Abstract: Preliminaries. Abstract Setting of the Optimal Shape Design Problem and Its Approximation. Optimal Shape Design of Systems Governed by a Unilateral Boundary Value State Problem the Scalar Case. Approximation of the Optimal Shape Design Problems by Finite Elements the Scalar Case. Numerical Realization of Optimal Shape Design Problems Associated with a Unilateral Boundary Value Problem the Scalar Case. Shape optimization in Unilateral Boundary Value Problems with a "Flux" Cost Functional. Optimal Shape Design Contact Problems the Elastic Case. Shape Optimization of Materially Non-linear Bodies in Contact. Shape Optimization in Problems with Inner Obstacles. Optimum Composite Material Design. Topology Optimization in Unilateral Problems. Appendices. Bibliography. Index.

Abstract: Shape design of a structure plays a significant part in deciding its least weight. In this paper, a fuzzy controlled genetic-based search technique for structural shape optimization is investigated. An automated optimal procedure based on the proposed approach is developed and used in the least-weight design of truss structures, which include their geometry as a design variable to be optimized. To increase the performance of the genetic-based approach for shape optimization problems, the design constraints related to member stress, joint displacement, and member buckling are described by using fuzzy set theory. A fuzzy rule-based system representing expert knowledge and experience is incorporated in the approach to control its optimal search process. Four examples for shape designs are presented to demonstrate the effectiveness and efficiency of the proposed hybrid approach in comparison with the use of pure genetic algorithms and other numerical methods. The examples show that the approach is flexible enough to deal with rigidly jointed structures.

Abstract: We suggest a shape optimization method for a non-linear and non-steady-state metal forming problem. It consists in optimizing the initial shape of the part as well as the shape of the preform tool during a two-step forging operation, for which the shape of the second operation is known. Shapes are described using spline functions and optimal parameter values of the splines are searched in order to produce, at the end of the forging sequence, a part with a prescribed geometric accuracy, optimal metallurgical properties and for a minimal production cost. The finite element method, including numerous remeshing operations, is used for the simulation of the process. We suggest using a least-squares-type algorithm for the unconstrained optimization method (based on external penalty) for which we describe the calculation of the derivatives of the objective function. We show that it can reduce to calculations which are equivalent to the derivative calculations of steady-state processes and to evolution equations. Therefore, the computational cost of such an optimization is quite reasonable, even for complex forging processes. Lastly, in order to reduce the errors due to the numerous remeshings during the simulation, we introduce error estimation and adaptive remeshing methods with respect to the calculation of derivatives.

Abstract: This paper is the second part of a two-part article about shape optimization of metal forming processes. This part is focused on numerical applications of the optimization method which has been described in the first paper. The main feature of this work is the analytical calculations of the derivatives of the objective function for a non-linear, non-steady-state problem with large deformations. The calculations are based on the differentiation of the discrete objective function and on the differentiation of the discrete equations of the forging problem. Our aim here is to show the feasibility and the efficiency of such a method with numerical examples. We recall the formulation and the resolution of the direct problem of hot axisymmetrical forging. Then, a first type of shape optimization problem is considered: the optimization of the shape of the initial part for a one-step forging operation. Two academic problems allow for checking the accuracy of the analytical derivatives, and for studying the convergence rate of the optimization procedure. Both constrained and unconstrained problems are considered. Afterwards, a second type of inverse problem of design is considered: the shape optimization of the preforming tool, for a two-step forging process. A satisfactory shape is obtained after few iterations of the optimization procedure.

Abstract: In this paper, a method is proposed for the design optimization of structural components where both shape and topology are optimized. The boundaries of the shape of the structure are represented using contours of a shape density function. The contour of the density function corresponding to a threshold value is defined as the boundary of the shape. The shape density function is defined over a feasible domain and is represented by a continuous piece-wise interpolation over the finite elements used for structural analysis. The values of the density function at the nodes serve as the design variables of the optimization problem. The advantage of this shape representation is that both shape and topology of the structure can be modified and optimized by the optimization algorithm. Unlike previous methods for shape and topology optimization, the material is not modeled as porous or composite using the homogenization method. Instead the material properties of the structure are assumed to depend on the density function and many approximate material property-density relations have been studied. The shape and topology of structural components are optimized with the objective of minimizing the compliance subject to a constraint on the total mass of the structure.

Abstract: Focuses on the inverse problems arising from the simulation of forming processes. Considers two sets of problems: parameter identification and shape optimization. Both are solved using an optimization method for the minimization of a suitable objective function. The convergence and convergence rate of the method depend on the accuracy of the derivatives of this function. The sensitivity analysis is based on a discrete approach, e.g. the differentiation of the discrete problem equations. Describes the method for non‐linear, non‐steady‐state‐forming problems involving contact evolution. First, it is applied to the parameter identification and to the torsion test. It shows good convergence properties and proves to be very efficient for the identification of the material behaviour. Then, it is applied to the tool shape optimization in forging for a two‐step process. A few iterations of the inverse method make it possible to suggest a suitable shape for the preforming tools.

Abstract: Investigates the efficiency of hybrid solution methods when incorporated into large‐scale topology and shape optimization problems and to demonstrate their influence on the overall performance of the optimization algorithms. Implements three innovative solution methods based on the preconditioned conjugate gradient (PCG) and Lanczos algorithms. The first method is a PCG algorithm with a preconditioner resulted from a complete or an incomplete Cholesky factorization, the second is a PCG algorithm in which a truncated Neumann series expansion is used as preconditioner, and the third is a preconditioned Lanczos algorithm properly modified to treat multiple right‐hand sides. The numerical tests presented demonstrate the computational advantages of the proposed methods which become more pronounced in large‐scale and/or computationally intensive optimization problems.

Abstract: Developed is an approach whereby aerodynamic shape sensitivity analysis and design optimization are pedormed on three-dimensional unstructured meshes. The advantage of unstructured grids (when compared with a structured-grid approach) is their inherent ability to discretize irregularly shaped domains with greater eficiency and less effort. Hence, this approach is ideally suited for geometrically complex configurations of practical interest. The nonlinear Euler equations are solved using a fullyimplicit, upwind, cell-centered, finite-volume scheme. The discrete, linearized systems which result from this scheme are solved iteratively by a preconditioned conjugate-gradient-like algorithm known as GMRES; a similar procedure is also used to solve the accompanying linear aerodynamic sensitivity equalions in incremental iterative form. As shown, this particular form of the sensitivity equations makes large-scale gradient-based aerodynamic optimization possible by taking advantage of memory eSJicient methods to construct exact matrix-vector products. Wing-planform parameterization is trccornplished via scaling and translation factors at pre-selected locations along the wing span, then linearly varying these factors between locations. Once the surface has been deformed, the unstructured grid is adapted by considering the mesh as a system of interconnected springs. Grid sensitivities are obtained by differentiating the surface parameterization and the grid adaptation algorithms with ADIFOR (which is an advanced automatic-diferentiation software tool). To evaluate this shape optimization procedure, the planform shape of an initially rectangular wing with uniform NACA-0012 cross-sections is optimized in a compressible, inviscid flow. 1. Introduct ion As recently noted by Reuther et al. [ 11 “while flow analysis has maturecl to the extent that Navier-Stokes calculations are routinely carried out over very complex configurations, direct CFD based design is only just beginning to be used in the treatment of moderately complex three-dimensional coizfgurations”. This is primarily due to the fact that to generate a single structured grid about such a configuration is difficult, if not impossible. Thus, to handle geometry of practical interest, some sort of domain decomposition scheme must be incorporated into the design code. For structured grid solvers, these techniques would include multiblocked, zonally patched, and overlapped (sometimes referred to as Chimera) grid algorithms. However, as the geometric flexibility of the method increases, so does the complexity of the underlying algorithm. Since the use of sensitivity analysis, to evaluate the needed gradients for a numerical optimizer, is still evolving, little work has been done toward extending these algorithms to include these domain decomposition methods. The research which has been accomplished has mostly concentrated on the use of niultiblocked grids. On this, Reuther et al. [ I ] have developed a multiblock-multigrid adjoint solver (“variational” or “control theory” approach [ 2 ] ) which was applied for the wing redesign of a transonic business jet. Eleshaky and Baysal [3] developed a multiblock “discrete” adjoint solver which was applied to a simple axisynlmetric nozzle near a flat plat. As for the use of the more advanced domain decomposition methods (zonal and overlapped grids), and combinations of the three various types, Taylor [4] has differentiated an advanced flow-analysis code to perform the discrete sensitivity analysis. * Graduate Research Assistant. Student Member, AIM # Associate Professor. Member, AIAA. Copyright

Abstract: This paper presents validated results of the optimization of cutouts in laminated carbon-fibre composite panels by adapting a recently developed optimization procedure known as Evolutionary Structural Optimization (ESO). An initial small cutout was introduced into each finite element model and elements were removed from around this cutout based on a predefined rejection criterion. In the examples presented, the limiting ply within each plate element around the cutout was determined based on the Tsai-Hill failure index. Plates with values below the product of the average Tsai-Hill number and a rejection ratio (RR) were subsequently removed. This process was iterated until a steady state was reached and the RR was then incremented by an evolutionary rate (ER). The above steps were repeated until a cutout of a desired area was achieved.

Abstract: The authors study the shape optimization of a complex cracked shell under complex criteria. The shell is one of various cases of a turboshaft, and optimization criteria are associated to the cost, the technology, and above all the working conditions for the turboshaft. The optimization criteria involved are of course the weight of the structure, but also the plastic instability and critical stress intensity factor. All computations have been made with the Ansys finite element program in which an optimization module exists.

Abstract: Studies the effect of parameters controlling the biological growth method by applying it to the classical optimization problem of a plate with a central hole under biaxial stress state. It has been found that the optimization character of the method depends strongly on the so‐called reference stress. Depending on the magnitude of this parameter either a local or global optimum is approached. A global optimum corresponds to the minimum possible v. Mises stress along the hole boundary (and hence in the plate), whereas a local optimum presents the modified shape of the hole yielding an uniform stress distribution whose magnitude is larger than the minimum possible value and which is equal to the specified reference stress. The magnification factor applied to the iterative displacement results influences the optimization speed. Too large factors lead to divergence of the solution. Furthermore, it has been found that the dimension of the optimization domain has a critical effect on the optimization result.

Abstract: A survey of problems of optimum design of structures and materials is presented with the main emphasis on fundamental aspects and on current methods and capabilities for topology and shape optimization.

Abstract: In this work we develop a new approach to designing curves and free-form surfaces on a computer. It is inspired by a style of pencil-and-paper design used for sculptured surfaces, in which the designer specifies the shapes of important curves (character lines) and indicates surfaces that pass through them smoothly, with no unnecessary bulges or wiggles (that is, the surfaces are fair). Unlike previous modeling approaches based on the notion of character lines, this approach allows surfaces to be cut apart and smoothly joined along arbitrary curves, so that the designer can build up complex shapes and topologies from simpler ones. Further, the surfaces are infinitely stretchy, so that the designer may add unlimited amounts of detail simply by indicating more control points and curves. Finally, portions of the surface may be made to copy externally controlled shape tools. This allows the designer to mix free-form and structured shapes within a single composite surface model of arbitrary topology. This kind of conceptually simple shape description ("give me a fair surface bordered by these curves that passes through those curves while touching that point") may be precisely interpreted as a functional minimization problem in the calculus of variations ("give me the surface coordinate function that maximizes this fairness integral subject to those geometric constraints"). The modeler described here represents curves and surfaces implicitly, as the solutions of such variational minimization problems. As the designer interacts directly with a surface, the modeler interprets these actions as changing the variational shape specification. Triangulated point sets are used to approximate these smooth variational surfaces in real time, using a novel finite-difference scheme over arbitrary-topology surface meshes along with an adaptive, interactive mesh refinement and re-triangulation scheme. Ultimately, all of these numerical details are hidden from the designer, who sees a pristine surface that may be grabbed at arbitrary points and along arbitrary curves, and whose shape changes in simple, predictable ways. The resulting ability to design variational shapes of arbitrary, mutable topology has never before been available in an interactive geometric modeler.

Abstract: The aim of this paper is twofold: on the one hand to present a new method for the numerical realization of optimal shape design problems, called a fictitious domain approach and on the other hand to describe the use of a genetic type algorithm in the above mentioned approach.

Abstract: This paper describes an optimal design method by means of the singular value decomposition and fuzzy inference in stabilizing the coefficient matrix We have already reported some optimal design methods with the boundary element method, and shown that an objective shape has been obtained with the iterative calculation that utilizes the least square method or the fuzzy inference method Because most coefficient matrices of optimization methods are ill-posed matrices, the analyses accompany much difficulty We investigate the coefficient matrix, and transform it into a well-posed one by using the singular value decomposition In this method, the ambiguous regularization parameter is determined with fuzzy inference As an example, singular value decomposition is applied to the ill-posed matrix of a synchrotron model for pole shape optimization

Abstract: This paper is devoted to shape optimization for Partial Differential Equations (PDE) systems related to Computational Fluid Dynamics (CFD). Numerical approximation of the PDE's relies on schemes satisfying discrete maximum principles and using unstructured meshes generated from the shape parameters. The theory of control is applied to the discrete design problem with the resulting constrained optimization problem solved by gradient based algorithms. An automatic differentiation procedure for Fortran codes is extensively used to carry out the CFD sensitivity analysis.

Abstract: This paper presents the shape optimization of windings to minimize the eddy current losses. The method used for this optimization combines the finite element analysis of the time harmonic electromagnetic field and gradient-based numerical optimization techniques. The objective of this type of optimization problem is to limit the losses by modifying the shape of the windings.

Abstract: Development of automatic shape design optimization algorithms requires the consideration of convergence issues. One issue is the approximation accuracy of the objective function and its gradient. This is of particular concern in shape optimization problems, where shape changes during the design iterations may drastically effect the flow solution. Thus, the discretization selected for the initial design guess may not be appropriate as the design evolves. It is natural to consider adaptive solvers for these problems. In this work, we consider the use of an adaptive finite element solver in computing flow variables and their sensitivities for incompressible, viscous flows. Introduction and Motivation The mathematical description of optimal design problems requires minimizing a design objective function over a set of admissible designs. In physical systems, the design often satisfies a constraint which is modeled as the solution of a partial differential equation (with appropriate boundary conditions) in the parameter dependent domain. We consider problems where the constraints are given by the two-dimensional Navier-Stokes equations. For most practical problems, the solution to the partial differential equation, and hence the solution to the optimal design problem, needs to be approximated. A survey of many approximation methods can be found in Frank and Shubin [11], We focus on implementation issues for a black-box method. This method uses an approximate-then-optimize approach, where the design objective function is first approximated, then an optimization algorithm is used to find parameters which minimize this approximate objective function. The convergence and efficiency of this method depend on the optimization algorithm used and the approximations used for the objective function (and its gradient). We consider the case where the Navier-Stokes equations are approximated by a finite element method. As the parameters are updated by the optimization algorithm, the original discretization may not be appropriate. This is particularly true when there are large changes in the shape of the domain. Since an automatic design program is desired, where there is no need for human intervention at the intermediate designs, it is natural to consider adaptive mesh refinement techniques in the approximations. These techniques change the discretization according to a measure of the error in the flow approximation. Since the exact flow solution is not known a priori, this error function is estimated using the computed solution [21]. There are three strategies for adaptivity, p-, r-, and h-refmement (and combinations of them). The prefinement relies on increasing the order of those finite elements where the error estimator is predicted to be large. This refinement scheme was considered in the context of structural shape optimization by [18] and [20]. This type of strategy does not readily apply to a mixed finite element formulation, such as that used to approximate the Navier-Stokes equations, due to the need to satisfy the LBB condition [7]. The r-refinement strategy consists of relocating the node points of the elements in order to uniformly distribute the error over all of the elements [14]. The h-refinement strategy subdivides elements in areas where the error is large. It has been Copjrijhl ©I9» by the «mhon. PublMnd by U* American Inslilule of Aeranjouiat and Astronxutlcs. Inc. with permiuioa. 1 American Institute of Aeronautics and Astronautics shown [21] that as the mesh is uniformly refined, that the error estimator will vanish. An h-remeshing adaptive finite element method has been demonstrated for solving complex flow problems, see [15], [16] and the references therein. This type of refinement scheme has been applied to structural shape optimization problems by [1], [8], [10] and [13]. When using a gradient-based optimization algorithm in the black-box method, the gradient needs to be computed efficiently in order to make the method practical. One technique is to use flow sensitivities, the derivative of the flow variables with respect to the design parameters. These sensitivities are commonly computed by differentiating the algorithm used to approximate the flow variables. Programs such as ADDFOR [2] can perform this differentiation automatically. However, since the discretization is parameter dependent, the derivatives of the discretization with respect to the design parameters (mesh sensitivities) need to be determined. For an adaptive solution scheme, these mesh sensitivities are not readily available, however an ad-hoc approach is demonstrated in [8]. We consider an alternate approach based on approximating the sensitivity equation, obtained by differentiating the partial differential equation and its boundary conditions with respect to the design parameters [4]. This equation is approximated efficiently using the same approximation scheme used to approximate the flow. Moreover, since the differentiation is performed before the approximation, there is no need to compute mesh sensitivities. Although the operations of approximation and differentiation don't commute in general, the convergence of the optimization algorithm using these computed sensitivities has been addressed using the notion of asymptotic consistency [3], [5]. In fact, this sensitivity equation technique has been successfully used in conjunction with a finite element approximation of the Navier-Stokes equations for a shape optimization problem [9]. This paper is organized as follows. In the next section, we introduce the sensitivity equations for the NavierStokes equations and the finite element approximation scheme. We then introduce the adaptive refinement strategy. We present two numerical examples, the first is a cylinder submerged in a channel where we demonstrate the accuracy of the sensitivity calculations and the second is a 2D shear layer example which motivates future work. In the following section, we provide our conclusions. Navier-Stokes Sensitivity Equations Navier-Stokes Equations In this paper, we consider optimal design problems where the constraint is described by the two-dimensional Navier-Stokes equations, V a = 0, (1) p (u • Vu) V • r(n) + V/» = F, (2) where p is the (constant) density, u = («, v) is the velocity, P is the pressure, r is the stress tensor defined by with (constant) viscosity fi and F is the external body force. The solution of (l)-(2) satisfies the appropriate Dirichlet or Neuman boundary conditions in the (possibly parameter dependent) domain £2. The finite element equations for the flow are obtained by writing equations (l)-(2) in weak form, i.e. (V • u, u;) = 0 (pu Vu, v) + a(n, v) (P, V • v) = (F, v) , where (-, •) is the usual L inner product, (u, v) = / u • v dfi Jn and a(-, •) is the bilinear form fl(u, v) = / r(u) : Vv di2. JQ These equations are solved in primitive variables using an augmented Lagrangian technique to treat the incompressibility. Discretization is performed with seven node Crouzeix-Raviart triangular elements (which use an enriched quadratic velocity approximation and a discontinuous linear pressure) [7]. The result is a set of nonlinear algebraic equations for the flow. Using Newton's method to solve these equations results in solving a linear system of the form 1 I A + -BM-B -(L(u") 1 AU" = [AH" + BM-'Bu" + N(u")u" f] (3) for the update Au", where the matrices A, B, L, M and N are formed using the current flow solution u". The flow at the next iteration is then obtained using the update = u" -|Au*. (4) American Institute of Aeronautics and Astronautics This process is repeated until the residual of the nonlinear equations satisfies a prescribed tolerance. We denote the final converged solution by p (s • Vu) + p (u • Vs)

Abstract: The objective of this paper is to demonstrate the effectiveness of recently developed multipoint function approximations in the context of structural size, configuration and shape optimization. The developments include approximations built using just two points and also more than two-point information of optimization iterations. Intervening variables are used to control the nonlinearity of the approximations. Several structural optimization problems with stress, displacement and buckling constraints are used to demonstrate the validity and accuracy of the multipoint approximations. These examples include the size optimization of a 40 member frame, the configuration design of a 25-bar space truss and the shape design of a torque arm and a plate with a hole.

Abstract: Publisher Summary This chapter presents two Shape Optimization problems for two dimensional airfoil designs. The first one is a reconstruction problem for an airfoil when the velocity of the flow is known on the surface of airfoil. The second problem is to minimize the shock drag of an airfoil at transonic regime. The flow is modeled by the full potential equations. The discretization of the state equation is done using the finite element method and the resulting non-linear system of equations is solved by using a multi-grid method. The non-linear minimization process corresponding to the shape optimization problems are solved by a parallel implementation of a genetic algorithm (GA). Some numerical experiments are computed on an IBM SP2 parallel computer. In these numerical examples the sequential quadratic programming (SQP) was more accurate and efficient since the problems were quite simple and well tailor made. The chapter provides a comparison of the results from the experiments with those obtained using a gradient based minimization method. It concludes that the designs obtained using the GAs are close to the optimal designs.

Abstract: The perimeter method for variable-topology shape optimization enforces an upper-bound constraint on the perimeter of the solid part of the structure The perimeter constraint ensures a well-posed design problem and allows the designer to control the number of holes in the optimal design and to establish their characteristic length scale Thus single-step procedures for topology design and detailed shape design are possible