Topological Derivative Research Articles

Improvement of a topological level-set approach to find optimal topology by considering body forces

PurposeBody forces are always applied to structures in the form of the weight of materials. In some cases, they can be neglected in comparison with other applied forces. Nevertheless, there is a wide range of structures in civil and mechanical engineering in which weight or other types of body forces are the main portions of the applied loads. The optimal topology of these structures is investigated in this study.Design/methodology/approachTopology optimization plays an increasingly important role in structural design. In this study, the topological derivative under body forces is used in a level-set-based topology optimization method. Instability during the optimization process is addressed, and a heuristic solution is proposed to overcome the challenge. Moreover, body forces in combination with thermal loading are investigated in this study.FindingsBody forces are design-dependent loads that usually add complexity to the optimization process. Some problems have already been addressed in density-based topology optimization methods. In the present study, the body forces in a topological level-set approach are investigated. This paper finds that the used topological derivative is a flat field that causes some instabilities in the optimization process. The main novelty of this study is a technique used to overcome this challenge by using a weighted combination.Originality/valueThere is a lack of studies on level-set approaches that account for design-dependent body forces and the proposed method helps to understand the challenges posed in such methods. A powerful level-set-based approach is used for this purpose. Several examples are provided to illustrate the efficiency of this method. Moreover, the results show the effect of body forces and thermal loading on the optimal layout of the structures.

Topology Derivation of Multiple-Port DC–DC Converters Based on Voltage-Type Ports

Multiple-port dc–dc converters are characterized by a variety of kinds and a large number of circuit topologies. In this article, we aim to reveal the intrinsic relationship among the topologies of multiple-port dc–dc converters and propose the topology derivation method. First, the voltage- and current-type ports are summarized from basic dc–dc converters, and the approach of converting current-type ports into voltage-type ports is discussed. Then, according to Kirchhoff's law, four types of multiple-input multiple-output converters named input-port-series output-port-series, input-port-series output-port-parallel, input-port-parallel output-port-series, and input-port-parallel output-port-parallel are presented. Second, the topology derivation method of multiple-port bidirectional dc–dc converters based on the voltage-type ports is studied in terms of power flow paths in various operation modes, and then the topology optimization method is proposed. Particularly, a flow diagram for the optimal design procedure is given to guide the topology derivation for some specific requirements. Based on the proposed approach, a family of multiple-port dc–dc converters can be derived, which provides lots of viable candidates for practical engineering. Furthermore, one derived converter named the parallel-type three-port bidirectional buck converter is analyzed in three operation modes to demonstrate the topology derivation. Finally, the effectiveness of the above theoretical analysis is verified by those experimental results.

Topology optimization considering the distortion in additive manufacturing

Additive manufacturing is a free-form manufacturing technique in which parts are built in a layer-by-layer manner. Laser powder bed fusion is one of the popular techniques used to fabricate metal parts. However, it induces residual stress and distortion during fabrication that adversely affects the mechanical properties and dimensional accuracy of the manufactured parts. Therefore, predicting and avoiding the residual stress and distortion are critical issues. In this study, we propose a topology optimization method that accounts for the distortion. First, we propose a computationally inexpensive analytical model for additive manufacturing that uses laser powder bed fusion and formulated an optimization problem. Next, we approximate the topological derivative of the objective function using an adjoint variable method that is then utilized to update the level set function via a time evolutionary reaction-diffusion equation. Finally, the validity and effectiveness of the proposed optimization method was established using two-dimensional design examples.

Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

A Shape Optimization Technique for Medical Images

Segmentation and restoration are among the primary objectives in the field of medical image processing. Among many methods available today, derivative method specifically domain based derivative method helps immensely in this field of medical imaging. There was a time-based improvement in the aspects of establishing results in a medical image with the aid of topological derivative. Advantage of this method lies in establishing a link between that of the perturbed domain and that of the original domain and solving for the required metrics in the problem. Treating the original domain as the original image and the diseased image to be the perturbed domain, topological derivative helps to solve the problem. Asymptotic analysis also helps in handling both large and small amount of data based on the big-O notation and small-o notation. It also establishes the supremacy of topological derived image metrics to that of other metrics. Here attempts are made for solving problems in medical images with the help of topological derivative method by adapting minimization procedure. Also, the end results are arrived in an outstanding way with help of the receiver operating characteristic curve.

Progress on different topology optimization approaches and optimization for additive manufacturing: a review

The progress of topology optimizations for the last decade has been reviewed based on the five main methods mentioned in Sigmund’s article in 2013: density, level-set, topological derivative, phase field, and evolutionary. The recent progress is mainly focusing on methods originated in density or level set method, with an inclination of combining them with the evolutionary method. The methods developed have a large number of applications, yet certain issues, e. g., guaranteeing the convergence of the results generated through evolutionary methods need further investigation. Besides, those studies specifically targeted for the optimization and designing for additive manufacturing are also discussed, as well as the ideas and guidelines for those designs. It is recognized that in order to cope with additive manufacturing, it is necessary to modify existing structures for the application of lattice structures and the reduction of support structures. Moreover, the introduction of the idea of designing for additive manufacturing (DfAM), as well as efforts to make the end-users understand and familiarize themselves with the dramatic differences in designs, are also desired changes that need more efforts to be applied on.

Prefrontal deep projection neurons enable cognitive flexibility via persistent feedback monitoring

Cognitive flexibility, the ability to alter strategy according to changing stimulus-response-reward relationships, is critical for updating learned behavior. Attentional set-shifting, a test of cognitive flexibility, depends on the activity of prefrontal cortex (PFC). It remains unclear, however, what role PFC neurons play to support set-shifting. Using optogenetics and two-photon calcium imaging, we demonstrate that medial PFC activity does not bias sensorimotor responses during set-shifting, but rather enables set-shifting by encoding trial feedback information, a role it has been known to play in other contexts. Unexpectedly, the functional properties of PFC cells did not vary with their efferent projection targets. Instead, representations of trial feedback formed a topological gradient, with cells more strongly selective for feedback information located further from the pial surface, where afferent input from the anterior cingulate cortex was denser. These findings identify a critical role for deep PFC projection neurons in enabling set-shifting through behavioral feedback monitoring.

A level set based topology optimization for finite unidirectional acoustic phononic structures using boundary element method

The paper presents a level set based topology optimization method for unidirectional phononic structures with finite layers of lattice cells. Boundary element method(BEM) is employed as the numerical approach to solve the acoustic problems governed by Helmholtz equation. A sized reduced coefficient matrix is derived due to the iteration forms for input and output quantities on the periodic boundary of unit cells. Topological derivatives are formulated by boundary integral equation combined with adjoint variable method and computed for each layer. An average topological sensitivity of a single design domain is proposed for the updating of the level set function(LSF) governed by an evolution equation. Numerical models with different number of layers are considered and several optimized structures of unit cells are obtained in concerned frequencies. A further investigation into the transmission of acoustic waves is carried out by employing more layers of the periodic structures between the input and output domains. The results demonstrate the effectiveness of the proposed optimization method for the finite unidirectional phononic structures.

A new adjoint problem for two-dimensional helmholtz equation to calculate topological derivatives of the objective functional having tangential derivative quantities

A new adjoint problem for two-dimensional helmholtz equation to calculate topological derivatives of the objective functional having tangential derivative quantities

Real-Time Topology Optimization in 3D via Deep Transfer Learning

The published literature on topology optimization has exploded over the last two decades to include methods that use shape and topological derivatives or evolutionary algorithms formulated on various geometric representations and parametrizations. One of the key challenges of all these methods is the massive computational cost associated with 3D topology optimization problems.We introduce a transfer learning method based on a convolutional neural network that (1) can handle high-resolution 3D design domains of various shapes and topologies; (2) supports real-time design space explorations as the domain and boundary conditions change; (3) requires a much smaller set of high-resolution examples for the improvement of learning in a new task compared to traditional deep learning networks; (4) is multiple orders of magnitude more efficient than the established gradient-based methods, such as SIMP. We provide numerous 2D and 3D examples to showcase the effectiveness and accuracy of our proposed approach, including for design domains that are unseen to our source network, as well as the generalization capabilities of the transfer learning-based approach. Our experiments achieved an average binary accuracy around 95% at real-time prediction rates. These properties, in turn, suggest that the proposed transfer-learning method may serve as the first practical underlying framework for real-time 3D design exploration based on topology optimization.

A robust topology optimization for enlarging working bandwidth of acoustic devices

AbstractWe propose a novel robust topology optimization for designing acoustic devices that are effective for broadband sound waves. Here, we define the objective function as the weighted sum of the acoustic response to an incident wave consisting of a single frequency and its standard deviation (SD) against the frequency perturbation. By approximating the SD, under the assumption that the incident frequency follows the normal distribution, with the high‐order Taylor expansion of the (conventional) objective function, we deal with significant frequency variations. To calculate such an approximation, we need to calculate the high‐order frequency derivatives of the state variable. Here, we define them by integral representations, which enables us to characterize them even when the state variable is defined in an unbounded domain as is often the case with wave scattering problems. We further show that, based on this definition, the high‐order derivatives can efficiently be computed by a combination of the boundary element method and automatic differentiation. We also present a derivation and calculation of the topological derivative for the newly defined objective function. We install all the proposed techniques into a topology optimization method based on the level‐set method to design wideband acoustic devices. The validity and effectiveness of the proposed topology optimization are confirmed through several numerical examples.

Topological gradient in structural optimization under stress and buckling constraints

Structural topology optimization aims to design mechanical structures by seeking the optimal material layout within a given design space. Within this framework, this paper addresses the minimization of the structural mass under stress and buckling constraints, formulated as a nonlinear combinatorial optimization problem. An algorithm is proposed for such a problem, that follows a topological gradient-based approach. The adjoint method is applied to efficiently compute the constraint gradients. An iterative algorithm for buckling analysis, featuring low memory requirements, is also proposed. Numerical results, including a real application arising in the aeronautical field, illustrate the efficiency of the two proposed algorithms.

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

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

Asymptotic analysis and topological derivative for 3D quasi-linear magnetostatics

In this paper we study the asymptotic behaviour of the quasilinearcurl-curlequation of 3D magnetostatics with respect to a singular perturbation of the differential operator and prove the existence of the topological derivative using a Lagrangian approach. We follow the strategy proposed in Gangl and Sturm (ESAIM: COCV26(2020) 106) where a systematic and concise way for the derivation of topological derivatives for quasi-linear elliptic problems inH1is introduced. In order to prove the asymptotics for the state equation we make use of an appropriate Helmholtz decomposition. The evaluation of the topological derivative at any spatial point requires the solution of a nonlinear transmission problem. We discuss an efficient way for the numerical evaluation of the topological derivative in the whole design domain using precomputation in an offline stage. This allows us to use the topological derivative for the design optimization of an electrical machine.

Principle and Topology Derivation of Integrated Cascade Bidirectional Converters for Centralized Charge Equalization Systems

This article presents a systematic topology derivation method for developing integrated cascaded bidirectional (ICB) converters, and the centralized charge equalization system can be optimized with the proposed family of converters. On the basis of the traditional centralized system, the proposed ICB converters are generated by integrating the polarity switches and different bidirectional dc–dc converters. Compared with existing methods, the ICB converter-based centralized equalization system obtains the advantages of high voltage gain, high efficiency, high reliability, and low cost, which is more attractive in engineering applications. Since different topology structures have various performances, a half-bridge-based ICB converter with a bidirectional switch is analyzed in detail as an example to validate the feasibility and effectiveness of the proposed topology family. The operation principles, design guidelines, and soft-switching conditions are discussed in more depth. Finally, the experimental results and comparative analysis are conducted to verify the advantages of the proposed ICB converter and prove the evident performance improvement compared with conventional methods.

Deriving a Topology Function to Be Used in the Reaction Diffusion Equation for Optimization of Structures with Thermal and Mechanical Boundary Conditions

A topological derivative of the Lagrangian is required for optimization of structures with thermal and mechanical boundary conditions by the level-set method using the reaction diffusion equation. In this study, drawing on the relationship between the shape derivative and the topological derivative, the topological derivative of the Lagrangian was obtained by Reynolds' transport theorem. Given that introducing holes to the topology creates boundaries, the derivative was found by incorporating the boundary integral into the Reynolds' transport theorem and analyzing the stress over the hole boundaries. The temperature was assumed to be dependent on topology in the present study under thermal and mechanical boundary conditions. Placing a hole in the structure affects the temperature of the remaining elements. Penalty factor is enforced on thermal conductivity for removed elements, and the result is taken into consideration in the Laplace's equation expressing the steady-state conductive heat transfer.

Topological-derivative-based design of stiff fiber-reinforced structures with optimally oriented continuous fibers

We use topological derivatives to obtain fiber-reinforced structural designs with non-periodic continuous fibers optimally arranged in specific patterns. The distribution of anisotropic fiber material within isotropic matrix material is determined for given volume fractions of void and material as well as fiber and matrix simultaneously, for maximum stiffness. In this three-phase material distribution approach, we generate a Pareto surface of stiffness and two volume fractions by adjusting the level-set plane in the topological sensitivity field. For this, we utilize topological derivatives for interchanging (i) isotropic material and void; (ii) fiber material and void; and (iii) isotropic and fiber materials, during iterative optimization. While the isotropic topological derivative is well known, the latter two required modification of the anisotropic topological derivative. Furthermore, we used the polar form of the topological derivative to determine the optimal orientation of the fiber at every point. Thus, in the discretized finite element model, we get element-wise optimal fiber orientation in the portions where fiber is present. Using these discrete sets of orientations, we extract continuous fibers as streamlines of the vector field. We show that continuous fibers are aligned with the principal stress directions as first reported by Pedersen. Three categories of examples are presented: (i) embedding fiber everywhere in the isotropic matrix without voids; (ii) selectively embedding fiber for a given volume fraction of the fiber without voids; and (iii) including voids for given volume fractions of fiber and matrix materials. We also present an example with multiple load cases. Additionally, in view of practical implementation of laying up or 3D-printing of fibers within the matrix material, we simplify the dense arrangement of fibers by evenly spacing them while retaining their specific patterns.

Two-Mode Controlled Single/Dual-Input DC–AC Inverter With Wide-Range DC Input

This article presents a two-mode controlled step-up inverter (TMCSI), which is capable of handling single or dual inputs with a wide range of dc input voltage. In comparison to traditional multistage dc/dc power conversion systems, the power conversion stage is reduced and the voltage/current stress of device is significantly alleviated without using high-voltage dc-bus capacitors. As a result, the reliability and overall conversion efficiency are improved accordingly. By combining the two working modes, the proposed inverter achieves uniform distribution of duty ratio under single/dual-input with a wide range of input voltage, and thereby is very suitable for systems having large input voltage variation. Moreover, based on the dual-input TMCSI, a novel power allocation method is also proposed to improve the system load-bearing ability. The power allocation method is controlled separately from two working modes that allows the two-input sources directly supply the ac load simultaneously. In this article, topology derivation, two-mode control strategy, characteristics of steady principle, and design criteria for the key circuit parameters are systematically analyzed, and important conclusions are obtained. Finally, the experimental results from the single/dual-input 500 VA 96-192 V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dc</sub> input and 220 V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ac</sub> /50 Hz output inverter prototype verify the effectiveness of the proposed TMCSI topology and its associated power regulation approach.

On the determination of ischemic regions in the monodomain model of cardiac electrophysiology from boundary measurements**In loving memory of Maria Assunta Pozio.

In this paper we consider the monodomain model of cardiac electrophysiology. After an analysis of the well-posedness of the model we determine an asymptotic expansion of the perturbed potential due to the presence of small conductivity inhomogeneities (modelling small ischemic regions in the cardiac tissue) and use it to detect the anomalies from partial boundary measurements. This is done by determining the topological gradient of a suitable boundary misfit functional. The robustness of the algorithm is confirmed by several numerical experiments.

Topological Derivative-Based Topology Optimization of Plate Structures Under Bending Effects

In this work, the topological derivatives of L2 and energy norms associated with the solution to Kirchhoff and Reissner-Mindlin plate bending models are introduced. Based on existing theoretical results, closed forms of the sensitivities are presented. A zero-order term is introduced in the equilibrium equations, which allows for adapting the obtained sensitivities to the context of topology optimization of plates under elastic support and free vibration condition as well. The resulting analytical formulae are used together with a level-set domain representation method to devise a simple topology design algorithm. Several finite element-based representative numerical experiments are presented showing its applications to the compliance minimization and eigenvalue maximization of Kirchhoff as well as Reissner-Mindlin plate structures under bending effects.