Optimal Shape Design Problems Research Articles

Curved Domains in Magnetics: A Virtual Element Method Approach for the T.E.A.M. 25 Benchmark Problem

In this paper, we are interested in solving optimal shape design problems. A critical challenge within this framework is generating the mesh of the computational domain at each optimisation step according to the information provided by the minimising functional. To enhance efficiency, we propose a strategy based on the Finite Element Method (FEM) and the Virtual Element Method (VEM). Specifically, we exploit the flexibility of the VEM in dealing with generally shaped polygons, including those with hanging nodes, to update the mesh solely in regions where the shape varies. In the remaining parts of the domain, we employ the FEM, known for its robustness and applicability in such scenarios. We numerically validate the proposed approach on the T.E.A.M. 25 benchmark problem and compare the results obtained with this procedure with those proposed in the literature based solely on the FEM. Moreover, since the T.E.A.M. 25 benchmark problem is also characterised by curved shapes, we utilise the VEM to accurately incorporate these “exact” curves into the discrete solution itself.

Nanoparticles optimal shape designing in the cylindrical coordinate to use for optical devices

Nanoparticles have unique optical properties that can enhance the performance of optical devices. In this regard size and shape of nanoparticles are very important factors that influence these properties. Although there are many different sample shapes of nanoparticles but researchers have not yet identified the best shape to use, especially in general. This article presents a mathematical model to determine the optimal shape of nanoparticles for use in various optical devices. We use the shape-measure method in cylindrical coordinates to solve this problem. This method has many advantages like it is independent of the initial shape and transforming the optimal shape design problem into a simple optimization one, even in strongly nonlinear cases. The use of cylindrical coordinates also makes calculations faster and more straightforward. First the optical properties of nanoparticles is described by using the discrete dipole method; then, simultaneously the optimal surface and image for the optimal shape and size of the nanoparticles are determined. Numerical simulations are also provided to demonstrate the effectiveness and benefits of the new approach.

Optimal Design of Sensors via Geometric Criteria

We consider a convex set Omega and look for the optimal convex sensor omega subset Omega of a given measure that minimizes the maximal distance to the points of Omega . This problem can be written as follows inf{dH(ω,Ω)||ω|=candω⊂Ω},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\inf \\{d^H(\\omega ,\\Omega ) \\ |\\ |\\omega |=c\\ \ ext {and}\\ \\omega \\subset \\Omega \\}, \\end{aligned}$$\\end{document}where cin (0,|Omega |),d^H being the Hausdorff distance. We show that the parametrization via the support functions allows us to formulate the geometric optimal shape design problem as an analytic one. By proving a judicious equivalence result, the shape optimization problem is approximated by a simpler minimization problem of a quadratic function under linear constraints. We then present some numerical results and qualitative properties of the optimal sensors and exhibit an unexpected symmetry breaking phenomenon.

Virtual Element Method and Optimal Shape Design in Magnetics

We propose an innovative technique for dealing with optimal shape design problems that exploits the flexibility of the virtual element method (VEM) in handling meshes with general polygonal and polyhedral elements. The shape synthesis of a magnetic pole is considered as the case study.

A free‐cutting mesh strategy for optimal shape synthesis in magnetics

The authors propose an innovative technique for dealing with optimal shape design problems that exploits the flexibility of the virtual element method in generating meshes composed of general polygonal and polyhedral elements. Virtual element method and finite element method can coexist on the same discretized domain; therefore, the possibility of dealing with hanging nodes and gluing sub-domain meshes is ensured. Accordingly, the shape synthesis of a magnetic pole is considered as the case study. It is shown that the proposed technique is effective in handling the shape variations dictated by an algorithm of evolutionary optimisation.

Adaptive level set method applied to drag minimization problems constrained by Stokes equations

Abstract We present a level set based adaptive mesh method for solving the drag minimization problem of incompressible flow governed by Stokes equations. Shape sensitivity analysis of the cost functional is derived. During the optimization process, two levels of meshes are adopted. A uniform coarse mesh for evolving the level set function is defined over the whole computational domain. The level set function also serves as the refinement indicator with an easy strategy. The coarse mesh that contains the interfaces then further divided into a uniform fine mesh and the computation is mainly near the interfaces. Therefore, the computational cost is significantly reduced compared with the uniform mesh over the whole domain that achieves the same resolution. Above all, the value of the shape derivative on the boundary can be obtained implicitly, which is a very difficult task in classical optimal shape design problems. The overall optimization procedure is presented and verified with benchmark examples.

Surrogate based blended-wing-body underwater glider shape optimization design

In order to improve the design quality and optimization efficiency for blended-wing-body underwater glider(BWBUG) shape design optimization problems, a surrogate-based blended-wing-body underwater glider shape optimization(SBUGSO) framework is proposed. The aim is to maximize the lift to drag ratio(LDR) of BWBUG with the constrain that the displacement volume of the optimal shape is larger than that of the initial shape. The LDR of the optimal BWBUG is improved by 24.32% with acceptable computational resources. The optimization results show that the present SBUGSO framework can efficiently decrease the computational resource, and improve the hydrodynamic performance and loading capacity of BWBUG. Comparing with the other optimization algorithms, SBUGSO framework shows the significant superiority.

Application of the method of fundamental solutions for designing the optimal shape in heat transfer

In this paper, we propose a meshless regularization technique for solving an optimal shape design problem (OSD) which consists of constructing the optimal configuration of a conducting body subject to given boundary conditions to minimize a certain objective function. This problem also can be seen as the problem of building a support for a membrane such that its deflection is as close as possible to 1 in the subset D of the domain. We propose a numerical technique based on the combination of the method of fundamental solutions and application of the Tikhonov’s regularization method to obtain stable solution. Numerical experiments while solving several test examples are included to show the applicability of the proposed method for obtaining the satisfactory results.

Designing the Optimal Shape of a Nozzle by the Method of Fundamental Solutions

In this paper, we propose a numerical technique based on the method of fundamental solutions (MFS) for solving a classical optimal shape design problem. The problem contains a free boundary condition which should be approximated to find the optimal domain for the solution of Laplace equation. For solving the considered optimization problem, we introduce a meshless regularization technique based on the combination of the MFS and application of the Tikhonov’s regularization method and reduce the problem to solve a system of nonlinear equations. A brief sensitivity analysis on model parameters including the position and the size of the subregion D as well the error with boundary conditions is discussed. Numerical simulations while solving several test examples are presented to show the applicability of the proposed method in obtaining satisfactory results.

On a numerical shape optimization approach for a class of free boundary problems

This paper is devoted to a numerical method for the approximation of a class of free boundary problems of Bernoulli’s type, reformulated as optimal shape design problems with appropriate shape functionals. We show the existence of the shape derivative of the cost functional on a class of admissible domains and compute its shape derivative by using the formula proposed in Boulkhemair (SIAM J Control Optim 55(1):156–171, 2017) and Boulkhemair and Chakib (J Convex Anal 21(1):67–87, 2014), that is, by means of support functions. On the numerical level, this allows us to avoid the tedious computations of the method based on vector fields. A gradient method combined with a boundary element method is performed for the approximation of this problem, in order to overcome the re-meshing task required by the finite element method. Finally, we present some numerical results and simulations concerning practical applications, showing the effectiveness of the proposed approach.

Optimal thickness distribution of stepped nonlinear Gao beam

We present an optimal shape design problem of a transversely and axially loaded nonlinear stepped elastic beam with piecewise constant thickness distribution. The nonlinear beam model proposed by D.Y. Gao in 1996 is considered for modelling the state problem. The objective of the optimization is to find the thickness distribution to maximize the stiffness of the beam subjected to geometrical constraints, namely the preservation of volume is imposed. The stiffness is characterized by the compliance cost functional. The existence of solutions of the optimization problem and convergence properties of discretization using finite element approach is analysed. The theory is illustrated by numerical results.

Shape Design Optimization of a Robot Arm Using a Surrogate-Based Evolutionary Approach

In the design optimization of robot arms, the use of simulation technologies for modeling and optimizing the objective functions is still challenging. The difficulty is not only associated with the large computational cost of high-fidelity structural simulations but also linked to the reasonable compromise between the multiple conflicting objectives of robot arms. In this paper we propose a surrogate-based evolutionary optimization (SBEO) method via a global optimization approach, which incorporates the response surface method (RSM) and multi-objective evolutionary algorithm by decomposition (the differential evolution (DE ) variant) (MOEA/D-DE) to tackle the shape design optimization problem of robot arms for achieving high speed performance. The computer-aided engineering (CAE) tools such as CAE solvers, computer-aided design (CAD) Inventor, and finite element method (FEM) ANSYS are first used to produce the design and assess the performance of the robot arm. The surrogate model constructed on the basis of Box–Behnken design is then used in the MOEA/D-DE, which includes the process of selection, recombination, and mutation, to optimize the robot arm. The performance of the optimized robot arm is compared with the baseline one to validate the correctness and effectiveness of the proposed method. The results obtained for the adopted example show that the proposed method can not only significantly improve the robot arm performance and save computational cost but may also be deployed to solve other complex design optimization problems.

The Harris hawks optimization algorithm, salp swarm algorithm, grasshopper optimization algorithm and dragonfly algorithm for structural design optimization of vehicle components

Abstract There is a growing interest in designing lightweight and low-cost vehicles. In this research, the Harris hawks optimization algorithm (the HHO), the salp swarm algorithm (SSA), the grasshopper optimization algorithm(GOA), and the dragonfly algorithm (DA) are introduced to solve shape optimization problems in the automotive industry. This research is the first application of the HHO, the SSA, the GOA, and the DA to shape design optimization problems in the literature. In this paper, the HHO, the SSA, and the DA algorithms are used for shape optimization of a vehicle brake pedal to prove how the HHO, the SSA, the GOA, and the DA can be used for solving shape optimization problems. The results show the ability of the HHO, the SSA, the GOA, and the DA to design better optimal components.

Design and Optimization of a Blended-Wing-Body Underwater Glider

The shape of blended-wing-body underwater glider (BWBUG) is an important factor in determining the hydrodynamic efficiency. In order to reduce the needed computation time and efforts during the search of the optimum shape, a new surrogate-based shape optimization framework is proposed to solve the complicated BWBUG shape design optimization problem in this paper. During the search, seven baseline airfoils are used to build the parametric geometric model of the BWBUG, with the planar surface being fixed. Moreover, a newly proposed ensemble of surrogates based global optimization algorithm using a hierarchical design space reduction method (ESGO-HSR) is employed to optimize each baseline airfoil. Then, the optimum shape of the BWBUG can be determined and rebuilt based on all baseline airfoils that are successful optimized. As the optimization target, the maximum lift to drag ratio of the initial design is increased by 13.72% under the given operating conditions. The results demonstrate that the presented surrogate-based shape optimization framework is efficient and capable in identifying the optimum shape of the BWBUG.

Research on Evolutionary Level Set Method and Gaussian Mixture Model Based Target Shape Design Optimization Problem

Evolutionary algorithms (EAs) have been successfully used in solving many design optimization problems. However, it generally requires expensive computational resources to tune the EA hyper-parameters and validate the performance. Since a single simulation takes a long time and multiple iterations are required, using EAs to real-world design structure optimization is becoming extremely time-consuming. Therefore, the target shape design optimization problem (TSDOP) has been proposed as a miniature model to replace real-world complex problems. There are three major components in developing EAs for TSDOP, i.e., a shape representation, a fitness evaluation, and an evolutionary strategy. For the shape representation, spline-based methods are frequently used in the research community. However, as their flexibility is dramatically limited by the fixed topology, they have difficulty in representing discontinuous shapes without adding any adjustment strategies. In addition, the spline-based methods will easily generate self-intersection (loop) problem that always increases the search difficulty and reduces the convergence speed during the evolutionary iterations. Therefore, in this paper, we first propose a level-set method integrated with a Gaussian mixture model (GMMLSM) as a shape representation method to overcome the fixed topology and loop problem in the existing spline-based methods. We also propose an improved chaotic evolution for the GMMLSM shape representation, namely, GMMLSM-CE, which integrates the ergodicity from the chaotic system and the good robustness from differential evolution (DE). To evaluate the efficiency and performance of the proposed GMMLSM-CE, experiments on two target shapes with three different EAs are conducted. The empirical results show that: 1) GMMLSM has the ability to represent continuous and discontinuous shapes and can naturally avoid self-intersection (loop) problem; 2) CE has a good performance for parameter tuning, and; 3) GMMLSM-CE has a good representation accuracy and fast convergence speed in terms of solving the TSDOP.

Designing polymer spin packs by tailored shape optimization techniques

We consider optimal shape design problems for polymer spin packs which are widely used in the production of synthetic fibers and nonwoven materials. The design goal is the minimization of the residence time of the polymer, which can be achieved by adjusting the wall shear stress along the boundary. Depending on the specific industrial setting we construct two tailored algorithms. First, we consider the design in three spatial dimensions based on a PDE constrained shape optimization problem. Here, the constraint is given by the Stokes flow. Second, we change the design goal and want to construct shapes in two spatial dimensions which allow for a lower bound on the wall shear stress. This can be incorporated as an additional state constraint. By relaxing this condition and employing the method of mapping we can pull-back the problem onto a fixed reference domain. We get an elegant formulation of this state constrained optimization problem, in which geometric constraints on the boundary can also be included. After discretization we end up with a large-scale NLP which can be handled by existing solvers. Finally, we present numerical results underlining the feasibility of our approach.

Combined size and shape optimization of truss structures using subset simulation optimization

This paper presents an approach to solve the combined size and shape design optimization problems using recently developed subset simulation optimization for both continuous and discrete design variables. Except for the componentwise Metropolis–Hasting algorithm, a recently developed adaptive conditional sampling algorithm is also employed as an alternative approach for generating new conditional samples (candidate designs) for each simulation level, which enhances the accuracy and stability of the optimization process. Besides, a double-criterion sorting algorithm is used to handle the design constraints and integrate them in the generation of conditional samples during the Markov Chain Monte Carlo simulation, and the inverse transform method is employed to deal with the discrete design variables. Totally, four numerical examples are considered, including a 15-bar 2D truss, an 18-bar 2D truss, a 39-bar 3D truss and a truss-type landing gear of an unmanned aerial vehicle. The optimal designs obtained from subset simulation optimization using either the componentwise Metropolis–Hasting algorithm or the adaptive conditional sampling algorithm succeed in substantially reducing the weights of the truss-type structures under design constraints in terms of the member stress, the Euler buckling and the nodal displacement. The computational results indicate the proposed method can be taken as an alternative tool for structural optimization design on truss structures when involving the combined size and shape design.

Optimal Design Problems for the First p-Fractional Eigenvalue with Mixed Boundary Conditions

Abstract In this paper, we study an optimal shape design problem for the first eigenvalue of the fractional p-Laplacian with mixed boundary conditions. The optimization variable is the set where the Dirichlet condition is imposed (that is restricted to have measure equal to a prescribed quantity α). We show existence of an optimal design and analyze the asymptotic behavior when the fractional parameter s converges to 1, and thus obtain asymptotic bounds that are independent of α.

A nonlinear eigenvalue optimization problem: Optimal potential functions

In this paper we study the following optimal shape design problem: Given an open connected set Ω⊂RN and a positive number A∈(0,|Ω|), find a measurable subset D⊂Ω with |D|=A such that the minimal eigenvalue of −div(ζ(λ,x)∇u)+αχDu=λu in Ω, u=0 on ∂Ω, is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution and we determine some qualitative aspects of the optimal configurations. For instance, we can get a nearly optimal set which is an approximation of the minimizer in ultra-high contrast regime. A numerical algorithm is proposed to obtain an approximate description of the optimizer.

Micro biogeography‐inspired multi‐objective optimisation for industrial electromagnetic design

In this Letter, the micro biogeography-inspired multi-objective optimisation (μBiMO) method, an extension of the biogeography-inspired multi-objective optimisation (BiMO) algorithm, is proposed. It is suitable for solving multi-objective optimisations in the field of industrial design because it is based on a small number of islands (hence the name μBiMO), i.e. few objective function calls are required. An analytical test function is used for evaluating the performances of the standard BiMO method and to show the need for the proposed μBiMO algorithm. To test its performances in industrial applications, an optimal shape design problem in electromagnetics, i.e. an electromagnet optimal design, has been chosen as a case study. To the best knowledge of the author, the main element of novelty of the Letter is that the BiMO algorithm has been applied for the first time to a multi-objective design with a reduced number of islands (μBiMO). The algorithm so modified leads to a strong reduction of computational time, and, at the same time, improves the quality of results.