Topological Derivative Research Articles

Investigation of a non-iterative technique based on topological derivatives for fast localization of small conductivity inclusions

In this contribution, we consider a topological derivative-based non-iterative technique for an inverse conductivity problem of localizing a small-conductivity inclusion completely embedded in a homogeneous domain via boundary measurement data. For this purpose, we derive the topological derivative by applying an asymptotic formula in the presence of a small-diameter conductivity inclusion and establish a mathematical structure of topological derivative. On the basis of the established structure, we propose a new imaging function with appropriate boundary conditions. Results of numerical simulation are presented to show the feasibility and limitations of the designed imaging function.

Topological optimization applied to a variable density and conductivity model of pollution in porous media

AbstractThis article deals with modeling and numerical study of pollutants transfer in porous media by topological optimization. Indeed we perform the modeling of pollution in porous media in non‐permanent cases by assuming the variability or not of some parameters such as the conductivity and the density. The model of pollutants transfer obtained in the case when the conductivity and the density are constant is similar to meadows of few coefficients to the one obtained by varying (with respect to the space) only the conductivity. By taking into account the space variability of density, conductivity, and diffusion under some assumptions we obtain a more realistic model of pollutants transfer. Topological optimization applied to these models allows reaching numerically the optimal design of a least squared functional. In fact, topological derivatives are used as descent direction and permit to creation of circular or cylindrical holes in order to decrease the considered cost function. Numerical results obtained are presented for Neumann and Dirichlet conditions on the boundary of the holes for 2 and 3 dimensions in nonpermanent cases of pollutants transfer.

Automated computation of topological derivatives with application to nonlinear elasticity and reaction–diffusion problems

While topological derivatives have proven useful in applications of topology optimization and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial differential equation (PDE) constraints. We present a systematic yet formal approach for the numerical computation of topological derivatives of a large class of PDE-constrained topology optimization problems with respect to arbitrary inclusion shapes. Scalar and vector-valued as well as linear and nonlinear elliptic PDE constraints are considered in two and three space dimensions including a nonlinear elasticity model and nonlinear reaction–diffusion problems. The systematic procedure follows a Lagrangian approach for computing topological derivatives. For problems where the exact formula is known, the numerically computed values show good coincidence. Moreover, by inserting the computed values into the topological asymptotic expansion, we verify that the obtained values satisfy the expected behavior also for other, previously unknown problems, indicating the correctness of the procedure.

A topology optimization of open acoustic waveguides based on a scattering matrix method

This study presents a topology optimization scheme for realizing a bound state in the continuum along an open acoustic waveguide comprising a periodic array of elastic materials. First, we formulate the periodic problem as a system of linear algebraic equations using a scattering matrix associated with a single unit structure of the waveguide. The scattering matrix is numerically constructed using the boundary element method. Subsequently, we employ the Sakurai–Sugiura method to determine resonant frequencies and the Floquet wavenumbers by solving a nonlinear eigenvalue problem for the linear system. We design the shape and topology of the unit elastic material such that the periodic structure has a real resonant wavenumber at a given frequency by minimizing the imaginary part of the resonant wavenumber. The proposed topology optimization scheme is based on a level-set method with a novel topological derivative. We demonstrate a numerical example of the proposed topology optimization and show that it realizes a bound state in the continuum through some numerical experiments.

Optimal shape design for a time-dependent Brinkman flow using asymptotic analysis techniques

In this paper, we consider the geometric inverse problem of recovering an obstacle ω immersed in a bounded fluid flow Ω governed by the time-dependent Brinkman model. We reformulate the inverse problem into an optimization problem using a least squares functional. We prove the existence of an optimal solution to the optimization problem. Then, we perform the asymptotic expansion of the cost function in a simple way using a penalty method. An important advantage of this method is that it avoids the truncation method used in the literature. To reconstruct the obstacle, we propose a fast algorithm based on the topological derivative. Finally, we present some numerical experiments in two- and three-dimensional cases showing the efficiency of the proposed method.

A topology optimisation of acoustic devices based on the frequency response estimation with the Padé approximation

We propose a topology optimisation of acoustic devices that work in a certain bandwidth. To achieve this, we define the objective function as the frequency-averaged sound intensity at given observation points, which is represented by a frequency integral over a given frequency band. It is, however, prohibitively expensive to evaluate such an integral naively by a quadrature. We thus estimate the frequency response by the Padé approximation and integrate the approximated function to obtain the objective function. The corresponding topological derivative is derived with the help of the adjoint variable method and chain rule. It is shown that the objective as well as its sensitivity can be evaluated semi-analytically. We present efficient numerical procedures to compute them and incorporate them into a topology optimisation based on the level-set method. We confirm the validity and effectiveness of the present method through some numerical examples.

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.

Multiphase Low Stresses High Step-Up DC–DC Converter With Self-Balancing Capacitor Voltages and Self-Averaging Inductor Currents

Voltage gain ratio is a key limitation faced by nonisolated dc–dc converters in practical applications. Low voltage and current stresses, small volume, simple sampling control circuit, low cost, suitability for high power, scalability, and modularization are optimization trends of the dc–dc converter design. To improve the above-mentioned performance, a multiphase high step-up dc–dc converter is proposed, with advantages of high voltage gain, low switching voltage stresses, low switching current stresses, low capacitor voltage stresses, low output voltage ripple, capacitor voltages self-balancing feature, inductor currents self-averaging feature, ease of extension, and being able to be modularized. The topology derivation, comparisons with other high step-up converters, the operating principle, characteristics, and mathematical models of the proposed converter are studied in detail. Finally, a laboratory prototype of the proposed dc–dc converter operating from 400 to 800 W is built and implemented with input being 24 V and output being 400 V. The experimental results verify the feasibility and superiority of the proposed converter. The proposed multiphase high step-up converter is suitable for the applications, such as fuel cells, electric vehicles, energy storage systems, photovoltaic systems, and microgrids.

Graph Theory-Based Programmable Topology Derivation of Multiport DC–DC Converters With Reduced Switches

Different from the typical topology derivation of power electronics converters (PECs) depending on manual effort, this article aims to explore a programmable method based on graph theory for multiport dc–dc converters with reduced switches. By mathematically modeling converters with graph theory and transforming their working criteria into the corresponding graph constraints, the topology derivation can be automatically and conveniently solved with the aid of computer program. In this article, first, the principle of the proposed graph theory-based programmable method is introduced in detail. Then, it is employed to derive multiport dc–dc converters with reduced switches, and a diversity of new topologies are simultaneously excavated, among which a favorable one is selected to be theoretically introduced and experimentally verified. The proposed method remains relatively simple with the increasing number of ports, as the graph constraints of working criteria are almost the same for different number of ports. More importantly, because PECs can be essentially regarded as graphs with different components and their connecting relationships, the proposed method is applicable for many various converters.

On the robustness of the topological derivative for Helmholtz problems and applications

Abstract We consider Helmholtz problems in two and three dimensions. The topological sensitivity of a given cost function J(u ∈) with respect to a small hole B ∈ around a given point x 0 ∈ B ∈ ⊂ Ω depends on various parameters, like the frequency k chosen or certain material parameters or even the shape parameters of the hole B ∈. These parameters are either deliberately chosen in a certain range, as, e.g., the frequencies, or are known only up to some bounds. The problem arises as to whether one can obtain a uniform design using the topological gradient. We show that for 2-d and 3-d Helmholtz problems such a robust design is achievable.

Design Techniques for High-Efficiency Envelope-Tracking Supply Modulator for 5th Generation Communication

This brief provides a brief tutorial on designing a high efficiency envelope-tracking supply modulator with recent techniques. Design challenges for 5G communication applications are given. Detailed techniques covering topology derivation, multi-level and fast-speed switching converter design considerations, voltage step-up and –down function implementations, linear amplifier bandwidth improvements and measurement setup are discussed, providing design guidelines and considerations.

Computer-Aided Systematic Topology Derivation of Single-Inductor Multi-Input Multi-Output Converters From Working Principle

In this paper, aiming to systematically derive single-inductor multi-input multi-output (SI-MIMO) converters having favorable merits of high-density and low-cost, a basic model is firstly extracted from the working principle. Theoretically, by enumerating all possible connections of the basic model and one-by-one manually judging their effectiveness according to working criterions, multiple viable SI-MIMO topologies can be obtained, but with heavy workload. To simplify the topology derivation process, this paper also proposes a computer-aided method to replace the manual effort. By modelling the basic model and working criterions with graph theory, MATLAB codes can be developed to automatically screen out the effective topologies. With the proposed method, 8 three-port, 28 four-port and 115 five-port converters with single inductor are conveniently obtained, including both the existing and new topologies. Moreover, by configuring ports and switches in each derived topology, multiple specific converters can be further expanded. These converters can provide more options for practical engineering applications, and a new topology is also introduced and experimentally verified in the paper.

A topologically gradient body centered lattice design with enhanced stiffness and energy absorption properties

Lattice structures have been widely used in biomedicine, transportation, infrastructure, and other engineering fields due to their high specific strength, strong energy absorption capacity, and tunable mechanical property. Existing researches mainly focus on the uniform lattice structure or gradient lattice structure with varying density, with the design potential of lattice structure not fully exploited. In this study, inspired by the functionally graded material, a novel topologically gradient lattice structure with varying cell topology is proposed to realize spacial adjustability of the mechanical properties. The new lattice structure evolves from the classical body-centered cubic (BCC) lattice, with the body center positions varying gradiently from cell to cell along the specified directions. The prototypes of the new lattice are additively manufactured by the Multi-Jet Fusion (MJF) technology and quasi-statically compressed by the universal testing machines. Experimental and numerical simulation results reveal that the proposed structure topology gradient can significantly improve both the stiffness and the energy absorption capacity compared with the classical BCC lattice structures. It is found that the mechanical properties of the topologically gradient lattice structures are sensitive to the gradient direction, the gradient magnitude and the loading direction. This study expands the design space of lattice structures by introducing the topology gradient of the composing cells across the space. In this way, further improvement in the mechanical properties of lightweight lattice structures can be achieved.

Fracture-based shape optimization built upon the topological derivative

In Silva et al. (2011) and Alidoost et al. (2020), the authors developed an approximation of the energy release rate field associated with a small edge or surface crack at any boundary location and with any orientation using the topological derivative. The approximation is computationally attractive because it requires only a single analysis on the non-cracked domain in contrast with conventional boundary-element and finite-element-based methods, which require a separate and costlier analysis for each crack length-location-orientation combination. In this work, a shape optimization scheme for fracture-resistant structures is developed using the energy release rate approximation. In the gradient-based optimization scheme, the domain and its boundary are defined implicitly using level-set functions. The level-set functions of arbitrary geometries are constructed using Boolean operations from the level-set functions of simple primitives. This geometrical representation has the dual advantage of (i) allowing shapes to intersect and/or separate during the optimization and (ii) simplifying the computation of the shape sensitivities.

Nonisolated high gain hybrid switched‐inductor DC‐DC converter with common switch grounding

AbstractThis paper presents a methodology that integrates the hybrid switched inductor (HSL) and split duty ratio techniques to synthesize a high gain converter with common switch grounding (HSL‐CSG). The DC‐DC converter's high gain feature is required for microsource power processing in low voltage renewable energy systems. The amplification in voltage gain of the proposed converter is achieved without using transformer, common core coupled inductors, voltage multipliers, intermediate capacitors, and voltage lifting methods so that the topology is simple in construction and operation. Moreover, with a split duty ratio the proposed converter switches are less prone to current stress due to reduction in extreme conducting duty intervals. The topological derivation, operating principle, analysis of both continuous conduction mode (CCM), and discontinuous conduction mode (DCM) are presented. The theoretical analysis is validated with the aid of a laboratory prototype.

Topology optimization for acoustic structures considering viscous and thermal boundary layers using a sequential linearized Navier–Stokes model

This study proposes a level set-based topology optimization method for designing acoustic structures with viscous and thermal boundary layers in perspective. Acoustic waves propagating in a narrow channel are damped by viscous and thermal boundary layers. To estimate these viscothermal effects, we first introduce a sequential linearized Navier–Stokes model based on three weakly coupled Helmholtz equations for viscous, thermal, and acoustic pressure fields. Then, the optimization problem is formulated, where a sound-absorbing structure comprising air and an isothermal rigid medium is targeted, and its sound absorption coefficient is set as an objective function. The adjoint variable method and the concept of the topological derivative are used to approximately obtain design sensitivity. A level set-based topology optimization method is used to solve the optimization problem. Two-dimensional numerical examples are provided to support the validity of the proposed method. Moreover, the mechanisms that lead to the high absorption coefficient of the optimized design are discussed.

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 Imaging Methods for the Iterative Detection of Multiple Impedance Obstacles

In this paper, we investigate shape inversion algorithms based on the computation of iterated topological derivatives for the detection of multiple particles coated by a complex surface impedance in two- and three-dimensional acoustic media. New closed-form formulae for the topological derivative of the misfit functional are derived when an approximate set of unknown particles has already been recovered. Proofs rely on the computation of shape derivatives followed by the topological asymptotic analysis of a boundary integral equation formulation of the forward and adjoint problems. The relevance of the theoretical results is illustrated by various 2D and 3D experiments using monochromatic imaging algorithms either fully or partially based on topological derivatives.

Analysis of single input dual output buck converter with reduced cross regulation using decoupled sliding mode control strategy

AbstractIn the present scenario, single input dual output (SIDO) converters are remarkably required for wide range of applications, as it is capable of handling two outputs with a single source. The general advantages of SIDO converters are reduced number of components in the topology; power rating can be increased or decreased depending on the application and it ease of control. However, it faces with a common problem of cross regulation occurring due to the change of load in anyone of the load terminals. This paper proposes a control technique for a SIDO buck converter (SIDO‐BC) to have a mitigated cross regulation scenario during load change. Decoupled sliding mode controller (D‐SMC) is developed for the SIDO‐BC by considering the boundary limits of the inductor current. The D‐SMC‐based SIDO‐BC provides better output regulation with mitigated cross regulation when compared to the conventional techniques. The topological small signal model derivation of the SIDO‐BC is firmly derived, and the controller is designed and developed in such way to provide a stabilized regulated output. The entire system is implemented in simulation and in experimental hardware to validate the performance.

The topological ligament in shape optimization: a connection with thin tubular inhomogeneities

In this article, we propose a formal method for evaluating the asymptotic behavior of a shape functional when a thin tubular is added between two distant regions of the boundary of a domain. In the contexts of the conductivity equation and the linear elasticity system, we relate this issue to a perhaps more classical problem of thin tubular inhomogeneities: we analyze the solutions to versions of the physical partial differential equations which are posed inside a fixed background medium, and whose material coefficients are altered inside a tube with vanishing thickness. Our main contribution from the theoretical point of view is to propose a heuristic energy argument to calculate the limiting behavior of these solutions with a minimum amount of effort. We retrieve known formulas when they are available, and we manage to treat situations which are, to the best of our knowledge, not reported in the literature (including the setting of the 3d linear elasticity system). From the numerical point of view, we propose three different applications of the formal ligament approach derived from these expansions. At first, it is an original way to account for variations of a domain, and it thereby provides a new type of sensitivity for a shape functional, to be used concurrently with more classical shape and topological derivatives in optimal design frameworks. Besides, it suggests new, interesting algorithms for the design of the scaffold structure sustaining a shape during its fabrication by a 3d printing technique, and for the design of truss-like structures. Several numerical examples are presented in two and three space dimensions to appraise the efficiency of these methods.