Existence and Regularity Results for Some Shape Optimization Problems

This study considers the numerical sensitivity calculation for discontinuous gradient- only optimization problems using the complex-step method. The complex-step method was initially introduced to differentiate analytical functions in the late 1960s, and is based on a Taylor series expansion using a pure imaginary step. The complex-step method is not subject to subtraction errors as with finite difference approaches when computing first order sensitiv- ities and therefore allows for much smaller step sizes that ultimately yields accurate sensitivi- ties. This study investigates the applicability of the complex-step method to numerically com- pute first order sensitivity information for discontinuous optimization problems. An attractive feature of the complex-step approach is that no real difference step is taken as with conven- tional finite difference approaches, since conventional finite differences are problematic when real steps are taken over a discontinuity. We highlight the benefits and disadvantages of the complex-step method in the context of discontinuous gradient-only optimization problems that result from numerically approximated (partial) differential equations. Gradient-only optimization is a recently proposed alternative to mathematical pro- gramming for solving discontinuous optimization problems. Gradient-only optimization was initially proposed to solve shape optimization problems that utilise remeshing (i.e. the mesh topology is allowed to change) between design updates. Here, changes in mesh topology result in abrupt changes in the discretization error of the computed response. These abrupt changes in turn manifests as discontinuities in the numerically computed objective and constraint func- tions of an optimization problem. These discontinuities are in particular problematic when they manifest as local minima. Note that these potential issues are not limited to problems in shape optimization but may be present whenever (partial) differential equations are ap- proximated numerically with non-constant discretization methods e.g. remeshing of spatial domains or automatic time stepping in temporal domains.

Multi-objective free-form optimization for shape and thickness of shell structures with composite materials

Two approaches are proposed for solving inverse problems in shape optimization. We are looking for the unknown position of a small hole in a domain Ω. First, the asymptotic analysis of the underlying p.d.e. defined in a perturbed domain is performed and the so-called topological derivative is defined. Then, in the first approach, the self-adjoint extensions of elliptic operators are used to model the solution of a partial differential equation defined in the singularly perturbed domain. A least-square functional is then minimized to identify the hole. In the second approach, neural networks are used to determine the inverse of the mapping which associates a set of shape functionals to the position of the unknown hole. In both approaches the topological derivatives are used to approximate the shape functionals.

In this paper we consider a shape optimization problem in which the data in the cost functional and in the state equation may change sign, and so no monotonicity assumption is satisfied. Nevertheless, we are able to prove that an optimal domain exists. We also deduce some necessary conditions of optimality for the optimal domain. The results are applied to show the existence of an optimal domain in the case where the cost functional is completely identified, while the right-hand side in the state equation is only known up to a probability $P$ in the space $L^2(D)$.

An improved marine predators algorithm for shape optimization of developable Ball surfaces

This paper is intended to overview on analytical and numerical methods in shape optimization. We compute and analyse the shape Hessian in order to distinguish well‐posed and ill‐posed shape optimization problems. We introduce different discretization techniques of the shape and present existence and convergence results of approximate solutions in case of well posedness. Finally, we survey on the efficient numerical solution of the state equation, including finite and boundary element methods as well as fictitious domain methods. Copyright © 2008 John Wiley & Sons, Ltd.

3R25. Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, Vol 146. - G Allaire (Center of Appl Math, Ecole Polytechnic, Paliseau Cedex, 91128, France). Springer-Verlag, New York. 2002. 456 pp. ISBN 0-387-95298-5. $79.95.Reviewed by MS Qatu (Ford Motor Co, EVB Bldg, MD X8, 20800 Oakwood Blvd, Dearborn MI 48124).This is mainly a research monograph on the theory and application of the homogenization method in shape optimization problems. Applications given in the book are in the fields of composite materials, conductivity, and elasticity with emphasis on the minimization of compliance. The audience for this book includes researchers and advanced graduate students in applied mathematics and engineers interested in the design of shapes of structures for conductive and elastic purposes. It also fits very well for material scientists with inclination towards theoretical aspects of design and analysis of composite materials. Based on the theoretical content and examples in the book, the word shape in the title should rather be understood as topology. In some engineering fields, shape is more related to boundary perturbation, however, the methods presented in this book are designed to define the boundary, rather than to perturb it. This concept of defining the boundary is closer to the idea of topology optimization as a material distribution problem. The book is divided into five chapters. The first chapter covers the subject of homogenization in quite details. Periodicity is first assumed and used to derive homogenized operators in elasticity by means of the two-scale asymptotic expansion. Definitions and proofs of H- and G-convergence are then presented in a rigorous way. The second chapter is titled mathematical modeling of composite materials. In this chapter, Allaire explains carefully the G-closure problem and its interpretations in the mechanical setting. Finding bounds on elastic properties of bi-phase materials is an old problem in mechanics that can be explained and treated mathematically with the help of the G-closure theory. Rank laminated materials are used to demonstrate the bounds derived in the book. Optimal design of materials under electric conductivity considerations is addressed in the third chapter. With an extensive number of proofs, this chapter explains the theory behind the computation of optimal topology (shape) using homogenization theory as a mean to relax the optimization problem. Composite materials are used to relax the problem, and homogenization of conductivity tensors is then used to go from an ill-posed problem to a well-posed one. Structural optimization has been an important subject of research since the days of Lagrange and Saint Venant. This chapter is the counterpart of the previous one, this time, in the field of elastic structures. Composite materials are again introduced to relax the optimization problem and homogenized elastic properties are computed in order to attain a solution to the problem. The minimization of compliance (maximization of stiffness) is addressed intensively in this chapter. The fifth and last chapter is titled Numerical Algorithms. Academic two-dimensional rectangular domains using quadrilateral elements in regular meshes are used to exemplify the theory presented in the previous chapters. Examples include minimization of compliance and maximization of eigenvalues. Multiple loadings and optimum topologies (shapes) of complaint mechanisms were considered. In summary, Shape Optimization by the Homogenization Method is an excellent book that complements very well the extensive literature in the field, including Sanchez-Palencia and Zaoui [1], Bendsøe [2], Cherkaev [3], and Bendsøe and Sigmund [4].

Shape optimization of automotive body frame using an improved genetic algorithm optimizer

In order to decrease the local high stress in the brackets which connect to the horizontal and vertical girders of an internal bulkhead and submersible pressure shell, the mathematical models for the shape optimization of the brackets are proposed. In the study, stress analysis of the pressure hull structure including an internal bulkhead and brackets with coarse mesh is firstly conducted, then the submodeling technique is further employed to analyze the refinement stress distribution of the brackets with refined mesh. The boundary shapes of the brackets are assumed as the design variables while the maximum stress of the bracket is treated as objective function to be minimized in the shape optimization problem. The proposed mathematical model is solved by using analysis code Hyperworks/Optistruct and optimal shapes of the brackets are obtained. Results of the shape optimization show that the optimized bracket types can effectively reduce the level of stress. Therefore, the proposed method can be referred to similar structure designs.

A crucial step for optimizing a system is to formulate the objective function, and part of it concerns the selection of the design parameters. One of the major goals is to achieve a fair trade-off between exploring feasible solutions in the design space and maintaining admissible computational effort. In order to achieve such balance in optimization problems with Computer Aided Engineering (CAE) models, the conventional constructive geometric representations are substituted by deformation methods, e.g. free form deformation, where the position of a few control points might be capable of handling large scale shape modifications. In light of the recent developments in the field of geometric deep learning, autoencoders have risen as a promising alternative for efficiently condensing high-dimensional models into compact representations. In this paper, we present a novel perspective on geometric deep learning modelsby exploring the applicability of the latent space of a point cloud autoencoder in shape optimization problems with evolutionary algorithms. Focusing on engineering applications, a target shape matching optimization is used as a surrogate to the computationally expensive CAE simulations required in engineering optimizations. Through the quality assessment of the solutions achieved in the optimization and further aspects, such as shape feasibility, point cloud autoencoders showed to be consistent and suitable geometric representations for such problems, adding a new perspective on the approaches for handling high-dimensional models to optimization tasks.

A methodology for structural optimisation with damage tolerance constraints

On the choice of source points in the method of fundamental solutions

Shape optimization is concerned about finding optimal designs under the aspect of some cost criteria often involving the solution of a partial differential equation (PDE) over the afore said unknown shape. In general, industrial cases involve a geometric model from Computer Aided Design (CAD). However, solving PDEs requires an analysis suitable working model, typically a Finite Element (FEM) triangulation. Hence, some of the geometric properties known from the CAD model may be lost during this format change. Therefore, we employ isogeometric analysis (IGA) instead, which has a tighter connection between geometry, simulation and shape optimization. In this paper, we present a self-contained treatment of gradient based shape optimization method with isogeometric analysis, focusing on algorithmic and practical aspects like computation of shape gradients in an IGA formulation and updating B-spline and NURBS geometries.

Shape and size optimization problems instructural design are addressed using the particle swarm optimization algorithm (PSOA). In our implementation of the PSOA, the social behaviour of birds is mimicked. Individual birds exchange information about their position, velocity and fitness, and the behaviour of the flock is then influenced to increase the probability of migration to regions of high fitness. New operators in the PSOA, namely the elite velocity and the elite particle, are introduced.

SHAPE ANALYSIS AND OPTIMIZATION IN DISTRIBUTED PARAMETER SYSTEMS