3d academic test case for a problem of compliance minimization.

Hydro-mechanical fracture modeling governed by topological derivatives

The process of fluid-driven crack propagation in permeable rocks is investigated by a simple hydro-mechanical model and the concept of topological derivative. Analytical and numerical computations are made to propose a promising model tested and validated on a series of bidimensional benchmark examples. The main guidelines for this model are the use of simple finite elements with a minimal number of user-defined algorithmic parameters.

Optimization of the electromagnetic scattering problem based on the topological derivative method.

A new optimization method based on the topological derivative concept is developed for the electromagnetic design problem. Essentially, the purpose of the topological derivative method is to measure the sensitivity of a given shape functional with respect to a singular domain perturbation, so that it has applications in many relevant fields such as shape and topology optimization for imaging processing, inverse problems, and design of metamaterials. The topological derivative is rigorously derived for the electromagnetic scattering problem and used as gradient descent direction to find local optima for the design of electromagnetic devices. We demonstrate that the resulting topology design algorithm is remarkably simple and efficient and naturally leads to binary designs, while depending only on the solution of the conventional finite element formulation for electrodynamics. Finally, several numerical experiments in two and three spatial dimensions are presented to illustrate the performance of the proposed formulation.

Analysis of topological derivative as a tool for qualitative identification

The concept of topological derivative has proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. Although for the most part, this approach remains based on a heuristic interpretation of the topological derivative, a first attempt toward its mathematical justification was done in Bellis et al (2013 Inverse Problems 29 075012) for the case of isotropic media with far field data and inhomogeneous refraction index. Our paper extends the analysis there to the case of anisotropic scatterers and background with near field data. Topological derivative-based imaging functional is analyzed using a suitable factorization of the near fields, which became achievable thanks to a new volume integral formulation recently obtained in Bonnet (2017 J. Integral Equ. Appl. 29 271–95). Our results include justification of sign heuristics for the topological derivative in the isotropic case with jump in the main operator and for some cases of anisotropic media, as well as verifying its decaying property in the isotropic case with near field spherical measurements configuration situated far enough from the probing region.

SIMP‐ALL: A generalized SIMP method based on the topological derivative concept

SummaryTopology optimization has emerged in the last years as a promising research field with a wide range of applications. One of the most successful approaches, the SIMP method, is based on regularizing the problem and proposing a penalization interpolation function. In this work, we propose an alternative interpolation function, the SIMP‐ALL method that is based on the topological derivative concept. First, we show the strong relation in plane linear elasticity between the Hashin‐Shtrikman (H‐S) bounds and the topological derivative, providing a new interpretation of the last one. Then, we show that the SIMP‐ALL interpolation remains always in between the H‐S bounds regardless the materials to be interpolated. This result allows us to interpret intermediate values as real microstructures. Finally, we verify numerically this result and we show the convenience of the proposed SIMP‐ALL interpolation for obtaining auto‐penalized optimal design in a wider range of cases. A MATLAB code of the SIMP‐ALL interpolation function is also provided.

A noniterative reconstruction method for the inverse potential problem with partial boundary measurements

In this paper, a noniterative reconstruction method for solving the inverse potential problem is proposed. The forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial or total boundary measurements of the associated potential. Since the inverse problem is written in the form of an ill‐posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional measuring the misfit between the solution obtained from the model and the data taken from the boundary measurements is minimized with respect to a set of ball‐shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting truncated expansion is trivially minimized with respect to the parameters under consideration that leads to a noniterative second order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to the noisy data and independent of any initial guess. Finally, some numerical experiments are presented showing the capability of the proposed method in reconstructing multiple anomalies of different sizes and shapes by taking into account complete or partial boundary measurements.

Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains

Mathematical analysis and numerical solutions of problems with unknown shapes or geometrical domains is a challenging and rich research field in the modern theory of the calculus of variations, partial differential equations, differential geometry as well as in numerical analysis. In this series of three review papers, we describe some aspects of numerical solution for problems with unknown shapes, which use tools of asymptotic analysis with respect to small defects or imperfections to obtain sensitivity of shape functionals. In classical numerical shape optimization, the boundary variation technique is used with a view to applying the gradient or Newton-type algorithms. Shape sensitivity analysis is performed by using the velocity method. In general, the continuous shape gradient and the symmetric part of the shape Hessian are discretized. Such an approach leads to local solutions, which satisfy the necessary optimality conditions in a class of domains defined in fact by the initial guess. A more general framework of shape sensitivity analysis is required when solving topology optimization problems. A possible approach is asymptotic analysis in singularly perturbed geometrical domains. In such a framework, approximations of solutions to boundary value problems (BVPs) in domains with small defects or imperfections are constructed, for instance by the method of matched asymptotic expansions. The approximate solutions are employed to evaluate shape functionals, and as a result topological derivatives of functionals are obtained. In particular, the topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, defects, source terms and cracks. This new concept of variation has applications in many related fields, such as shape and topology optimization, inverse problems, image processing, multiscale material design and mechanical modeling involving damage and fracture evolution phenomena. In the first part of this review, the topological derivative concept is presented in detail within the framework of the domain decomposition technique. Such an approach is constructive, for example, for coupled models in multiphysics as well as for contact problems in elasticity. In the second and third parts, we describe the first- and second-order numerical methods of shape and topology optimization for elliptic BVPs, together with a portfolio of applications and numerical examples in all the above-mentioned areas.

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

The framework of topological sensitivity analysis in singularly perturbed geometrical domains, presented in the first part of this series of review papers, allows the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, source terms and cracks. This new concept in shape sensitivity analysis generalizes the shape derivatives from the domain boundary to its interior for admissible domains in two and three spatial dimensions. Therefore, the concept of topological derivative is a powerful tool for solving shape–topology optimization problems. There are now applications of topological derivative in many different fields of engineering and physics, such as shape and topology optimization in structural mechanics, inverse problems for partial differential equations, image processing, multiscale material design and mechanical modeling including damage and fracture evolution phenomena. In this second part of the review, a topology optimization algorithm based on first-order topological derivatives is presented. The appropriate level-set domain representation method is employed within the iterations in order to design an optimal shape–topology local solution. The algorithm is successfully used for numerical solution of a wide class of shape–topology optimization problems.

Large-scale stochastic topology optimization using adaptive mesh refinement and coarsening through a two-level parallelization scheme

Topology optimization under uncertainty of large-scale continuum structures is a computational challenge due to the combination of large finite element models and uncertainty propagation methods. The former aims to address the ever-increasing complexity of more and more realistic models, whereas the latter is required to estimate the statistical metrics of the formulation. In this work, the computational burden of the problem is addressed using a sparse grid stochastic collocation method, to calculate the statistical metrics of the topology optimization under uncertainty formulation, and a parallel adaptive mesh refinement method, to efficiently solve each of the stochastic collocation nodes. A two-level parallel processing scheme (TOUU-PS2) is proposed to profit from parallel computation on distributed memory systems: the stochastic nodes are distributed through the distributed memory system, and the efficient computation of each stochastic node is performed partitioning the problem using a domain decomposition strategy and solving each subdomain using an adaptive mesh refinement method. A dynamic load-balancing strategy is used to balance the workload between subdomains, and thus increasing the parallel performance by reducing processor idle time. The topology optimization problem is addressed using the topological derivative concept in combination with a level-set method. The performance and scalability of the proposed methodology are evaluated using several numerical benchmarks and real-world applications, showing good performance and scalability up to thousands of processors.

Crack Growth Control Based on the Topological Derivative of the Rice’s Integral

In fracture mechanics, an important question concerns the useful life of mechanical components. Such components are, usually, submitted to the actions of external forces and/or degrading agents which can trigger the crack nucleation and propagation process. In particular, when a mechanical component is already partially cracked, the question is how to extend its remaining useful life. In this work, a simple and efficient methodology aiming to extend the remaining useful life of cracked elastic bodies is proposed. More precisely, we want to find a way to retard or even avoid the triggering of the crack propagation process by nucleating hard and/or soft inclusions far from the crack tip. The main idea consists in minimize a shape functional based on the Rice’s integral with respect to the nucleation of inclusions by using the concept of topological derivative. The obtained sensitivity, which corroborates with the famous Eshelby theorem, is used to indicate the regions where the controls have to be inserted. According to the Griffith’s energy criterion, this simple procedure allows for increasing the remaining useful life of the cracked body. Finally, some numerical experiments are presented showing the applicability of the proposed methodology.

Topology optimization of magnetic shielding using level-set function combined with element-based topological derivatives

PurposeShape optimization using the level-set method is one of the most effective automatic design tools for electromagnetic machines. While level-set method has the advantage of being able to suppress unfeasible shape, it has a weakness of being unable to handle complex topology changes such as perforate at material region. With this method, it is only possible to define simple connected topology, and it is difficult to determine the optimal shape which has holes. Therefore, it is important to efficiently expand the searching area in the optimization process with level-set method.Design/methodology/approachIn this paper, the authors introduce the newly defined hole sensitivity which is based on concept of topological derivatives, and combine it with level-set method to effectively create holes in the search process. Furthermore, they consider a variable bandwidth of gray scale, which indicates the transition width between air and magnetic body and combine it with the hole creation method described above. With these methods, the authors aim to expand the searching area in comparison with the conventional level-set method.FindingsAs a result of applying the proposed methods to a magnetic shielding problem, the multi-layered shielding which effectively reduces the magnetic flux in the target area, is successfully produced.Originality/valueThe proposed methods enable us to effectively create a hole and to expand the searching area in the topology optimization process unlike in the case of conventional level-set method.

Noniterative Reconstruction Method for an Inverse Potential Problem Modeled by a Modified Helmholtz Equation

ABSTRACTThis paper deals with an inverse potential problem posed in two dimensional space whose forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial measurements of the associated potential. Since the inverse problem, we are dealing with, is written in the form of an ill-posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional is defined to measure the misfit of the solution obtained from the model and the data taken from the partial measurements. This shape functional is minimized with respect to a set of ball-shaped anomalies using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration which leads to a noniterative second-order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to noisy data and independent of any initial guess. Finally, some numerical experiments are presented to show the effectiveness of the proposed reconstruction algorithm.