Topological Derivative Concept Research Articles

A simplified model of fracking based on the topological derivative concept

This paper deals with a simplified hydraulic fracture model based on the concept of topological derivatives. It means that we consider a two dimensional idealization in which the rock is assumed to be impermeable, while the fracturing process is activated by a given pressure acting within the existing geological faults. The basic idea consists in adapting the Francfort–Marigo damage model to the context of hydraulic fracture. The Francfort–Marigo damage model is a variational approach to describe the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damaged regions under an irreversibility constraint. In our model problem, the loading comes out from a pressurized damaged region embedded into the rock, which is used to trigger the hydraulic fracturing process. In particular, a shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipation energy term is minimized with respect to a set of ball-shaped pressurized inclusions by using the topological derivative concept. Thus, the topological asymptotic expansion of the shape functional with respect to the nucleation of a circular inclusion endowed with non-homogeneous transmission condition on its boundary is obtained. The associated topological derivative, which corroborates with the famous Eshelby theorem, is used to devise a simple topology optimization algorithm specifically designed to simulate the whole nucleation and propagation process of hydraulic fracturing. To assess our model, some numerical examples are presented, showing typical features of hydraulic fracture phenomenon, including the characterization of the fault-activation pressure and specific crack path growth, allowing for kinking and bifurcations.

Reliability‐based topology optimization of structures under stress constraints

SummaryIn this paper, we propose an approach for reliability‐based design optimization where a structure of minimum weight subject to reliability constraints on the effective stresses is sought. The reliability‐based topology optimization problem is formulated by using the performance measure approach, and the sequential optimization and reliability assessment method is employed. This strategy allows for decoupling the reliability‐based topology optimization problem into 2 steps, namely, deterministic topology optimization and reliability analysis. In particular, the deterministic structural optimization problem subject to stress constraints is addressed with an efficient methodology based on the topological derivative concept together with a level‐set domain representation method. The resulting algorithm is applied to some benchmark problems, showing the effectiveness of the proposed approach.

Inverse acoustic scattering using high-order small-inclusion expansion of misfit function

This article concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems involving the identification of penetrable obstacles, whereby the featured data-misfit cost function \begin{document} $\mathbb{J}$ \end{document} is expanded in powers of the characteristic radius \begin{document} $a$ \end{document} of a single small inhomogeneity. The \begin{document} $O(a^6)$ \end{document} approximation \begin{document} $\mathbb{J}_6$ \end{document} of \begin{document} $\mathbb{J}$ \end{document} is derived and justified for a single obstacle of given location, shape and material properties embedded in a 3D acoustic medium of arbitrary shape. The generalization of \begin{document} $\mathbb{J}_6$ \end{document} to multiple small obstacles is outlined. Simpler and more explicit expressions of \begin{document} $\mathbb{J}_6$ \end{document} are obtained when the scatterer is centrally-symmetric or spherical. An approximate and computationally light global search procedure, where the location and size of the unknown object are estimated by minimizing \begin{document} $\mathbb{J}_6$ \end{document} over a search grid, is proposed and demonstrated on numerical experiments, where the identification from known acoustic pressure on the surface of a penetrable scatterer embedded in a acoustic semi-infinite medium, and whose shape may differ from that of the trial obstacle assumed in the expansion of \begin{document} $\mathbb{J}$ \end{document} , is considered.

Higher order topological derivatives for three-dimensional anisotropic elasticity

This article concerns an extension of the topological derivative concept for 3D elasticity problems involving elastic inhomogeneities, whereby an objective function 𝕁 is expanded in powers of the characteristic size a of a single small inhomogeneity. The O (a 6 ) approximation of 𝕁 is derived and justified for an inhomogeneity of given location, shape and elastic properties embedded in a 3D solid of arbitrary shape and elastic properties; the background and the inhomogeneity materials may both be anisotropic. The generalization to multiple small inhomogeneities is concisely described. Computational issues, and examples of objective functions commonly used in solid mechanics, are discussed.

Design optimization of laminar flow machine rotors based on the topological derivative concept

Flow machines are very important to industry, being widely used on various processes. Thus, performance improvements are relevant and can be achieved by using topology optimization methods. In particular, this work aims to develop a topological derivative formulation to design radial flow machine rotors by considering laminar flow. Based on the concept of traditional topology optimization approaches, in the adopted topological derivative formulation, solid or fluid material is distributed at each point of the domain. This is achieved by combining Navier–Stokes equations on a rotary referential with Darcy’s law equations. This strategy allows for working in a fixed computational domain, which leads to a topology design algorithm of remarkably simple computational implementation. In the optimization problem formulation, a multi-objective function is defined, aiming to minimize the energy dissipation, vorticity and power considering a volume constraint. The constrained optimization problem is rewritten in the form of an unconstrained optimization problem by using the Augmented Lagrangian formalism. The resulting multi-objective shape functional is then minimized with help of the topological derivative concept. In the context of this article, the topological derivative represents the exact sensitivity with respect to the nucleation of an inclusion within the design domain and the obtained analytical (closed) formula can be evaluated through a simple post processing of the solutions to the direct and adjoints problems. Both mentioned features allow for obtaining the optimized designs in few iterations by using a minimal number of user defined algorithm parameters. All equations and the derived continuous adjoint equations are solved through finite element method. As a result, two-dimensional designs of flow machine rotors are obtained by using this methodology. Their performance is analyzed by evaluating velocity and pressure distributions inside rotor.

Topological derivative-based fracture modelling in brittle materials: A phenomenological approach

The Griffith-Francfort-Marigo damage model describes the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damage regions. It is based on the minimization of a shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipated energy, with respect to the distribution of the healthy and damaged phases, under an irreversibility constraint. A natural approach to deal with such a minimization problem consists in considering the topological derivative concept to nucleate small damaged regions and the shape gradient to propagate them. In contrast to such an approach, in this paper the Griffith-Francfort-Marigo damage model is revisited by using the sole tool of topological derivative. In particular, we propose a striking simple numerical scheme based on the computation of the topological derivative field to determine damage nucleation as well as crack/damage propagation. In other words, the topological derivative is used as descent direction to minimize the Francfort-Marigo functional indicating, in each iteration, the regions that have to be damaged. Therefore, the proposed topology optimization algorithm is able to capture the whole nucleation and propagation damaging process, including important features like kinking and bifurcations. These properties are confirmed through several numerical experiments and by comparison with available laboratory experiments.

Structural weight minimization under stress constraints and multiple loading

In this paper we present an approach for structural weight minimization under von Mises stress constraints and multiple load-cases. The minimization problem is solved by using the topological derivative concept, which allows the development of efficient and robust topology optimization algorithms. Since we are dealing with multiple loading, the resulting sensitivity is obtained as a sum of the topological derivatives associated with each load-case. The derived result is used together with a level-set domain representation method to devise a topology design algorithm. Several numerical examples are presented showing the effectiveness of the proposed approach.

A new non-iterative reconstruction method for the electrical impedance tomography problem

The electrical impedance tomography (EIT) problem consists in determining the distribution of the electrical conductivity of a medium subject to a set of current fluxes, from measurements of the corresponding electrical potentials on its boundary. EIT is probably the most studied inverse problem since the fundamental works by Calderón from the 1980s. It has many relevant applications in medicine (detection of tumors), geophysics (localization of mineral deposits) and engineering (detection of corrosion in structures). In this work, we are interested in reconstructing a number of anomalies with different electrical conductivity from the background. Since the EIT problem is written in the form of an overdetermined boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional measuring the misfit between the boundary measurements and the electrical potentials obtained from the model is minimized with respect to a set of ball-shaped anomalies by using the concept of topological derivatives. It means that the objective functional is expanded and then truncated up to the second order term, leading to a quadratic and strictly convex form with respect to the parameters under consideration. Thus, a trivial optimization step leads to a non-iterative 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, in order to show the effectiveness of the devised reconstruction algorithm, some numerical experiments into two spatial dimensions are presented, taking into account total and partial boundary measurements.

Topological derivative-based topology optimization of structures subject to design-dependent hydrostatic pressure loading

In this paper the topological derivative concept is applied in the context of compliance topology optimization of structures subject to design-dependent hydrostatic pressure loading under volume constraint. The topological derivative represents the first term of the asymptotic expansion of a given shape functional with respect to the small parameter which measures the size of singular domain perturbations, such as holes, inclusions, source-terms and cracks. In particular, the topological asymptotic expansion of the total potential energy associated with plane stress or plane strain linear elasticity, taking into account the nucleation of a circular inclusion with non-homogeneous transmission condition on its boundary, is rigorously developed. Physically, there is a hydrostatic pressure acting on the interface of the topological perturbation, allowing to naturally deal with loading-dependent structural topology optimization. The obtained result is used in a topology optimization algorithm based on the associated topological derivative together with a level-set domain representation method. Finally, some numerical examples are presented, showing the influence of the hydrostatic pressure on the topology of the structure.

Obstacles reconstruction from partial boundary measurements based on the topological derivative concept

In this work a new method for obstacles reconstruction from partial boundary measurements is proposed. For a given boundary excitation, we want to determine the quantity, locations and sizes of a number of holes embedded within a geometrical domain, from partial boundary measurements related to such an excitation. The resulting inverse problem is written in the form of an ill-posed and over-determined boundary value problem. The idea therefore is to rewrite it as an optimization problem where a shape functional measuring the misfit between the boundary measurement and the solution to an auxiliary boundary value problem is minimized with respect to a set of ball-shaped holes. The topological derivative concept is used for solving the associated topology optimization problem, leading to a second-order reconstruction algorithm. The resulting algorithm is non-iterative --- and thus very robust with respect to noisy data --- and also free of initial guess. Finally, some numerical results are presented in order to demonstrate the effectiveness of the proposed reconstruction algorithm.

A new method for topology design of electromagnetic antennas in hyperthermia therapy

The topological derivative concept has been proved to be useful in many relevant applications such as topology optimization, inverse problems, image processing, multi-scale constitutive modeling, fracture mechanics and damage evolution modeling. In this work, we develop a new optimization method based on the topological derivative concept applied to the cancer treatment by hyperthermia. Hyperthermia therapy is a non-invasive medical treatment in which body tissue is artificially heated through electromagnetic waves, focusing the heat in cancerous cells undergoing apoptosis. The basic idea, therefore, consists in finding a distribution of heat source generated by electromagnetic antenna aiming to increase the temperature in the region occupied by the tumor, while keeping the temperature in the remainder part of the body. Numerical results are presented illustrating possible application of the proposed methodology to treatment of cancer by hyperthermia.

Level set-based topology optimization for the design of acoustic metamaterial using two-phase material model

Acoustic metamaterials are artificial materials composed of periodic structures made of elastic and fluid material. They could be used for noise reduction problems and efficient sound propagation control by their unusual properties, such as negative bulk modulus, mass density, and refractive index. The performance of acoustic metamaterial is strongly affected by its designs rather than its material properties. Therefore, topology optimization method could be used to obtain a desired property of acoustic metamaterial. To model acoustic-elastic coupled system, we introduce a two-phase material model where the solid and fluid phases are mixed. While coupling boundary conditions are required in a usual acoustic-elastic coupled analysis, this model allows us to analyze the system without imposing such boundary conditions since their conditions are satisfied naturally. In this research, we propose a level set-based topology optimization method for the design of acoustic metamaterial. First, we introduce the two-phase material model and level set-based topology optimization method. An optimization problem for an acoustic metamaterial that has negative mass density is formulated and sensitivity analysis is performed based on the concept of topological derivative. After verifying the topological derivative by comparing it to numerical differences, we present two-dimensional numerical examples for negative mass acoustic metamaterial.

Topology design of compliant mechanisms with stress constraints based on the topological derivative concept

Compliant mechanisms are mechanical devices composed by one single piece that transforms simple inputs into complex movements. This kind of multi-flexible structure can be manufactured at a very sm...

Topology Optimization of Electric Motor Using Topological Derivative for Nonlinear Magnetostatics

We aim at finding an optimal design for an interior permanent magnet electric motor by means of a sensitivity-based topology optimization method. The gradient-based ON/OFF method has been successfully applied to optimization problems of this form. We show that this method can be improved by considering the mathematical concept of topological derivatives (TDs). TDs for optimization problems constrained by linear partial differential equations (PDEs) are well understood, whereas little is known about TDs in combination with nonlinear PDE constraints. We derive the TD for an optimization problem constrained by the equation of nonlinear 2-D magnetostatics, illustrate its advantages over the sensitivities used in the ON/OFF method, and show numerical results for the optimization of an interior permanent magnet electric motor obtained by a level-set algorithm, which is based on the TD.

Robust compliance topology optimization based on the topological derivative concept

SummaryIn this paper, we present an approach for robust compliance topology optimization under volume constraint. The compliance is evaluated considering a point‐wise worst‐case scenario. Analogously to sequential optimization and reliability assessment, the resulting robust optimization problem can be decoupled into a deterministic topology optimization step and a reliability analysis step. This procedure allows us to use topology optimization algorithms already developed with only small modifications. Here, the deterministic topology optimization problem is addressed with an efficient algorithm based on the topological derivative concept and a level‐set domain representation method. The reliability analysis step is handled as in the performance measure approach. Several numerical examples are presented showing the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.

Imaging of a fractal pattern with time and frequency domain topological derivative

While fractal boundaries and their ability to describe irregularity have been intensively studied, only a few studies are available on wave propagation in a medium where a fractal or quasi fractal pattern is embedded. Acoustical propagation in 1D or 2D domains can be modeled using Time Domain Finite Differences or in Frequency domain with Finite element methods. The fractal object is then considered as a subwavelength set of scatterers, and the problem becomes a multiple scattering one leading to acoustic localization. So, as some important part of the energy remains trapped inside the fractal pattern, imaging such a medium becomes difficult and imaging such complex media with classical tools as B-scan or comparable methods is not sufficient. The resolution of the inverse problem of wave propagation can then be achieved with the help of the more efficient imaging methods related with Time Reversal. Using the concept of topological derivative as defined in Time Domain Topological Energy and Fast Topological Imaging Method, respectively, in Time domain and Frequency domain is powerful in that case. Examples in 1D and 2D will be presented including the image obtained for a sliced sponge.

Energy change to insertion of inclusions associated with a diffusive/convective steady‐state heat conduction problem

The topological derivative concept has been successfully applied in many relevant physics and engineering problems. In particular, the topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial differential equations. In this paper, the topological asymptotic analysis of the energy shape functional associated with a diffusive/convective steady‐state heat equation is developed. The topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form with help of a non‐standard adjoint state. Finally, we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justification for the derived formulas. The obtained result is new and can be applied in the context of topology design of heat sinks, for instance. Copyright © 2015 John Wiley & Sons, Ltd.

Energy change to insertion of inclusions associated with the Reissner–Mindlin plate bending model

The topological derivative concept has been proved to be useful in many relevant applications such as topology optimization, inverse problems, image processing, multi-scale constitutive modeling, fracture mechanics and damage evolution modeling. The topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial differential equations. On the other hand, the topological derivatives associated with coupled problems have been derived only in their abstract forms. In this paper, therefore, we deal with the Reissner–Mindlin plate bending model, which is written in the form of a coupled system of partial differential equations. In particular, the topological asymptotic analysis of the associated total potential energy is developed and the topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form. Finally, we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justification for the derived formulas.

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.

Topological Derivative for the Inverse Conductivity Problem: A Bayesian Approach

The employment of topological derivative concept is considered to propose a new optimization algorithm for the inverse conductivity problem. Since this inverse problem is nonlinear and ill-posed it is necessary to incorporate a prior knowledge about the unknown conductivity. In particular, we apply the Bayes theorem to add the assumption that we have just one small ball-shaped inclusion, which must be at a certain distance from the boundary of the domain. As the main emphasis of this paper is to investigate numerically the proposed approach, we shall use the meshless method of fundamental solutions to present some numerical results.