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

To meet the demand for light-weight structures, many researchers have investigated TPMS (Triply Periodic Minimal Surface) structures for various micro design applications. This study proposes a multi-scale, non-parametric shape and thickness optimization method for TPMS-based micro-shell structures embedded within a macrostructure. The effective properties of these porous micro-shells are evaluated using the NIAH (New Implementation of Asymptotic Homogenization) method. The distributed thickness and shape of the TPMS micro-shell structures are used as design variables. By minimizing the squared error norm between actual and target displacements at arbitrary macrostructure points, the method controls displacement while satisfying a total volume constraint that includes the micro-shell volume. The optimization problem is formulated as a distributed parameter optimization, with sensitivity functions derived using the numerical derivative and adjoint variable methods. These sensitivities are then integrated into a scalar-type H1 gradient method for thickness optimization and a vector-type H1 gradient method for shape optimization. Numerical examples confirm the method’s effectiveness in determining optimal shape and thickness distributions of TPMS-based micro-shell structures to achieve desired displacement control within a macrostructure.

This paper presents design data defining optimal freeform notch shapes for use in tensile loading conditions for a range of geometries, notably the notch aspect ratio and plate widths. Here, an iterative two-dimensional finite element gradientless shape optimization procedure was used to obtain the optimal shapes, which offer the lowest possible stress concentration, subject to the geometric constraints. Precise non-dimensional coordinates are given, for a range of notch aspect ratios and plate widths, and are presented in a tabular form that allows them to be easily used by designers. All cases have been solved with a minimum radius of curvature constraint, which results in a ‘robust’ optimal shape that has no sharp corners. The optimal shapes were found to provide up to a 26 per cent reduction in local peak stress over circular shapes of the same aspect ratio. Optimal geometries are also given for limiting notches that provide a net section stress equal to unity, at a finite notch aspect ratio. The results obtained are applicable to both the initial design of notched components and the shape reworking of existing components to remove damage.

This chapter is the printed matter of a series of lectures on shape optimization given in Udine (Italy) in June 2012. It is mainly focused on the optimal design problem in elasticity.It is well known, since the pioneer work of Tartar and Murat in the late 70’s, that this problem – as well as the scalar shape optimization problem in conductivity – is ill posed and does not always admit solutions. If the goal is to construct numerical methods to address this optimal design problem, this illposedness will lead to algorithms showing instabilities and erratic behaviours.Two ways are possible to slightly change the initial problem so that it may be solved in an efficient and reliable way by numerical methods. The first one consists in enlarging the set of admissible shapes and allow, as solutions, “generalized shapes” that may contain fine mixtures of the initial material and void. This process is called “relaxation” and the main mathematical tool involved is the homogenization theory. It leads to the so called “homogenization method for shape optimization”. It is described in Section 2, both from theoretical and numerical point of views, including practical issues necessary to write a numerical code.The second approach consists in restricting the set of admissible solutions. A few theorems prove the existence of solutions under further geometrical, topological or regularity constraints. The classical domain variation method is revisited using the level set representation, that may be seen as a constraint on the set of admissible solutions. The main tool is here the shape derivative. The level set algorithm is presented in Section 3.Of course this chapter is too short to develop new results and methods and present extensive proofs. Its goal is rather to give an overview on different parts of our work on this topics. Interested readers should check the references cited to have further details.KeywordsTopology OptimizationShape OptimizationHomogenization MethodAdmissible SolutionOptimal Design ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Metamaterials, which exhibit unconventional physical behaviors such as negative Poisson’s ratio and negative thermal expansion (NTE), have garnered significant attention for their potential applications in various engineering fields. However, conventional design methods made it difficult to systematically achieve optimal performance. This study proposes a H¹ gradient method-based shape and size optimization method for lattice structures, aiming at tailoring linear elastic behaviors. This approach treats both the geometric shape (curvature) and the cross-sectional areas of lattice members as continuous design variables. The objective function is formulated as a weighted sum of the squared error between target and actual displacements (corresponding to properties such as Poisson’s ratio or thermal expansion) and the structural compliance. Using adjoint variable methods, sensitivities are derived precisely with regard to both shape and size variations. The proposed method provides a unified and efficient framework for the systematic design of lattice metamaterials exhibiting auxetic and NTE properties.

Motivated by the works on non-gradient techniques in the domain of shape optimization of the structure, the present work intends to suggest a novel non-gradient procedure for shape optimization of structures and compare it to an existing gradient-based method. The presented technique optimizes the shape of structural parts using a fuzzy controlled integrated zero-order methodology incorporating the notion of design elements and automated mesh construction with mesh refinement at each iteration. The movement of nodes and convergence monitoring is taken care of using the triangular fuzzy membership function. The changes in shape occur according to the selected target maximum shear stress (σt) with a view of reaching as near to the target as possible at all the points. The present methodology is packaged in a piece of software termed GSO (Gradientless shape optimization) coded in FORTRAN language. To explain the efficacy of the current approach, a few basic structural shapes have been optimized under various constraints, and the results of the same are compared to those obtained using Optistruct (a part of software suite HyperWorks from Altair engineering), which works on gradient descent method. The proposed approach works well and produces more industry fabricable results than what is produced by the gradient descent method in Optistruct.

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.

A numerical method of structural dynamic shape optimization is presented by using reproducing kernel particle method (RKPM), by which the mesh distortion that exists in shape optimal method based on finite element can be eliminated completely and the optimal model for structural dynamic optimization design is built. The discreteness-based design sensitivity analysis in both natural frequency and dynamic response is proposed by using direct differentiation method and discrete derivatives on the basis of structural dynamic analysis, in which the penalty method is employed into imposing the essential boundary conditions, and the derivatives of shape functions with respect to design variables are derived. The algorithm of dynamic sensitivity analysis is testified by numerical example, and the numerical results obtained are in good agreement with those obtained using semi-analytical method and global finite differences method. Finally, by integrating the algorithm mentioned based on RKPM with parameterized descriptive method of boundary shape, two examples for structural dynamic shape optimization are performed.

In today's society, with an increasing need for energy worldwide and at the same time a growing awareness of its negative consequences, optimizing and reducing energy consumption has become important for sustainability. The process of optimizing the appliances in large industry and household scale contributes to the minimization of the global energy consumption. In order to increase the efficiency of these appliances, various optimizations are employed, such as the shape optimization. In this thesis, a shape optimization method is used to modify the geometry of an existing heat exchanger. The heat exchanger is used in a domestic boiler and the shape optimization is a crucial step for its fast development. Currently, the shape of the heat exchanger is highly dependent on its manufacturing process. Thus, more complex shapes are investigated and their effects on the heat transfer and pressure drop are evaluated in this thesis. Two aspects need to be investigated for this heat exchanger optimization: the complex internal flow within the heat exchanger, and the performance of the optimization method as well as its resulting shapes. Since the heat exchanger consists of different flow regimes due to its complex geometry, a detailed study on these regimes (ReD = 2000, 100, and 10) is conducted prior to the optimization procedure. Furthermore, the complex geometry of this heat exchanger leads to different models used in this thesis. Therefore this thesis is divided into three parts based on the models: the three-dimensional single cylinder, the two-dimensional cylinder arrays, and the three-dimensional cylinder array. In optimizing the heat exchanger, the shape optimization method is justified as the most suitable method. Using the ANSYS Fluent's adjoint shape optimization method, the pressure drop and the heat transfer of the heat exchanger are optimized. This method has been used in the past for optimizing various geometries, yet to the author's knowledge, no detailed study on the cylinder array has been published. Consequently, a study on the adjoint parameters and its effect on the overall performance of the optimization procedure is of importance. Furthermore, best practices and recommendations for future work regarding the adjoint optimization procedure are elaborated.

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.

Polymer extrusion is one of the most widely utilized manufacturing processes across many industries including automotive, architecture, aerospace etc. However, in order to maintain normal operations, polymer extrusion dies are conventionally designed with large dimensions and thick walls which results in the overweight of them. In this paper, a shape optimization method is proposed to reduce the weight of polymer extrusion dies without sacrificing the required performances of extrudate. Firstly, Finite element simulation of the extrusion process is conducted using the commercial software HyperXtrude to study both the essential flow characteristics of polymer melts and the deformation and stress distribution of extrusion die. Secondly, shape optimization is conducted to find the minimum weight of extrusion die while satisfying the required properties and productivity of polymer product. The extrusion die is then redesigned according to the result of shape optimization and compared with the original one. A Medium-sized polymer profile extrusion die is selected as case study, the result of which shows that the weight of the extrusion die is reduced by 31.6%,though the maximal deformation and stress of the die are increased by 1.7% and 16.1% respectively. The proposed approach is demonstrated to be effective for the lightweight design of polymer extrusion die.

GA-based optimization to control the thickness distribution in components manufactured via superplastic forming

Mathematical shape optimization of hip prosthesis design

Dangers inherited in shells optimized with linear assumptions

Free vibration analysis and shape optimization of variable thickness plates and shells—I. Finite element studies

P03-149 - Study on the executive function and learning ability of the first-episode schizophrenia