Abstract: Active Shape Models (ASM) use an iterative algorithm to match statistically defined models of known but variable objects to instances in images. Each iteration of ASM search involves two steps: image data interrogation and shape approximation. Here we consider the shape approximation step in detail. We present a new method of shape approximation which uses directional constraints. We show how the error term for the shape approximation problem can be extended to cope with directional constraints, and present iterative solutions to the 2D and 3D problems. We also present an efficient algorithm for the 2D problem in which a modification of the error term permits a closed-form approximate solution which can be used to produce starting estimates for the iterative solution.

Abstract: Finite-Element-Based Engineering Design Sensitivity Analysis and Optimizations, N. Olhoff, E. Lundt. Structural Design Sensitivity Analysis - Continuum and Discrete Approaches, J.S. Arora. A View on Nonlinear Optimization, J. Herskovits. Large Scale Structural Optimization, G.N. Vanderplaats. What is Meaningful in Topology Design? An Engineer's Viewpoint, G.I.N. Rozvany. Optimal Shape and Topology Design of Vibrating Structures, N. Kikuchi, Hsien-Chie Cheng, Zheng-Dong Ma. Material Optimizations - An Engineering View, P. Pedersen. Integrated Optimization of Intelligent Structures, A.E. Sepulveda. Contact Shape Optimization, J. Haslinger. First and Second Order Design Sensitivity at a Bifurcation Point, Z. Mroz, J. Piekarski. Shell Optimization, Some Remarks, B. Rousselet, S. Mehrez, A. Myslinski, J. Piekarski. Application of Automatic Differentiation to Optimal Shape Design M. Masmoudi, Ph. Guillaume, C. Broudiscou. Large Scale Tracked Vehicle Concurrent Engineering Environment, K.K. Choi, J. Kirk Wu, Kuang-Hua Chang, Jun Tang, Jia-Yi Wang, E.J. Haug. Multidisciplinary Design Optimization - an Emerging New Engineering Discipline, J. Sobieszczanski-Sobieski.

Abstract: This paper presents a general parametric design approach for 2-D shape optimization problems. This approach has been achieved by integrating practical design methodologies into numerical procedures. It is characterized by three features : (i) automatic selection of a minimum number of shape design variables based on the CAD geometric model ; (ii) integration of sequential convex programming algorithms to solve equality constrained optimization problems ; (iii) efficient sensitivity analysis by means of the improved semi-analytical method. It is shown that shape design variables can be either manually or systematically identified with the help of equality constraints describing the relationship between geometric entities. Numerical solutions are performed to demonstrate the applicability of the proposed approach. A discussion of the results is also given :

Abstract: An improved shape annealing algorithm for truss topology generation and optimization, based on the techniques of shape grammars and simulated annealing, is introduced. The algorithm features a shape optimization method using only simulated annealing with a shape grammar move set; while no traditional gradient-based techniques are employed, the algorithm demonstrates more consistent convergence characteristics. By penalizing the objective function for violated constraints, the algorithm incorporates geometric constraints to avoid obstacles. The improved algorithm is illustrated on various structural examples taking into account stress, Euler buckling and geometric constraints, generating a variety of solutions based on a simple grammar.

Abstract: In an effort to improve and automate the fluid dynamic design of rotary blood pumps, a coupled computational fluid dynamics (CFD) shape optimization methodology has been developed and implemented. This program couples a finite element flow simulation with a gradient-based optimization routine to modify automatically the shape of an initial candidate blood path, according to a variety of desired fluid dynamic criteria, including shear stress, vorticity/circulation, and viscous dissipation. Preliminary results have led to both intuitive and nonintuitive transformations of the initial blood flow paths for both internal and external flows. This application of computer design optimization offers the ability to explore a much broader design space much more efficiently than would be possible with traditional parametric methods. It is believed that this computer tool can assist developers of rotary blood pumps in designing blood-wetted components that minimize thrombosis and hemolysis while simultaneously providing maximum flow performance.

Abstract: An efficient numerical approach for the design of optimal aerodynamic shapes is presented in this paper. The objective of any optimization problem is to find the optimum of a cost function subject to a certain state equation (governing equation of the flow field) and certain side constraints. As in classical optimal control methods, the present approach introduces a costate variable (Lagrange multiplier) to evaluate the gradient of the cost function. High efficiency in reaching the optimum solution is achieved by using a multigrid techniique and updating the shape in a hierarchical manner such that smooth (low frequency) changes are done separately from high-frequency changes. Thus, the design variables are changed on a grid where their changes produce nonsmooth (high frequency) perturbations that can be damped efficiently by the multigrid. The cost of solving the optimization problem is approximately two to three times the cost of the equivalent analysis problem.

Abstract: This paper presents a Genetic Algorithm approach to two-dimensional shape optimization. Shapes are represented as arrays of boolean pixels (material/void), or bit-arrays. The inadequacy of the (one-dimensional) bitstring representation is emphasized, both a priori and experimentally. This leads to the design of crossover operators adapted to the two-dimensional representation. Similarly, some non standard mutation operators are introduced and studied. A strategy involving evolutionary choice among these different operators is finally proposed. All experiments are performed on a simple test-problem of Optimum Design, as the computational cost of real-world problems forbids extensive experimental tests.

Abstract: Two different formulations for shape design sensitivity analysis are provided for heat conducting solids. The material derivative concept and the adjoint variable method of analysis are the common techniques utilized in the two formulations. It is found that complementary boundary integral equations arise in the case of the integral sensitivity analysis. Numerical solutions of the primary and adjoint equations are achieved by the BEM, whilst shape configurations are discretized by means of ray and height functions. Several 2-D constrained shape optimization problems are solved by the proposed numerical solution techniques.

Abstract: For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard dierential calculus in order to get second derivatives by a straight forward ”repetition of dierentiation”. Moreover, by using boundary value characerizations for more regular data, a complete boundary integral representation of the second derivative of the objective is possible. Basing on this, one easily obtains that the second derivative contains only normal components for stationary domains, i.e. for domains, satisfying the first order necessary condition for a free optimum. Moreover, the nature of the second derivative is discussed, which is helpful for the investigation of sucient optimality

Abstract: Equivalent plate structural modeling and doublet point lifting surface unsteady aerodynamics are used to obtain analytic sensitivities of aeroelastic and aeroservoelastic response with respect to wing and control surface planform shape parameters. Rational function approximations for unsteady aerodynamic forces, their shape sensitivities, and the resulting linear time invariant state space models of aeroservoelastic systems and their shape sensitivities are examined. The goal is to develop effective and numerically efficient approximation techniques for wing shape optimization for use with nonlinear programming and approximation concepts as a multidisciplinary optimization strategy. Effects of structural and unsteady aerodynamic modeling errors are studied. Examination of approximation accuracy using alternative approximation techniques (and the resulting move limits) provide insight and experience on the way to realistic wing/control surface shape optimization with active controls and aeroservoelastic constraints.

Abstract: The aim of this paper is to present basic concepts and selected finite element based methods and tools for sensitivity analysis and rational engineering design and optimization of mechanical structures and components. The main emphasis is devoted to sensitivity and optimization problems that involve shape and sizing design variables.

Abstract: The shape optimization of coupled systems by a gradients-based method is presented. The design specifications are in one system, while the critical parameters are in both systems. The method is demonstrated using an induction heating system. The magnetic and thermal models coexist in the same geometry. The eddy currents calculated from the electromagnetic solution is used as the thermal sources for the thermal finite element analysis. The objective is to achieve a required steady state temperature profile by modifying the geometry of the magneto-thermal domain. The objective function, defined as a function of the state variable temperature is no longer linked directly to the design parameters of the magneto-thermal system through the classic design differential equation, describing the electromagnetic fields in sensitivity analysis but through the "coupled" one. The proposed algorithm allows the calculation of the gradient of the object function with respect to the design parameters. >

Abstract: The shape of conducting post in H-plane waveguide tee-junction is optimized to maximize the transmitting power. The method employs an edge finite element technique with magnetic fields as state variables. Sensitivity analysis and steepest descent algorithm are used for the optimization. >

Abstract: The multigrid one shot method for optimal shape design problems, governed by elliptic systems, is introduced for the infinite dimensional design space. In this case the design variable is a function whose discrete representation involves increasing number of variables with grid refinement. The minimization algorithm uses Lagrange multipliers to calculate sensitivity gradients. A gradient descent algorithm is accelerated by a set of coarse grids. It optimizes for different scales in the representation of the boundary position on different discretization levels. Numerical results have been obtained for an optimal shape design of a two dimensional geometry with a finite element discretization. The computer time for solving the optimization problem, up to the discretization error, was of the same order of magnitude as solving the constraint PDE just a few times.

Abstract: A finite element approach to shape optimization in a 2D frictionless contact problem for two different cost functions is presented in this work. The goal is to find an appropriate shape for the contact boundary, performing an almost constant contact-stress distribution. The whole formulation, including the mathematical model for the unilateral problem, sensitivity analysis and geometry definition is treated in a continuous form, independently of the discretization in finite elements. Shape optimization is performed by a direct modification of the geometry throughB-spline curves and an automatic mesh generator is used at each new configuration to provide the finite element input data. Augmented-Lagrangian techniques (to solve the contact problem) and an interior-point mathematical-programming algorithm (for shape optimization) are used to obtain numerical results.

Abstract: A number of examples on design optimization from different disciplines such as structural mechanics, manufacturing costs, fluid flow, acoustics, topology, mechanical systems and environmental load are presented. All examples have Swedish or Danish origin. Techniques to solve certain special optimization problems are discussed. Some of those are multiple loading cases in shape optimization problems involving contact zones, production tolerance sensitivity design, volume dependence and fail safe design.

Abstract: The p method of analysis, when used with shape optimization, has certain advantages over the h method. The method of h adaptivity results in fine meshes with small elements in areas of stress concentration. The presence of these small elements prevents the shape from changing significantly during optimization due to mesh distortion. The p-adaptive method uses relatively large sized elements, and adaptivity is achieved by increasing the polynomial order of the shape functions rather than by introducing small elements as in the h method. Thus, the p-adaptive method allows for greater shape changes without sacrificing the accuracy of the analysis. A methodology is developed in this paper to integrate p-adaptive analysis with shape optimization for general velocity fields. The key contributions lie in an easily implementable shape update strategy and an adaptive scheme to locally increase the polynomial order based on error indicators. Numerical results show that significant shape change is achievable prior to serious mesh distortion. In addition, need for an h-p adaptive strategy rather than a pure p-adaptive strategy is also highlighted.

Abstract: The problem of aerodynamic shape optimization in Euler flow is addressed. B-splines are used for parametrization of the shape. Cheap gradient calculation is obtained via sensitivity analysis and the solution of an adjoint equation; pseudo secondorder spatial accuracy is achieved by means of a semianalytical formulation. As a validation of the approach, several inverse and constrained optimization test problems are presented with emphasis on civil engine nacelle design. The handling of nondifferentiable quantities (such as maxima) in cost functions is allowed for via the use of the Kreisselmeier-Steinhauser function.

Abstract: In this paper structural and sensitivity analysis for the optimization of laminated axisymmetric shells subjected to static constraints and arbitrary loading is presented. The shell thickness, radial coordinate of a nodal point, lamina thickness and the angle of orientation of the fibers are the design variables. The objective of the design optimization is the minimization of the volume of the shell or the strain energy. The model is based on a three-node axisymmetric finite element with 24 degrees of freedom. A higher-order theory is developed for the nonlinear distribution of the meridional displacement component through the thickness of the shell. The sensitivities of the discrete model developed are evaluated analytically using a symbolic manipulator. The efficiency and accuracy of the proposed model is discussed with reference to the applications.