Abstract: Two types of solutions may be considered in generalized shape optimization. Absolute minimum weight solutions, which are rather unpractical, consist of solid, empty and porous regions. In more practical solutions of shape optimization, porous regions are suppressed and only solid and empty regions remain. This note discusses this second class of problems and shows that a solid, isotropic microstructure with an adjustable penalty for intermediate densities is efficient in generating optimal topologies.

Abstract: Shape and layout optimization represent some of the most useful but also most difficult classes of problems in structural design, which have been investigated in detail only during the last few years. Shape optimization is concerned with the optimal shape of boundaries of continua or of interfaces between two materials in composites. Layout optimization deals with the simultaneous optimization of the topology, geometry and cross-sectional sizes of structural systems. In spite of its complextiy, layout optimization is a very rewarding task, because it results in much greater savings than the optimization of cross-sectional sizes only. Because of their important role in shape and layout optimization, the book also covers in detail new optimality criteria methods, which are capable of handling many thousand design variables and active design contraints. Shape and layout optimization is becoming an indispensable tool in the design of aeroplanes, space structures, cars, ships, building and civil engineering structures, power stations, chemical plants, artificial organs, sporting equipment, and all other solid systems where stresses and deformations play an important role.

Abstract: Boundary problems in geochemistry, J. Chadam shape derivatives and differentiability of min max, M.C. Delfour some free boundary problems with industrial applications, A. Fasano problemes de surfaces libres en mecanique des fluides, M. Fortin numerical structural optimization via a relaxed formulation, R.V. Kohn optimal shape design with applications to aerodynamics, O. Pironneau approximation and localization of attractors, K. Promislow and R. Temam shape sensitivity analysis of variational inequalities, J. Sokolowski diffusion with strong absorption, I. Stakgold an introduction to the mathematical theory of the porous medium equation, J.L. Vazquez asymptotic behaviour near extinction points for a semilinear equation with strong absorption, J.J.L. Velazquez introduction to shape optimization problems and free boundary problems, J. -P. Zolesio.

Abstract: This paper presents a constructive solid geometry approach to generic three-dimensional shape optimization. The problem definition and shape control are based on constructive solid geometry whereas the assets of boundary representation are exploited to specify the physics of the problem and for meshing the object. This approach is strongly coupled to an automatic mesh generator and uses to its advantage the explicit association of the finite element data with the model geometry for performing shape sensitivity analysis. Hybrid approximation methods are used to minimize the number of finite element analyses. A classical example of a cantilevered plate with a hole and a realistic aircraft turbine disk problem are solved for optimum shape using this new approach.

Abstract: Introduction T HE technique for associating design variables with mesh data is the most crucial factor in three-dimensional shape optimization. Previously, work in three-dimensional shape optimization involved specifying design variables by associating parameters directly with grid points on an existing mesh. For realistic problems this can be a very tedious (and errorprone) process. In the past, shape optimization capabilities have been developed based on a variety of design/analysis capabilities ranging from associating parameters with a mesh created manually to associating parameters with control points of a mapped mesh generator. Special techniques have also been developed to properly move internal grid points during sensitivity calculations. More recently, a capability based on constructive solid geometry (CSG) has been developed, but CSG representations are not particularly suitable for design optimization. A new optimization approach is demonstrated in this Note that uses a new, fully automatic mesh generation capability. The design model is developed based on design-oriented geometric primitives that represent recognizable features of a part and can be assembled into complete solid models that are defined in terms of a small set of design parameters.

Abstract: The continuity and differentiability of object functions is a basic prerequisite for the application of gradient methods in optimization. However, for parameters defining the shape of an electromagnetic device, the finite element discretization in the field analysis introduces discontinuities into the object function which slow down the convergence rate. Additionally, depending on the geometric parametrizaiion employed, the optimization frequently yields shape contours that are impracticable for manufacturing purposes. This paper investigates the problems inherent in geometric parametrization and shows that the discontinuities in the object function are caused by changes in mesh topology as the geometric parameters vary; these changes inevitably follow from the use of free meshing algorithms. As a solution to these shortcomings a structural mapping technique is outlined that maps surface displacements onto the parameters of the finite element mesh and obtains the parameter dependent geometric variations without a change in mesh topology. This resulting geometric parametrization yields continuous object functions without artificial local minima and results in smooth surface contours of the optimized device. Using this new parametrization technique, design sensitivity analysis, is shown to be a reliable and essential part in the efficient application of gradient methods for shape optimization.

Abstract: Design velocity fields affect every stage of the shape optimization process. The progress of the optimization process, distortion of the finite element mesh, and final shape are sensitive to the quality of velocity fields. It is important to identify and generate effective velocity fields at the beginning of the process. This paper provides several criteria to determine the effectiveness of velocity fields. A systematic approach for generating these velocity fields using deformation fields is developed. The use of interactive procedures is shown to be indispensable for ensuring the effectiveness and quality of design velocity fields. General strategies and guidelines for generating velocity fields are given. Concepts of weight-reducing, stress-reducing, form-preserving, and smooth basis shapes are presented. Normalization of velocity fields is discussed. A method for controlling mesh distortion during the shape optimization process is given based on an explicit limit on the design change to prevent the Jacobian from vanishing. Two- and three-dimensional design problems are solved.

Abstract: In this study, a numerical scheme is developed for shape-sensitivity analysis and design optimization of linear, quasistatic, thermoelastic solids. In this scheme, the finite-element method is used as the analyzer for analyzing stress, temperature, shape sensitivity, and design velocity field. Based upon the method of material derivatives, both the techniques of the direct-differentiation method and the adjoint-variable method are applied to derive the shape-sensitivity equations. The shape-optimization formulations discussed here include boundary integrals of displacements and heat fluxes as well as domain integrals of stresses and areas. Numerical results show that the proposed scheme works well in terms of accuracy.

Abstract: A new feature based shape optimization technique is presented that is capable of modifying the topology (configuration) and shape to reduce the area of 2-D components based on the stress distribution in the component. Shape optimization attempts to maximize material usage to achieve a uniform stress distribution near the allowable limit of the material. Features can be added to the component, or can be modified, in order to optimize the material usage. By using features as a basis for shape modification, the problem of component connectivity can be handled in a consistent, intelligent manner, and the problem of smoothness is eliminated. A program was written to implement the optimization technique and was applied to two example problems, including one from the literature that used a different modification technique. The other example illustrates shape modification capabilities with more complicated geometry. Results from both examples are compared to results obtained using other topological modification techniques.

Abstract: This paper deals with structural shape and thickness optimization of axisymmetric shell structures loaded symmetrically. In the finite element stress analysis use is made of newly developed linear, quadratic, and cubic, variable thickness, C(0) elements based on axisymmetric Mindlin‐Reissner shell theory. An integrated approach is used to carry out the whole shape optimization process in a fully automatic manner. A robust, versatile and flexible mesh generator is incorporated with facilities for generating either uniform or graded meshes, with constant, linear, or cubic variation of thickness, pressure etc. The midsurface geometry and thickness variations of the axisymmetric shell structure are defined using cubic splines passing through certain key points. The design variables are chosen as the coordinates and/or the thickness at the key points. Variable linking procedures are also included. Sensitivity analysis is carried out using either a semi‐analytical method or a global finite difference method. The objective of the optimization is the weight minimization of the structure. Several examples are presented illustrating optimal shapes and thickness distributions for various shells. The changes in the bending, membrane and shear strain energies during the optimization process are also monitored.

Abstract: It is the intention of the present article to point out a new method for computer aided tool optimization as part of computer integrated tool manufacturing. Based on the results of finite element (FE) analysis and subsequent tool failure simulation, it is possible to optimize the FE model of a tool already at the stage of construction, in order to enhance the service life and process reliability. The permissible degree of freedom for any shape correction, of course, is mainly limited by constructive constraints of the tool and the properties of the material flow during the extrusion process. Thus the resulting optimized geometry has to he considered as a possible constructive alternative. However, analytical as well as practical solutions already show that a parabolical or elliptical curved surface contour, replacing a regular radius, not only improves the fatigue resistance but may have a positive influence on material flow behaviour, friction forces and resulting tool loads as well (1).The influence imp...

Abstract: — A cylindrical bar subjected to bending has a rectangular hole for functional reasons. The initial design failed occasionally due to fatigue cracks originating at the corners of the hole under service conditions. Therefore, a shape optimization was performed using CAO (computer-aided optimization) based on the computer-simulation of biological growth. A significant reduction of large notch stresses was achieved. Prototypes manufactured with the shape-optimized design endured 40 times longer in a fatigue test than the previous design. The example demonstrates that the CAO method is a very powerful and straightforward method for the designer who wants a light-weight and fatigue-resistant design.

Abstract: We are concerned with existence results in shape optimization as well as with necessary conditions for optimality. In the first section we give existence results for a weak shape formulation of Bernoulli-like free boundary problems for stationary potential flows. In the second section it is shown how the Bounded Perimeter-constraint can apply to give an existence result for control in the Transient Wave Equation. The third section deals with the very definition of shape deri vatives and with results on the structure of the derivatives. The fourth section deals with the shape variational free boundary problem associated with the Stokes stationary fluid. It underlines that the free boundary condition cannot be achieved in such a linearized modelling. Also, we give existence and continuity results obtained by a penalty approach (via transmission “two-fluid” problems) which apply also to unilateral problems. Finally, the last section extends an existence result for eigenvalues of the Laplace operator.

Abstract: The paper addresses the problem of finding an optimum thickness distribution for a rectangular, isotropic plate of given volume and plan dimensions (length and width) that would maximize its uniaxial buckling load, loosely referred to as shape optimization. Earlier studies suggest that optimal profiles are not only characterized by a concave thickness distribution with higher values near the edges compared to the center, but also by a convex distribution with very high thickness at the center compared with the edges. This paradox regarding the nature of the optimal thickness distribution is the subject of the present investigation. It is established that the qualitative nature of optimal thickness distribution is dependent on the assumptions made regarding the prebuckling loading state, that is, whether the uniaxial stress or force per unit length remains constant. The paper also highlights the fact that shape optimization is seriously limited by local buckling considerations and illustrates the interacti...

Abstract: Optimal shape design problems for an elastic body made from physically nonlinear material are presented. Sensitivity analysis is done by differentiating the discrete equations of equilibrium. Numerical examples are included.

Abstract: This chapter presents a review of mathematical programming methods used in the design of skeletal elastic structures in which the possibility of altering the shape, position or layout of the members is considered. Virtually every type of optimization procedure including linear, non-linear, and dynamic programming has been applied to this design problem. These methods have been implemented using three main approaches. The first, referred to as the ‘ground structure’ approach, is one in which members are removed from a highly connected structure to derive an optimum subset of bars. In the second approach the co-ordinates of the joints of the structure are treated as design variables and moved during the optimization procedure to enable an optimum layout to be designed. The third type of method includes those which allow for topological considerations at certain points during the design process and generally keeps the design variables in two separate groups. The paper discusses the way in which each of the mathematical programming methods has been applied to these approaches.

Abstract: A design sensitivity analysis for optimal shape design of electromagnetic devices is described. The design sensitivity is explicitly derived for two-dimensional electromagnetic systems by using implicit differentiation and direct boundary element methods. The proposed design sensitivity analysis is applied to the optimal shape designs of the yoke of an electromagnet and magnetic pole, and desired distributions of magnetic induction in the air-gap are then obtained. >

Abstract: The introduction of second order sensitivities to behaviour approximation in structural optimization is applied and investigated. Numerical computation of second derivatives is considered and evaluated. Two approaches in shape optimization are involved, namely shape finding by moving boundaries and shape genesis by distributing mass in a specified design domain. A series of examples is engaged to work out the benefits of second order information in both the above fields.

Abstract: A reliability-based shape optimization problem is formulated with the total expected cost as the objective function and requirements for the element or systems reliability measures as constraints. As design variables both sizing and shape variables are used.

Abstract: A three-dimensional shape optimization algorithm is developed by combining design sensitivity analysis and the boundary element method (BEM). The design sensitivity is derived by implicitly differentiating the boundary integral equation with respect to the design variables. The proposed algorithm is validated by applying it to the pole shape optimization of three-dimensional electromagnets. The objective function has been observed to monotonically converge, and the final optimized shape is obtained within a reasonable number of iterations. >

Abstract: Gradient method is applied to the optimal control of a system for which each simulation is expensive. Instead of solving completely the flow equation as a state equation in the shape optimization, we use a "one-shot method" which solves simultaneously the system optimality. It is tested for the problem of shape optimization of a nozzle in a 2D Euler flow.

Abstract: Component shape optimization normally requires a parameterized geometric representation or a generic model for the solid geometry which evolves to an optimal design. Generic models for large-scale three-dimensional components are difficult to build. The difficulties result from the lack of robust automatic mesh generation and the availability of a parametric model. To remedy this problem, a basis function concept used in mathematics for representing an arbitrary function is employed for geometric representation of solids. This approach does not require automatic mesh generation or parametric models for geometric representation and thus is suitable for large-scale complicated components. Numerical examples are used to demonstrate the applicability of this approach to realistic problems.