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This paper is devoted to the shape optimization of the internal structure of an electric motor, and more precisely of the arrangement of air and ferromagnetic material inside the rotor part with the aim to increase the torque of the machine. The governing physical problem is the time-dependent, nonlinear magneto-quasi-static version of Maxwell’s equations. This multiphase problem can be reformulated on a 2D section of the real cylindrical 3D configuration; however, due to the rotation of the machine, the geometry of the various material phases at play (the ferromagnetic material, the permanent magnets, air, etc.) undergoes a prescribed motion over the considered time period. This original setting raises a number of issues. From the theoretical viewpoint, we prove the well-posedness of this unusual nonlinear evolution problem featuring a moving geometry. We then calculate the shape derivative of a performance criterion depending on the shape of the ferromagnetic phase via the corresponding magneto-quasi-static potential. Our numerical framework to address this problem is based on a shape gradient algorithm. The nonlinear time periodic evolution problems for the magneto-quasi-static potential is solved in the time domain, with a Newton–Raphson method. The discretization features a space-time finite element method, applied on a precise, meshed representation of the space-time region of interest, which encloses a body-fitted representation of the various material phases of the motor at all the considered stages of the time period. After appraising the efficiency of our numerical framework on an academic problem, we present a quite realistic example of optimal design of the ferromagnetic phase of the rotor of an electric machine. [ABSTRACT FROM AUTHOR]

Triply periodic minimal surfaces (TPMS) are highly versatile porous formations that can be defined by formulas. Computationally based, load-specific shape optimization enables tailoring these structures for their respective application areas and thereby enhance their potential. In this investigation, individual sheet-based gyroid structures with varying porosities are specifically optimized with respect to their stiffness. A modified phase-field method is employed to establish a simulation framework for the shape optimization process. Despite constant volume and the preservation of the periodicity of the unit cells, volume redistribution occurs through displacement of the interfaces. The phase-field-based optimization process is detailed using unidirectional loading on three gyroidal unit cells with porosities of 75 %, 80 %, and 85 %. Subsequently, the gyroidal unit cell with a porosity of 85 % is shape-optimized under multidirectional loading. A subsequent experimental validation of the unidirectionally loaded cells confirms that the shape-optimized structures exhibit, on average, higher stiffness than the non-optimized structures. The highest increase of 40 % in effective modulus is achieved with the gyroid structure having a porosity of 75 %, while maintaining minimal alteration to the surface-to-volume ratio and preserving periodicity. Additionally, the experimental data show that the optimization process resulted in a shift in the linear elasticity and plasticity range. In summary, the phase-field method proves to be a valid optimization technique for complex porous structures, allowing the preservation of characteristic properties. [ABSTRACT FROM AUTHOR]

The Vertex Morphing method is a node-based shape parameterization that uses an explicit filtering approach to regularize the optimization problem and generate smooth shapes. It has been successfully applied to shape optimization problems of industrial size in recent years. This work investigates in detail how irregular discretizations, design surface boundaries, and complex geometries can influence the progress of a gradient-based optimization using the standard Vertex Morphing formulation. A sensitivity weighting approach based on the available shape morphing functions is presented, which eliminates all of the aforementioned influences. Subsequently, a design variable scaling strategy is developed that transforms the optimization problem into an alternative design space and allows the use of arbitrary, even highly irregular surface discretizations in combination with black-box optimization algorithms for shape optimization with the Vertex Morphing method. Illustrative academic examples and an application case of an additively manufactured part are presented to support the work. [ABSTRACT FROM AUTHOR]

In this paper we investigate the Riemannian extensibility of saturation phenomena treated first in the Euclidean framework by Brandolini et al. [Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv Math (N Y). 2011;228(4):2352–2365.]. The saturation problem is formulated in terms of the first eigenvalue of the perturbation of the Laplace-Beltrami operator by the integral of the unknown function: the first eigenvalue increases with the weight affecting the integral up to a finite critical value and then remains constant, i.e. it saturates. Given a Riemannian manifold with certain curvature constraints, by using symmetrization arguments and sharp isoperimetric inequalities, we reduce the general problem to a variational one, formulated on either positively or negatively curved Riemannian model spaces; in addition, the possible scenarios for the optimal domains turn to be either geodesic balls or the union of two disjoint geodesic balls. We then explicitly compute the eigenvalues and eigenfunctions in terms of the radii, curvature and weight. A sufficient condition (incompatibility of a system of nonlinear equations involving special functions) is given that implies similar saturation phenomena to the Euclidean case. Due to its highly nonlinear character of the reduced problem (arising from the presence of curvature and special functions), we provide only partial answers to the original problem. However, both analytical computations and numerical tests suggest that the required incompatibility always persists. In addition, in the limit cases when the curvature tends to zero (for both positive an negative curvature), our results reduce to the Euclidean version. [ABSTRACT FROM AUTHOR]

• We present an effective isogeometric shape optimization method to optimize phononic band gaps of periodic plates. • The particle swarm optimization is employed to solve the constrained dynamic maximization problem. • The thickness, i.e., z-component of control point of B-spline surface is defined as optimal design variable. • Several numerical examples for finding optimal phononic band gaps of periodic plates are studied. This work presents an effective isogeometric shape optimization approach for finding and widening the phononic band gaps of single-phase Mindlin plate structures. As the single-phase material phononic plate is easy to be manufactured by additive manufacturing, it has been investigated here by optimized its thickness profile to find and widen the band gaps. The proposed method utilizes a coarse B-spline surface to model the thickness profile of periodic plate and a fine B-spline surface to model the mid-surface of plate structure for simulation. The optimal design variables are the thickness variables, i.e., the z-components of the control points of the coarse B-spline surface. To avoid specifying the initial control point locations manually, the constrained dynamic maximization problem is solved by a particle swarm optimization (PSO) algorithm here. Various numerical examples demonstrate the effectiveness and reliability of the proposed method in finding optimal phononic band gaps of periodic plates. And the numerical results show that there is no band gap for h max = 3 h min , and the band gaps can be found and widen for h max ≥ 5 h min. The obtained band range is 638.9–823.6 Hz with a decreased central frequency of 731.25 Hz for h max = 5 h min and the width and the number of band gaps are increased as the maximum allowable thickness increases. Finally, one optimized design is fabricated through additive manufacturing, and the experimental frequency response is consistent with the results based on isogeometric analysis. [ABSTRACT FROM AUTHOR]

We consider a shape optimization based method for finding the best interpolation data in the compression of images with noise. The aim is to reconstruct missing regions by means of minimizing a data fitting term in an L p -norm, for 1 ⩽ p < + ∞, between original images and their reconstructed counterparts using linear diffusion PDE-based inpainting. Reformulating the problem as a constrained optimization over sets (shapes), we derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small ball (a single pixel) using the adjoint method. Based on the achieved distributed topological shape derivatives, we propose a numerical approach to determine the optimal set and present numerical experiments showing the efficiency of our method. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression. [ABSTRACT FROM AUTHOR]

The density distribution in an urban matrix is one of the significant issues which affects other urban living factors such as building lighting, energy consumption and residents' interactions. The research toward achieving the optimum density distribution has received attention for the last decade. However, developing a generative approach that provides more freedom for the formation of the plans and incorporates adaptability in different land blocks is still missing. To address such a gap, this study proposes an adaptable approach developing the formation of residential blocks. This formation is according to the pre-defined size and shape of the land, and sought performance objectives. Hence, a suite of applications including Grasshopper, Python and Ladybug were applied in a residential block of Tehran as a case study. The purpose is to develop a new density distribution increasing view quality, visual privacy, and solar gain. For the optimization process, a genetic algorithm was applied utilizing the topology optimization technique. The results of the optimization process highlight the significance of this research since the developed alternatives are more efficient in terms of improving the view quality, visual privacy and increasing the solar gain. This achievement expands the potential of this research to be applied in different case studies and with different design and development objectives in order to develop better shape plans of building blocks. [ABSTRACT FROM AUTHOR]

In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically. [ABSTRACT FROM AUTHOR]

In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BV one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BV regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases. [ABSTRACT FROM AUTHOR]

The paper studies the fractional order version of the reinforced membrane problem introduced in [A. Henrot and H. Maillot, 2001]. Existence and uniqueness of the solutions of the corresponding non-local equations has been proven for the relaxed problem. In addition, for the radial symmetric case the existence of the optimal domain has been shown. [ABSTRACT FROM AUTHOR]

Studies on aerodynamic controls of central-slotted box decks primarily focused on mitigating vortex-induced vibrations (VIV), as this type of deck typically performs well against flutter instability. However, as the span length increases, the critical wind speed of aerodynamic static instability (U cr ) might be lower than flutter critical wind speed. Thus, U cr will determine the overall aerodynamic performance of such bridges. Investigating this instability through wind tunnel testing methods and numerical simulation can be expensive and time-consuming. In this paper, surrogate models using machine learning approaches, specifically artificial neural network (ANN) and extreme gradient boosting (XGBoost), were developed and optimized for fast and reliable prediction for U cr based on wind tunnel tests and simulation data. The results demonstrated that the built surrogate models can predict U cr accurately. The parametric study results showed that the height ratio of wind fairing apex (a/b), wind angle of attack (α), and length of the main span (L) have the most influence on the U cr compared with other parameters. Finally, based on the developed ANN surrogate model and the artificial bee colony (ABC) optimization algorithm, an optimized section was proposed to enhance the section's performance against aerodynamic static instability. [ABSTRACT FROM AUTHOR]

We investigate a phase-field version of the Faber-Krahn theorem based on a phase-field optimization problem introduced by Garcke et al. in their 2023 paper formulated for the principal eigenvalue of the Dirichlet-Laplacian. The shape that is to be optimized is represented by a phase-field function mapping into the interval [0,1]. We show that any minimizer of our problem is a radially symmetric-decreasing phase-field attaining values close to 0 and 1 except for a thin transition layer whose thickness is of order ε>0. Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a Γ-convergence result which allows us to recover a variant of the Faber-Krahn theorem for sets of finite perimeter in the sharp interface limit. [ABSTRACT FROM AUTHOR]

Flywheels used for energy storage may be assembled by press fitting or bonding of concentric rotor parts. During operation, each part is subjected to a superposition of stress fields associated with centrifugal loading and with contact at the component interfaces. To capture the highest overall energy storage capacity, design properties must be selected to achieve the maximum moment of inertia, subject to constraints on the stress occurring within each part at the maximum operating speed. This paper proposes a mathematical formulation for parametric shape optimization problems which can be used to design a multi-disk flywheel rotor to maximize the energy storage capacity. It is shown that the energy storage function for each part of the flywheel can be expressed as a product of two fundamental shape-factors: one that accounts for the stress distribution and the other accounts for the mass distribution. These factors are expressed analytically for cases involving hyperbolic disks, leading to a finite dimension shape optimization problem for a multi-disk rotor. Case studies are presented based on the derived analytical solutions that show how the theory can be usefully applied in flywheel design optimization problems. The results from the case study show significant improvements in specific energy compared to previous research results. This advancement is particularly relevant to develop high-performance, cost-effective flywheel systems, offering potential for widespread application in energy storage technologies. [ABSTRACT FROM AUTHOR]

The shape of the connecting section of the shaft spillway directly affects the flow pattern of the tunnel, the vibration characteristics of the shaft and the energy dissipation effect. At present, the research on the shape of connecting section is mostly single factor, and few multi-factor studies. In engineering design practice, the optimization of shaft shape mostly refers to similar engineering experience, and there is no more systematic optimization method. In this paper, the numerical orthogonal test is used to analyze the sensitivity of the length-width ratio, the contraction ratio of the exit pressure slope and the compression slope ratio of the connecting section. The results show that the length-width ratio has the most significant effect on the outlet turbulent kinetic energy, followed by the compression slope ratio and the lowest contraction ratio. The research results can provide some reference for the optimization of the joint shape of the shaft spillway tunnel in engineering design. [ABSTRACT FROM AUTHOR]

Shape optimization approaches to inverse design offer low‐dimensional, physically‐guided parameterizations of structures by representing them as combinations of primitives. However, on fixed grids, computing the gradient of a user objective via the adjoint variables method requires a product of forward/adjoint field solutions and the Jacobian of the simulation material distribution with respect to the structural shape parameters. Shape parameters often perturb global parts of the simulation grid resulting in many non‐zero Jacobian entries. These are often computed by finite‐difference (FD) in practice, and hence can be non‐trivial. In this work, the gradient calculation is accelerated by invoking automatic differentiation (AD) in instantiations of structural material distributions, enabled by the development of extensible differentiable feature‐mappings from parameters to primitives and differentiable effective logic operations (denoted AutoDiffGeo or ADG). ADG can also be used to accelerate FD‐based shape optimization by efficient boundary selection. AD‐enhanced shape optimization is demonstrated using three integrated photonic examples: a blazed grating coupler, a waveguide transition taper, and a polarization‐splitting grating coupler. The accelerations of the gradient calculation by AD relative to FD with boundary selection exceed 10×$\times$, resulting in total optimization wall time accelerations of 1.4×$1.4\times$–3.8×$3.8\times$ on the same hardware with no compromise to device figure‐of‐merit. [ABSTRACT FROM AUTHOR]

In this article, we present the design and experimental validation of a labyrinthine metamaterial for vibro-acoustic applications. Based on a two-dimensional unit cell, different designs of finite-size metamaterial specimens in a sandwich configuration including two plates are proposed. The design phase includes an optimization based on Bloch-Floquet analysis with the aims of maximizing the band gap and extruding the specimens in the third dimension while keeping the absorption properties almost unaffected. By manufacturing and experimentally testing finite-sized specimens, we assess their capacity to mitigate vibrations in vibro-impact tests. The experiments confirm a band gap in the low- to mid-frequency range. Numerical models are employed to validate the experiments and to examine additional vibro-acoustic load cases. The metamaterial's performances are compared with benchmark solutions, usually employed for noise and vibration mitigation, showing a comparable efficacy in the band gap region. To eventually improve the metamaterial's performance, we optimize its interaction with the air and test different types of connections between the metamaterial and the homogeneous plates. This finally leads to metamaterial samples largely exceeding the benchmark performances in the band gap region and reveals the potential of interfaces for performance optimization of composed structures. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'. [ABSTRACT FROM AUTHOR]

The aerodynamic drag of high-speed trains has a negative effect on their running stability and energy efficiency. Since the shape of the high-speed train head closely influences its surrounding airflow, optimizing the head shape is the primary way to reduce the aerodynamic drag. However, existing optimization methods have limitations in parametrically describing the train head with enough details and fewer parameters. In this paper, we propose a novel parametric modeling method based on the approximate analytical partial differential equation (PDE) for the aerodynamic shape optimization of high-speed train heads. With this method, the detailed shape of the train head is controlled by four design parameters. To enhance the optimization efficiency, a two-stage multilayer perceptron (MLP) surrogate model is proposed to predict the aerodynamic drag coefficients of the high-speed train, and a classic genetic algorithm (GA) is adopted to optimize the total drag coefficient and generate the train head shape with good aerodynamic performance. The effectiveness of the proposed method is demonstrated through several comparison experiments. [ABSTRACT FROM AUTHOR]

The reduction of wheel-rail wear is a fundamental task in railway engineering that significantly affects the operating performance in the lifecycle. To improve the dynamic response and profile wear evolution performance of wheel-rail interaction, a shape optimization procedure for the railway wheel profile is proposed. First, the geometry modeling method, which ensures the continuity of first-order derivation of the wheel profile, is introduced to generate a large number of candidate profiles, and multibody dynamics simulation is conducted to analyze the dynamics response of the wheel profiles, including wear index, lateral force, lateral acceleration of the frame and derailment coefficient. Then, the Kriging model is constructed to establish the relationship between the design variables and objectives obtained by multibody dynamics simulation, and particle swarm optimization (PSO) is employed to evaluate the optimal parameters for wheel profile that simultaneously considers wheel wear, stability, and lateral force. Finally, the performance of the wheel-rail interaction is evaluated to demonstrate the effectiveness of the proposed method. The numerical simulation result indicates that the optimized wheel profile not only has good performance, including contact state, pressure, and friction at the design stage, but also the physical performance is acceptable after a long-term profile evolution during service, which the maximum wear depth of the optimal wheel profile averagely decreases over 10 % in long-term wear evolution. [ABSTRACT FROM AUTHOR]

The Vortex-Induced Vibration (VIV) phenomenon can cause fatigue damage. Usually, countermeasures are implemented to mitigate the VIV amplitudes. However, its cost is high. Therefore, approaches have been developed to avoid VIV at an early stage of design. Most such studies have focused on avoiding the coincidence of the vortex shedding frequency with the natural frequency of the structure and delaying VIVs by including geometrical changes to the leading and trailing edges in order to divide the oncoming flow. In this study, a new framework is proposed to find the optimal deck cross-sectional shape of Trans-Tokyo Bay Bridge, based on Reliability-Based Design Optimization (RBDO) and aiming to mitigate the VIV effects under uncertainty. Validation of the optimal designs is conducted via Computational Fluid Dynamics (CFD) simulations for different target reliability indices. [ABSTRACT FROM AUTHOR]

Modern underwater warfare necessitates the development of high-speed supercavitating torpedoes. Achieving supercavitation involves integrating a cavitator at the torpedo's front, making cavitator design a critical research area. The present study simulated supercavity formation by cavitators of various shapes attached to a heavyweight torpedo. The study involves simulations of thirteen cavitator designs with various geometrical configurations at different cavitation numbers. The simulations employ the VOF multiphase model along with the Schnerr and Sauer cavitation model to analyze supercavitation hydrodynamics. The study examines the supercavity geometry and drag characteristics for individual cavitator designs. The results reveal a significant reduction in skin friction drag by a majority of cavitators. Notably, a disc cavitator at a cavitation number of 0.09 demonstrates a remarkable 92% reduction in the coefficient of skin friction drag. However, the overall drag reduces when incorporating a cavitator, but it introduces additional pressure drag. The study found that the cavitators generating larger supercavities also yield higher pressure drag. Therefore, the supercavity should just envelop the entire torpedo, as excessively small supercavities amplify skin friction drag, while overly large ones elevate pressure drag. Ultimately, the study concludes that selecting the ideal cavitator entails a comprehensive evaluation of factors such as supercavity and torpedo geometry, reductions in skin friction drag and increments in pressure drag. [ABSTRACT FROM AUTHOR]

A comprehensive review of recent advancements in applying deep reinforcement learning (DRL) to fluid dynamics problems is presented. Applications in flow control and shape optimization, the primary fields where DRL is currently utilized, are thoroughly examined. Moreover, the review introduces emerging research trends in automation within computational fluid dynamics, a promising field for enhancing the efficiency and reliability of numerical analysis. Emphasis is placed on strategies developed to overcome challenges in applying DRL to complex, real-world engineering problems, such as data efficiency, turbulence, and partial observability. Specifically, the implementations of transfer learning, multi-agent reinforcement learning, and the partially observable Markov decision process are discussed, illustrating how these techniques can provide solutions to such issues. Finally, future research directions that could further advance the integration of DRL in fluid dynamics research are highlighted. [ABSTRACT FROM AUTHOR]

In this work, the lumped model of Rogowski coils is briefly reviewed to illustrate that the reduction in stray capacitance expands the bandwidth and improves the high-frequency performance. Then, a network model as well as explicit formulas for the stray capacitance of Rogowski coils are established, and the influence of geometrical parameters of the skeletons on the stray capacitance is investigated. It is found that the stray capacitance of Rogowski coils is approximately proportional to the perimeter of the skeleton cross-section. Based on the above discussion, optimization of the shape of the skeleton cross-section, aiming at minimizing the perimeter without affecting the magnetic flux and mutual inductance of the coils, is carried out. The widely adopted circular and rectangular cross-sections are discussed first. Then, the cross-section of an arbitrary smooth and convex shape is optimized by solving a constrained variational problem, leading to an explicit equation for the optimal skeleton cross-section of Rogowski coils. Numerical results demonstrate that, compared with the common circular and rectangular skeleton cross-sections, the proposed optimal cross-section exhibits the shortest perimeter and thus the highest upper cutoff frequency under fixed magnetic flux. The optimization method developed by this work can provide a theoretical basis and guidance for the design of Rogowski coils. [ABSTRACT FROM AUTHOR]

With the increase in running speed, the aerodynamic characteristics of high-speed trains have a significant impact on running stability, energy consumption and passenger comfort. Since the shape of the high-speed train head can directly influence the surrounding airflow, optimizing the head shape is the primary way to improve the aerodynamic performance of the train. This paper reviews current research studies on the surrogate-based aerodynamic shape optimization of high-speed train heads, aiming to provide a comprehensive reference for designers to enhance design efficiency and optimization performance. The entire optimization process is divided into four essential steps, and the key optimization technologies in each step are discussed, including parametric modeling, computational fluid dynamics (CFD) simulation, surrogate model and optimization algorithm. By introducing the practical applications of these technologies, we summarize their advantages and disadvantages and suggest four potential research directions for the future. [ABSTRACT FROM AUTHOR]

In the context of shape optimization, this article proposes a numerical method for designing wheel rims as linear elastic structures. Our approach is based on a combination of the classical shape derivative and the Lagrangian method. The compliant shape derivative with respect to the domain variation has been computed by using Céa fast derivation method. With a novel utilization of the gradient shape optimization strategy, we present a numerical scheme for the optimal design of wheel rims. A remeshing technique is implemented to achieve the regularization of shapes. Design applications are examined on a general vehicle in various circumstances to illustrate the efficiency of the presented scheme. The proposed method applies to different wheel designs due to its simplicity and generic nature. [ABSTRACT FROM AUTHOR]

In this paper, we deal with some shape optimization geometrical inverse spectral problems involving the first eigenvalue and eigenfunction of a p-Laplace operator, over a class of open domains with prescribed volume. We first briefly show the existence of the optimal shape design for the L p norm of the eigenfunctions. We carried out the shape derivative calculation of this shape optimization problem using deformation of domains by vector fields. Then we propose a numerical method using lagrangian functional, Hadamard's shape derivative and gradient method to determine the minimizers for this shape optimization problem. We investigate also numerically the problem of minimizing the first eigenvalue of the p-Laplacian-Dirichlet operator with volume-constraint on domains, using constrained and unconstrained shape optimization formulations. The resulting proposed algorithms of the optimization process are based on the inverse power algorithm (Biezuner et al. 2012) and the finite elements method performed to approximate the first eigenvalue and related eigenfunction. Numerical examples and illustrations are provided for different constrained and unconstrained shape optimization formulations and for various cost functionals to show the efficiency and practical suitability of the proposed approach. [ABSTRACT FROM AUTHOR]

Flywheels used for energy storage may be assembled by press fitting or bonding of concentric rotor parts. During operation, each part is subjected to a superposition of stress fields associated with centrifugal loading and with contact at the component interfaces. To capture the highest overall energy storage capacity, design properties must be selected to achieve the maximum moment of inertia, subject to constraints on the stress occurring within each part at the maximum operating speed. This paper proposes a mathematical formulation for parametric shape optimization problems which can be used to design a multi-disk flywheel rotor to maximize the energy storage capacity. It is shown that the energy storage function for each part of the flywheel can be expressed as a product of two fundamental shape-factors: one that accounts for the stress distribution and the other accounts for the mass distribution. These factors are expressed analytically for cases involving hyperbolic disks, leading to a finite dimension shape optimization problem for a multi-disk rotor. Case studies are presented based on the derived analytical solutions that show how the theory can be usefully applied in flywheel design optimization problems. The results from the case study show significant improvements in specific energy compared to previous research results. This advancement is particularly relevant to develop high-performance, cost-effective flywheel systems, offering potential for widespread application in energy storage technologies.

We consider the inverse problem of identifying an unknown portion of the boundary of a two-dimensional body $ \Omega $ by using a pair of Cauchy data $ (f, g) $ on the accessible portion $ \Sigma $ associated with the solution of the advection-diffusion problem – with constant velocity and diffusivity coefficient – denoted as $ u $ in $ \Omega $. We propose to solve the considered inverse problem using shape optimization methods, which are well-suited to this type of problem. In this direction, we demonstrate that the state variable corresponding to the advection-diffusion problem is differentiable with respect to the shape, and we rigorously derive its material derivative. The problem is recast into two different shape optimization formulations, and the corresponding shape derivatives of the cost functions – in boundary integral forms – are obtained by introducing appropriate adjoint systems. The shape gradient information is then used in a gradient-based scheme to approximate a solution to the optimization problems. Numerical results are provided to illustrate the feasibility of the proposed numerical methods in two spatial dimensions. [ABSTRACT FROM AUTHOR]

We introduce a Monte Carlo method for computing derivatives of the solution to a partial differential equation (PDE) with respect to problem parameters (such as domain geometry or boundary conditions). Derivatives can be evaluated at arbitrary points, without performing a global solve or constructing a volumetric grid or mesh. The method is hence well suited to inverse problems with complex geometry, such as PDE-constrained shape optimization. Like other walk on spheres (WoS) algorithms, our method is trivial to parallelize, and is agnostic to boundary representation (meshes, splines, implicit surfaces, etc.), supporting large topological changes. We focus in particular on screened Poisson equations, which model diverse problems from scientific and geometric computing. As in differentiable rendering, we jointly estimate derivatives with respect to all parameters---hence, cost does not grow significantly with parameter count. In practice, even noisy derivative estimates exhibit fast, stable convergence for stochastic gradient-based optimization, as we show through examples from thermal design, shape from diffusion, and computer graphics. [ABSTRACT FROM AUTHOR]

In response to the shortcomings of particle swarm optimization (PSO), such as low execution efficiency and difficulty in overcoming local optima, this paper proposes a multi-strategy PSO method incorporating snow ablation operation (SAO), known as SAO-MPSO. Firstly, Cubic initialization is performed on particles to obtain a good initial environment. Subsequently, SAO and PSO are combined in parallel, and a balanced search mechanism led by multiple sub-populations is devised, significantly improving the search efficiency of overall population. Finally, the degree day method of SAO is introduced, and particles are endowed with memory of environmental changes to prevent premature convergence of PSO, while balancing the exploration and exploitation (ENE) capabilities in later phases. All adaptive parameters are used throughout this method in place of fixed parameters to improve the robustness and adaptability. For a comprehensive analysis of SAO-MPSO, its good ENE ability is verified on CEC 2020 and CEC 2022 and this method is compared with existing improved PSO versions on both test sets. The results show that SAO-MPSO has certain advantages in the comparison of similar improved algorithms. In order to further validate the strength of SAO-MPSO in dealing with nonlinear optimization problems (OPs) with strong constraints, firstly, based on the ball Wang-Ball (BWB) curve, a combined BWB (CBWB) curve is constructed, and a construction method for CBWB curves that satisfy G1 and G2 continuity is derived. Then, with the energy minimization and scale parameters of the CBWB curve as the optimization objective and variables respectively, a shape optimization model that satisfies G2 continuity is established. Finally, three numerical optimization examples based on this model are solved using SAO-MPSO and compared with 10 other methods. The results show that the energy obtained by SAO-MPSO is the smallest, which verifies the effectiveness of this method applied to shape OPs of CBWB curve. [ABSTRACT FROM AUTHOR]

This paper presents a parameterized model to optimize the texture units using an improved particle swarm optimization. The texture units are optimized under three lubrication states: boundary lubrication, mixed lubrication, and hydrodynamic lubrication, by utilizing the Stribeck curve. The optimal texture units can be applied to journal bearings and thrust bearings in different operating conditions to achieve maximum load-carrying capacity. The paper applies the optimal textures under the three lubrication states to the start-up stage of journal bearings to examine the lubrication performance and dynamic characteristics of the textured journal bearing system. The results demonstrate that compared to the untextured journal-bearing system, the introduction of optimal textures improves the tribological performance of the journal-bearing system, and the optimal textured units under hydrodynamic lubrication have the best friction reduction effect. Furthermore, it is found that the introduction of textures has no significant effect on the dynamic characteristics of the journal bearings. [ABSTRACT FROM AUTHOR]

The Rayleigh Conjecture for the bilaplacian consists in showing that the clamped plate with least principal eigenvalue is the ball. The conjecture has been shown to hold in 1995 by Nadirashvili in dimension 2 and by Ashbaugh and Benguria in dimension 3. Since then, the conjecture remains open in dimension d ≥ 4 . In this paper, we contribute to answer this question, and show that the conjecture is true in any dimension as long as some special condition holds on the principal eigenfunction of an optimal shape. This condition regards the mean value of the eigenfunction, asking it to be in some sense minimal. This main result is based on an order reduction principle allowing to convert the initial fourth order linear problem into a second order affine problem, for which the classic machinery of shape optimization and elliptic theory is available. The order reduction principle turns out to be a general tool. In particular, it is used to derive another sufficient condition for the conjecture to hold, which is a second main result. This condition requires the Laplacian of the optimal eigenfunction to have constant normal derivative on the boundary. Besides our main two results, we detail shape derivation tools allowing to prove simplicity for the principal eigenvalue of an optimal shape and to derive optimality conditions. Finally, because our first result involves the principal eigenfunction of a ball, we are led to compute it explicitly. [ABSTRACT FROM AUTHOR]

Modern underwater warfare necessitates the development of high-speed supercavitating torpedoes. Achieving supercavitation involves integrating a cavitator at the torpedo's front, making cavitator design a critical research area. The present study simulated supercavity formation by cavitators of various shapes attached to a heavyweight torpedo. The study involves simulations of thirteen cavitator designs with various geometrical configurations at different cavitation numbers. The simulations employ the VOF multiphase model along with the Schnerr and Sauer cavitation model to analyze supercavitation hydrodynamics. The study examines the supercavity geometry and drag characteristics for individual cavitator designs. The results reveal a significant reduction in skin friction drag by a majority of cavitators. Notably, a disc cavitator at a cavitation number of 0.09 demonstrates a remarkable 92% reduction in the coefficient of skin friction drag. However, the overall drag reduces when incorporating a cavitator, but it introduces additional pressure drag. The study found that the cavitators generating larger supercavities also yield higher pressure drag. Therefore, the supercavity should just envelop the entire torpedo, as excessively small supercavities amplify skin friction drag, while overly large ones elevate pressure drag. Ultimately, the study concludes that selecting the ideal cavitator entails a comprehensive evaluation of factors such as supercavity and torpedo geometry, reductions in skin friction drag and increments in pressure drag.

This article is concerned with the minimization of peak stresses occurring in linear elasticity. We propose to minimize the maximal von Mises stress of the elastic body. This leads to a non-smooth shape functional. We derive the shape derivative and associate it with the Clarke sub-differential. Using a steepest descent algorithm, we present numerical simulations. We compare our results to the usual p -norm regularization and show that our algorithm performs better in the presented tests. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class C4. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'. [ABSTRACT FROM AUTHOR]

An optimized divertor design is crucial to maximize the lifetime of plasma‐facing components and reduce costs of future fusion power plants. Numerical shape optimization could be a powerful tool to obtain improved designs in an automated way. However, it is not trivial to apply due to the need of a field‐aligned and boundary‐fitted grid for simulating the plasma and quantifying the heat load. This paper shows how a grid deformation tool can automate the gridding process while safeguarding the grid quality. Additionally, sensitivities of shape parameters are computed using finite differences and compared to those obtained by remeshing using standard tools such as CARRE2. The plasma and neutral behavior is simulated using the unstructured SOLPS‐ITER code with the latest advanced fluid neutrals model for an ASDEX Upgrade test case. The comparison shows that, contrary to the remeshing strategy, the grid deformation approach yields smoother sensitivities. Furthermore, it is shown that the deformed grids have better mesh quality in terms of poloidal cell size ratio compared to the grid generated with CARRE2, which improves the accuracy of the simulation. This supports the use of the grid deformation tool for automated shape design in future work. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a new formula for the computation of the shape derivative of boundary integral cost functionals using Minkowski deformation of star-shaped domains by convex ones. The formula is expressed by means of the support function of the convex domain. The proof uses some geometrical tools in addition to an analysis of star-shapedness involving gauge functions. Finally, in order to illustrate this result, the formula is applied for solving an optimal shape design problem of minimizing a surface cost functional constrained to elliptic boundary value problem, using the gradient method performed by the finite element approximation. [ABSTRACT FROM AUTHOR]

A novel figure of merit based on the concept of signature curve for cold formed steel (CFS) cross sections is used to improve the structural member's overall behavior regardless of length and boundary conditions. The objective is defined as the area under the signature curve, plus a penalty function that ensures improved performance over standard sections at specified lengths. Charged system search (CSS), a meta-heuristic optimization algorithm, is used to search the design space. End-use and other geometrical constraints suggested by previous studies are considered to arrive at practical cross sections. This includes limiting the fold angles to minimize sharp corners in the optimized cross sections, which might result in residual stresses that diminish axial capacity. Such nonlinear constraints are also taken into account using penalty functions to facilitate integration with the heuristic optimization process. The proposed strategy is examined through a couple of illustrative examples and is shown to yield higher axial capacity at all points when combined with the proper penalization. The optimized cross sections are also analyzed in simple-simple and clamped-clamped boundary conditions showing improved axial capacity compared to the standard lipped-channel sections with the same coil width. [ABSTRACT FROM AUTHOR]

We consider numerical shape optimization of a fluid–structure interaction model. The constrained system involves multiscale coupling of a two-dimensional unsteady Navier–Stokes equation and a one-dimensional ordinary differential equation for fluid flows and structure, respectively. We derive shape gradients for both objective functionals of least-squares type and energy dissipation. The state and adjoint state equations are numerically solved on the time-dependent domains using the Arbitrary-Lagrangian–Eulerian method. Numerical results are presented to illustrate effectiveness of algorithms. [ABSTRACT FROM AUTHOR]

Modern fighter aircraft increasingly need to conjugate aerodynamic performance and low observability. In this paper, we showcase a methodology for a gradient-based bidisciplinary aero-stealth optimization. The shape of the aircraft is parameterized with the help of a CAD modeler, and we optimize it with the SLSQP algorithm. The drag, computed with the help of a RANS method, is used as the aerodynamic criterion. For the stealth criterion, a function is derived from the radar cross-section in a given cone of directions and weighed with a function whose goal is to cancel the electromagnetic intensity in a given direction. Stealth is achieved passively by scattering back the electromagnetic energy away from the radar antenna, and no energy is absorbed by the aircraft, which is considered as a perfect conductor. A Pareto front is identified by varying the weights of the aerodynamic and stealth criteria. The Pareto front allows for an easy identification of the CAD model corresponding to a chosen aero-stealth trade-off. [ABSTRACT FROM AUTHOR]

The knowledge of energy consumption of different machine tool production processes leading to products is necessary for energy labeling of machined parts in the increasingly sustainability-aware world thus the need for better machining energy modeling techniques. The milling process dynamics is complicated thus numerical and averaging techniques are hitherto usually applied in the cutting energy modeling thus limiting decision-making. This work proposes a generalized force-based closed-form model for the milling process cutting energy. To the best of the author's knowledge, the model is the first closed-form cutting energy model for milling which not only applies to the conventional cylindrical milling tools with constant helix angle but also to cylindrical milling tools with any helix angle variation. The demonstrated applications of the proposed model include modeling of milling machine electrical energy consumption, modeling/optimization of milling project energy/efficiency and helix angle optimization for passive reduction of cutting energy. The proposed model is checked with experimentally-verified results in literature. For example, the model agrees with numerically computed cutting energy in literature by absolute error of 0.0320%–0.4025% and modeling of milling machine electrical energy consumption using the proposed model recorded the goodness-of-fit indices of 0.9980 R 2 -value and −0.1271 mean percentage error compared to a published experimental data. A parametric plot and an optimization based on genetic algorithm showed that increase of helix angle increases cutting energy due to increased influence of edge forces, and the effect is more pronounced at higher helix angles. Various potential applications of the presented model are highlighted in the concluding section. [ABSTRACT FROM AUTHOR]

The problem of covering a two-dimensional bounded set with a fixed number of minimum-radius identical disks is studied in the present work. Bounds on the optimal radius are obtained for a certain class of nonsmooth domains, and an asymptotic expansion of the bounds as the number of disks goes to infinity is provided. The proof is based on the approximation of the set to be covered by hexagonal honeycombs and on the thinnest covering property of the regular hexagonal lattice arrangement in the whole plane. The dependence of the optimal radius on the number of disks is also investigated numerically using a shape-optimization approach, and theoretical and numerical convergence rates are compared. An initial point construction strategy is introduced, which, in the context of a multistart method, finds good-quality solutions to the problem under consideration. Extensive numerical experiments with a variety of polygonal regions and regular polygons illustrate the introduced approach. Funding: This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 2013/07375-0, 2016/01860-1, 2018/24293-0, and 2019/25258-7] and Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 302682/2019-8, 303243/2021-0, 304258/2018-0, and 408175/2018-4]. [ABSTRACT FROM AUTHOR]

Structural element shape optimization based on the biological growth method is increasingly used nowadays. This method consists of two main methods: topological optimization (soft kill option—SKO) and shape optimization (computer-aided optimization—CAO). This paper presents the solution procedures for both shape optimization and topological optimization. In applying these methods, first of all, a certain stress norm must be established, where the most appropriate and most used criterion is the equivalent stress according to von Mises. The application of the mentioned optimization methods is illustrated by several examples. The aim was to compare the change in volume or mass and the maximum stress of the structural elements between the different designs: the initial design, the design after topological optimization, and the design after shape optimization. [ABSTRACT FROM AUTHOR]

This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a nonmagnetic bulk dielectric. The shape optimization approach is based on a homogenization theory for time-harmonic Maxwell's equations that describes effective material parameters for the propagation of electromagnetic waves through the metamaterial. The control parameter of the optimization is a deformation field representing the deviation of the microscale geometry from a reference configuration of the cell problem. This allows for describing the homogenized effective permittivity tensor as a function of the deformation field. We show that the underlying deformed cell problem is well-posed and regular. This, in turn, proves that the shape optimization problem is well-posed. In addition, a numerical scheme is formulated that utilizes an adjoint formulation with either gradient descent or BFGS as optimization algorithms. The developed algorithm is tested numerically on a number of prototypical shape optimization problems with a prescribed effective permittivity tensor as the target. [ABSTRACT FROM AUTHOR]

Mechanical periodic structures exhibit unusual dynamic behavior thanks to the periodicity of their structures, which can be attributed to their cellular arrangement. The source of this periodicity may result from periodic variations of material properties within their cells and/or variations in the cell geometry. The authors present the results of their studies on the optimization of physical parameters of a three-dimensional axisymetrical periodic beam in order to obtain the desired vibroacoustic properties. The aim of the optimization process of the unit cell shape was to obtain band gaps of a given width and position in the frequency spectrum. [ABSTRACT FROM AUTHOR]

Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface–dependent kernels. We derive a novel shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques. The code for obtaining the results in this paper is published at (https://github.com/schustermatthias/nlshape). [ABSTRACT FROM AUTHOR]

This study revisits the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It investigates two shape optimization reformulations employing least-squares boundary-data-tracking cost functionals. Firstly, it rigorously addresses the existence of optimal shape solutions, thus filling a gap in the literature. The argumentation utilized in the proof strategy is contingent upon the specific formulation under consideration. Secondly, it demonstrates the ill-posed nature of the two shape optimization formulations by establishing the compactness of the Riesz operator associated with the quadratic shape Hessian corresponding to each cost functional. Lastly, the study employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary. Numerical experiments in two and three dimensions illustrate the numerical procedure relying on Sobolev gradients proposed herein. [ABSTRACT FROM AUTHOR]

This work is devoted to show the efficiency of a new numerical approach in solving geometrical shape optimization problems constrained to partial differential equations, on a family of convex domains. More precisely, we are interested to an improved numerical optimization process based on the new shape derivative formula, using the Minkowski deformation of convex domains, recently established in Boulkhemair and Chakib (J. Convex Anal. 21( n ∘ 1 ), 67–87 2014), Boulkhemair (SIAM J. Control Optim. 55( n ∘ 1 ), 156–171 2017). This last formula allows to express the shape derivative by means of the support function, in contrast to the classical one expressed in term of vector fields Henrot and Pierre 2005, Delfour and Zolésio 2011, Sokolowski and Zolesio 1992. This avoids some of the disadvantages related to the classical shape derivative approach, when one use the finite elements discretization for approximating the auxiliary boundary value problems in shape optimization processes Allaire 2007. So, we investigate here the performance of the proposed shape optimization approach through the numerical resolution of some shape optimization problems constrained to boundary value problems governed by Laplace or Stokes operator. Notably, we carry out a comparative numerical study between its resulting numerical optimization process and the classical one. Finally, we give some numerical results showing the efficiency of the proposed approach and its ability in producing good quality solutions and in providing better accuracy for the optimal solution in less CPU time compared to the classical approach. [ABSTRACT FROM AUTHOR]

The mixed area of a Reuleaux polygon and its symmetric with respect to the origin is expressed in terms of the mixed area of two explicit polygons. This gives a geometric explanation of a classical proof due to Chakerian. Mixed areas and volumes are also used to reformulate the minimization of the volume under constant width constraint as isoperimetric problems. In the two dimensional case, the equivalent formulation is solved, providing another proof of the Blaschke–Lebesgue theorem. In the three dimensional case the proposed relaxed formulation involves the mean width, the area and inclusion constraints. [ABSTRACT FROM AUTHOR]