Abstract: An aerodynamic shape optimization method that treats the design of complex aircraft configurations subject to high fidelity computational fluid dynamics (CFD), geometric constraints and multiple design points is described. The design process will be greatly accelerated through the use of both control theory and distributed memory computer architectures. Control theory is employed to derive the adjoint differential equations whose solution allows for the evaluation of design gradient information at a fraction of the computational cost required by previous design methods. The resulting problem is implemented on parallel distributed memory architectures using a domain decomposition approach, an optimized communication schedule, and the MPI (Message Passing Interface) standard for portability and efficiency. The final result achieves very rapid aerodynamic design based on a higher order CFD method. In order to facilitate the integration of these high fidelity CFD approaches into future multi-disciplinary optimization (NW) applications, new methods must be developed which are capable of simultaneously addressing complex geometries, multiple objective functions, and geometric design constraints. In our earlier studies, we coupled the adjoint based design formulations with unconstrained optimization algorithms and showed that the approach was effective for the aerodynamic design of airfoils, wings, wing-bodies, and complex aircraft configurations. In many of the results presented in these earlier works, geometric constraints were satisfied either by a projection into feasible space or by posing the design space parameterization such that it automatically satisfied constraints. Furthermore, with the exception of reference 9 where the second author initially explored the use of multipoint design in conjunction with adjoint formulations, our earlier works have focused on single point design efforts. Here we demonstrate that the same methodology may be extended to treat complete configuration designs subject to multiple design points and geometric constraints. Examples are presented for both transonic and supersonic configurations ranging from wing alone designs to complex configuration designs involving wing, fuselage, nacelles and pylons.

Abstract: Compliant mechanisms are jointless mechanical devices that take advantage of elastic deformation to achieve a force or motion transformation. An important step toward automated design of compliant mechanisms has been the development of topology optimization techniques. The next logical step is to incorporate size and shape optimization to perform dimensional synthesis of the mechanism while simultaneously considering practical design specifications such as kinematic and stress constraints. An improved objective formulation based on maximizing the energy throughput of a linear static compliant mechanism is developed considering specific force and displacement operational requirements. Parametric finite element beam models are used to perform the size and shape optimization. This technique allows stress constraints to limit the maximum stress in the mechanism. In addition, constraints which restrict the kinematics of the mechanism are successfully applied to the optimization problem. Resulting optimized mechanisms exhibit efficient mechanical transmission and meet kinematic and stress requirements. Several examples are given to demonstrate the effectiveness of the optimization procedure.

Abstract: SUMMARY A multiobjective multidisciplinary design optimization (MDO) of two-dimensional airfoil is presented. In this paper, an approximation for the Pareto set of optimal solutions is obtained by using a genetic algorithm (GA). The first objective function is the drag coefficient. As a constraint it is required that the lift coefficient is above a given value. The CFD analysis solver is based on the finite volume discretization of the inviscid Euler equations. The second objective function is equivalent to the integral of the transverse magnetic radar cross section (RCS) over a given sector. The computational electromagnetics (CEM) wave field analysis requires the solution of a two-dimensional Helmholtz equation which is obtained using a fictitious domain method. Numerical experiments illustrate the above evolutionary methodology on an IBM SP2 parallel computer. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract: In this paper the topological derivative for an arbitrary shape functional is defined. Examples are provided for elliptic equations and the elasticity system in the plane. The topological derivative can be used for solving shape optimization problems in structural mechanics.

Abstract: This paper deals with the constrained reconstruction of 3D geometric models of objects from range data. It describes a new technique of global shape improvement based upon feature positions and geometric constraints. It suggests a general incremental framework whereby constraints can be added and integrated in the model reconstruction process, resulting in an optimal trade-off between minimization of the shape fitting error and the constraint tolerances. After defining sets of constraints for planar and special case quadric surface classes based on feature coincidence, position and shape, the paper shows through application on synthetic model that our scheme is well behaved. The approach is then validated through experiments on different real parts. This work is the first to give such a large framework for the integration of geometric relationships in object modelling. The technique is expected to have a great impact in reverse engineering applications and manufactured object modelling where the majority of parts are designed with intended feature relationships.

Abstract: The form of topological derivatives of arbitrary shape functionals depending on solutions of the three-dimensional Laplace equation is derived. The derivatives can be used for solving shape optimization problems involving diffusion or heat transfer.

Abstract: This paper discusses the use of optimization techniques for structural design. This discussion begins with a brief historical review of this technology from the initial concept to modern approximation techniques. The present state of the art is reviewed and some examples are offered to demonstrate the state of the art. Finally, future needs are addressed to indicate some of the challenges that lie ahead. It is concluded that the state of the art is now reasonably well refined. The challenge now is to assimilate this technology into the practicing design environment.

Abstract: This paper presents a new method, Reverse Adaptivity, for automatically generating solutions to initial design and redesign problems. The method is based on a combination of existing adaptive finite element methods and evolutionary structural optimization methods. The usual difficulties inherent in structural optimization problems, and the shortcomings of the evolutionary methods in tackling these difficulties, are reviewed as a prelude to discussing Reverse Adaptivity. Once the initial finite element problem is defined, the method proceeds with reverse adaptive analysis, which refines low stress regions of the finite element mesh by element subdivision. Following this, any low stress subdivided elements are removed and the process is repeated. With successive decrements of adapted element size, the process satisfies many of the shortcomings of existing evolutionary optimization methods, yet is simple to understand and can be readily implemented. The results produced by the method are superior to those produced by existing methods, yet can be obtained with highly practicable computational resources. As a demonstration, solutions to a number of well- known classical problems are presented, and highlight the method's ability to distinguish new classes of solutions for some problems. Full implementation and parameter details are also presented. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract: This report approaches the question of multi-objective optimization for optimum shape design in aerodynamics. The employed optimizer is a semi-stochas- tic method, more precisely a Genetic Algorithm (GA). GAs are very robust optimization algorithms particularly well suited for problems in which (1) the initialization is not intuitive, (2) the parameters to be optimized are not all of the same type (boolean, integer, real, functionnal), (3) the cost functional may present several local minima, (4) several criteria should be accounted for simultaneously (multiphysics, efficiency, cost, quality, ...). In a multi-objective optimization problem, there is no unique optimal solution but a whole set of potential solutions since in general no solution is optimal w.r.t. all criteria simultaneously ; instead, one identifies a set of non-dominated solutions, referred to as the Pareto optimal front. After making these concepts precise, genetic algorithms are implemented and first tested on academic examples ; then a numerical experimentation is conducted to solve a multi-objective shape optimization problem for the design of an airfoil in Eulerian flow.

Abstract: This paper presents an iterative gradientless method for the shape optimization of stress concentrators, from the context of structural fatigue life extension. The method has been implemented to interface with the finite element code PAFEC, which does not normally have an optimization capability. The key feature of the approach is to achieve constant boundary stresses, in regions of interest, by moving nodes on the stress concentrator boundary by an amount dependent on the sign and magnitude of the local hoop stress obtained from a previous iteration of a standard finite element analysis. The results of example problems are presented which include the optimization of hole shapes in flat plates and the optimization of the design of bonded reinforcements with a focus on minimizing adhesive stress while maintaining the effectiveness of the reinforcement. In all cases, it was found that significant stress reductions were achieved by way of the local shape changes due to the optimization. The method pr...

Abstract: A class of optimal shape design problems is considered where a part of the boundary of the domain represents the free parameter. The variable domain is parametrized by a class of functions in such a way that the optimal design problem results in an optimal control problem on a fixed domain. The functions for the parametrization of the domain are used as controls, and the corresponding states are then given by the solution of an elliptic boundary value problem on a fixed domain. Discretizing this control problem normally leads to a large-scale optimization problem, where the corresponding solution methods are characterized by the requirement of solving many boundary value problems. In spite of this interesting numerical challenge, until now little work has been done to derive more efficient algorithms by taking advantage of the specific structure of this kind of problem. In this report, Newton's method in function space is derived, resulting in an efficient algorithm for the discretized optimization problems. By using the specific structure of these optimal shape design problems, an efficient implementation of the numerical algorithm is introduced. The properties of this algorithm are compared with those of the gradient method using illustrative numerical examples.

Abstract: A practical methodology to optimize electrical devices, based on a combination of numerical simulation and the experimental design method, is illustrated on the shape optimization of a permanent magnet motor. Firstly, significant parameters are identified. Secondly, these parameters are used to improve the performance of the machine.

Abstract: A new approach for optimal shape design based on a CAD-free framework for shape and unstructured mesh deformations, automatic differentiation for the gradient computation and mesh adaption by metric control in 2D is presented. The CAD-free framework is shown to be particularly convenient for optimization when the mesh connectivities and control space size are variable during optimization. Constrained optimization for a transonic regime has been investigated in both 2D and 3D

Abstract: A natural velocity field method for shape optimization of reinforced concrete (RC) flexural members has been demonstrated. The possibility of shape optimization by modifying the shape of an initially rectangular section, in addition to variation of breadth and depth along the length, has been explored. Necessary shape changes have been computed using the sequential quadratic programming (SQP) technique. Genetic algorithm has been used to optimize the diameter and number of main reinforcement bars. A limit-state design approach has been adopted for the nonprismatic RC sections. Such relevant issues as formulation of optimization problem, finite-element modeling, and solution procedure have been described. These design examples—a simply supported beam, a cantilever beam, and a two-span continuous beam, all under uniformly distributed loads—have been optimized. The results show a significant savings (40–56%) in material and cost and also result in aesthetically pleasing structures. This procedure will lead to considerable cost saving, particularly in cases of mass-produced precast members and a heavy cast-in-place member such as a bridge girder.

Abstract: Additive/Subtractive Solid Freeform, Fabrication (SFF) integrates material addition (deposition) and removal (machining) to build up three-dimensional objects incrementally. This class of processes offers sophisticated design flexibility with engineering materials, three-dimensional layer building, and the ability to fabricate complex engineering devices and multi-material objects. However, planning for such processes exhibits rigorous challenges due to process flexibility and highly demanding planning automation. These challenges, moreover, can not be sufficiently tackled via common boundary representation of geometric models. In this thesis, various techniques based on the Medial Axis Transform (MAT) are presented. The medial axis transform encodes intrinsic shape characteristics into a lower dimensional metric. The MAT together with the boundary representation empowers shape manipulation and geometric reasoning. Although numerous algorithms have been proposed to recognize the MAT of polygonal objects, a robust model for arbitrarily shaped regions, especially suitable for engineering designs, is still an art of research. The approaches presented in this thesis utilize representation of the MAT in terms of clearance functions on the object boundary. The clearance functions are computed via a divide-and-conquer methodology. Various planning techniques based on the proposed MAT representation are developed to tackle three process planning problems in additive/subtractive SFF. First, an automated manufacturability analysis approach is presented to assist in evaluation of part build sequences. Such a method allows fast identification of feasible build sequences that permit cutting tool access at all build stages. Second, a shape optimization scheme is developed to compute optimal layer geometry for material deposition. The shape is so optimized that the connected and smooth deposition paths can be generated. Third, an efficient cutter selection strategy is proposed for shaping near-net deposition as well as for bulk material removal. A procedure based on the histograms of shape thickness is presented to efficiently compute an optimal set of cutters for achieving the minimal machining time. The proposed MAT representation and planning approaches apply not only to additive/subtractive solid freeform fabrication but also to various conventional manufacturing processes. The potential of utilizing such techniques for geometric reasoning and process planning is yet to be explored.

Abstract: The topological derivative is introduced for the extremal values of cost functionals for control problems. The optimal control problem considered in the paper is defined for the elliptic equation which models the deflection of an elastic membrane. The derivative measures the sensitivity of the optimal value of the cost with respect to changes in topology. A change in topology means removing a small ball from the interior of the domain of integration. The topological derivative can be used for obtaining the numerical solutions of the shape optimization problems.

Abstract: The topics include: sparse matrix systems from finite element applications dual domain decomposition block diagonal preconditioners for the Schur complement method domain decomposition techniques hybrid mixed finite element models for the characterization in Reissner/Mindlin plates parallel solution techniques for hybrid mixed finite element models parallel dynamic relaxation formfinding mesh optimality criteria and remeshing strategies for singular point problems parallel dynamic relaxation mesh optimality criteria and remeshing strategies for singular point problems parallel adaptive mesh generation and geometric modelling using NURBS innovative computational methods for structural optimization some studies on integrating topology and shape optimization genetic algorithms and evolution strategies computer aided design of profile extrusion dies and automatic design of reinforced concrete structures with parallel computing.

Abstract: Usually mechanical laws are applied to determine the structural response, for example deflections and stress state, while loads, boundary conditions, and geometry of the structure, i.e. the topology and the shape, are given. However, the mechanical principles can also be used to determine topology and shape of a structure for a prescribed structural response. This inverse method is called structural optimization. Since structural optimization deals in general with nonlinear and implicit functionals, only numerical methods have a chance to solve application-orientated problems in engineering design. Structural optimization can be distinguished into material, shape, and topology optimization depending on what is varied in the optimization process. The most challenging task is to determine the basic geometrical layout by topology optimization. In particular, recently the so-called material topology optimization of continuous structures has gained substantial interest both by mathematicians as well as engineers. The present contribution tries to consolidate these developments from an engineering point of view. In order to overcome problems of conventional numerical modeling techniques, material topology optimization is extended to geometrically adaptive methods. The adaptive discretization of the geometrical model in topology optimization turns out to be an appropriate way to obtain reliable results and simultaneously to reduce the numerical effort. This is verified by topology optimization problems with stress constraints and considering elastoplastic material behavior. The optimization based on either a linear or a nonlinear structural response leads to completely different results and shows the relevance of an appropriate mechanical model in the optimization process.

Abstract: The objective in this aerodynamic shape design effort is to minimize total pressure loss across the two-dimensional linear airfoil cascade row while satisfying a number of constraints. They included fixed axial chord, total torque, inlet and exit flow angles, and blade cross-section area, while maintaining thickness distribution greater than a minimum specified value. The aerodynamic shape optimization can be performed by using any available flow-field analysis code. For the analysis of the performance of intermediate cascade shapes we used an unstructured grid based compressible Navier-Stokes flow-field analysis code with k-e turbulence model. A robust genetic optimization algorithm was used for optimization and a constrained sequential quadratic programming was used enforcement of certain constraints. The airfoil geometry was parameterized using conic section parameters and B-splines thus keeping the number of geometric design variables to a minimum while achieving a high degree of geometric flexibility and robustness. Significant reductions of the total pressure loss were achieved using this constrained method for a supersonic exit flow axial turbine cascade.Copyright © 1999 by ASME

Abstract: This paper presents a method for shape optimization of flat or curved 3D shell structures, that takes advantage of the geometric modelling and automatic meshing capabilities of an existing parametric/associative CAD system It is an extension of a method previously proposed by the authors for shape optimization of 2D and 3D solid structures The implementation of the shell elements used is outlined, as well as the calculation of analytical sensitivities with the discrete method The validity of the method is demonstrated with the optimization of two complex 3D shell structures

Abstract: A voxel-based shape representation when integrated with an evolutionary algorithm offers a number of potential advantages for shape optimization. Topology need not be predefined, geometric constraints are easily imposed and, with adequate resolution, any shape can be approximated to arbitrary accuracy. However, lack of boundary smoothness, length of chromosome, and inclusion of small holes in the final shape have been stated as problems with this representation. This paper describes two experiments performed in an attempt to address some of these problems. First, a design problem with only a small computational cost of evaluating candidate shapes was used as a testbed for designing genetic operators for this shape representation. Second, these operators were refined for a design problem using a more costly finite element evaluation. It was concluded that the voxel representation can, with careful design of genetic operators, be useful in shape optimization.

Abstract: We consider the problem of aerodynamic airfoil shape optimization where the shape of an airfoil is to be determined such that a priori specified design criteria will be met to the best possible extent. The design criteria are formulated by defining an objective or cost function, the minimum of which represents the solution to the design problem. A survey is given of developments at NLR applying the adjoint operator approach, utilizing a compressible inviscid flow model based on the Euler equations and a compressible viscous flow model based on the Reynolds-averaged Navier-Stokes equations. Computational results are presented for a two-point drag-reduction design problem

Abstract: A variant of the usual boundary element method (BEM), called the boundary contour method (BCM), has been presented in the literature in recent years. In the BCM in three dimensions, surface integrals on boundary elements of the usual BEM are transformed, through an application of Stokes’ theorem, into line integrals on the bounding contours of these elements. The BCM employs global shape functions with the weights, in the linear combinations of these shape functions, being defined piecewise on boundary elements. A very useful consequence of this approach is that stresses, at suitable points on the boundary of a body, can be easily obtained from a post-processing step of the standard BCM. The subject of this paper is shape optimization in three-dimensional (3D) linear elasticity by the BCM. This is achieved by coupling a 3D BCM code with a mathematical programming code based on the successive quadratic programming (SQP) algorithm. Numerical results are presented for several interesting illustrative examples.