Abstract: Topology optimization of fluid dynamic systems is a comparatively young optimal design technique. Its central ingredient is the computation of topological sensitivity maps. Whereas, for finite element solvers, implementations of such sensitivity maps have been accomplished in the past, this study focuses on providing this functionality within a professional finite volume computational fluid dynamics solver. On the basis of a continuous adjoint formulation, we derive the adjoint equations and the boundary conditions for typical cost functions of ducted flows and present first results for two- and three-dimensional geometries. Emphasis is placed on the versatility of our approach with respect to changes in the objective function. We further demonstrate that surface sensitivity maps can also be computed with the implemented functionality and establish their connection with topological sensitivities. Copyright © 2008 John Wiley & Sons, Ltd.

Abstract: This paper presents an effective parametric approach by extending the conventional level set method to structural shape and topology optimization using the compactly supported radial basis functions (RBFs) and the optimality criteria (OC) method. The structural design boundary is first represented implicitly by embedding into a higher-dimensional level set function as its zero level set, and the RBFs of a favorable smoothness are then applied to interpolate the level set function. The original initial value problem is thus converted to a parametric optimization, with the expansion coefficients of the interplant posed as the design variables. The OC method is then applied to advance the structure boundary in terms of the velocity field derived from the parametric optimization. Hence, the structural shape and topology optimization is now transformed into a process of iteratively finding coefficients to update the level set function to achieve an optimal configuration. The numerical considerations of the conventional level set method, including upwind schemes, velocity extension, and reinitialization, are eliminated. The proposed scheme is capable of addressing structural shape fidelity and topology change simultaneously and of keeping the boundary smooth during the optimization process. Furthermore, numerical convergence is expected to be improved. A widely investigated example, in the framework of structural stiffness designs, is applied to demonstrate the efficiency and accuracy of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.

Abstract: The methodology of eXtended finite element method is applied to the measurement of displacements through digital image correlation. An algorithm, initially based on a finite element decomposition of displacement fields, is extended to benefit from discontinuity and singular enrichments over a suited subset of elements. This allows one to measure irregular displacements encountered, say, in cracked solids, as demonstrated both in artificial examples and experimental case studies. Moreover, an optimization strategy for the support of the discontinuity enables one to adjust the crack path configuration to reduce the residual mismatch, and hence to be tailored automatically to a wavy or irregular crack path. Copyright © 2007 John Wiley & Sons, Ltd.

Abstract: A major issue in surrogate model-based design optimization is the modeling fidelity. An effective approach is to employ multiple surrogates based on the same training data to offer approximations from alternative modeling viewpoints. This approach is employed in a compressor blade shape optimization using the NASA rotor 37 as the case study. The surrogate models considered include polynomial response surface approximation, Kriging, and radial basis neural network. In addition, a weighted average model based on global error measures is constructed. Sequential quadratic programming is used to search the optimal point based on these alternative surrogates. Three design variables characterizing the blade regarding sweep, lean, and skew are selected along with the three-level full factorial approach for design of experiment. The optimization is guided by three objectives aimed at maximizing the adiabatic efficiency, as well as the total pressure and total temperature ratios. The optimized compressor blades yield lower losses by moving the separation line toward the downstream direction. The optima for total pressure and total temperature ratios are similar, but the optimum for adiabatic efficiency is located far from them. It is found that the most accurate surrogate did not always lead to the best design. This demonstrated that using multiple surrogates can improve the robustness of the optimization at a minimal computational cost.

Abstract: A novel domain element shape parameterization method is presented for computational fluid dynamics-based shape optimization. The method is to achieve two aims: (1) provide a generic 'wrap-around' optimization tool that is independent of both flow solver and grid generation package and (2) provide a method that allows high-fidelity aerodynamic optimization of two- and three-dimensional bodies with a low number of design variables. The parameterization technique uses radial basis functions to transfer domain element movements into deformations of the design surface and corresponding aerodynamic mesh, thus allowing total independence from the grid generation package (structured or unstructured). Independence from the flow solver (either inviscid, viscous, aeroelastic) is achieved by obtaining sensitivity information for an advanced gradient-based optimizer (feasible sequential quadratic programming) by finite-differences. Results are presented for two-dimensional aerofoil inverse design and drag optimization problems. Inverse design results demonstrate that a large proportion of the design space is feasible with a relatively low number of design variables using the domain element parameterization. Heavily constrained (in lift, volume, and moment) two-dimensional aerofoil drag optimization has shown that significant improvements over existing designs can be achieved using this method, through the use of various objective functions.

Abstract: We propose an alternative to the classical post-treatment of the homogenization method for shape optimization. Rather than penalize the material density once the optimal composite shape is obtained (by the homogenization method) in order to produce a workable shape close to the optimal one, we macroscopically project the microstructure of the former through an appropriate procedure that roughly consists in laying the material along the directions of lamination of the composite. We have tested our approach in the framework of compliance minimization in two-dimensional elasticity. Numerical results are provided.

Abstract: This paper describes the topology optimization of thermoelastic structures, using level set method. The objective is to minimize the mean compliance of a structure with a material volume constraint. In level set method, free boundary of a structure is considered as design variable, and it is implicitly represented via level set model. Objective function of the optimization problem is defined as a function of the shape of a structure. Sensitivity analysis based on continuum model is conducted with respect to the free boundary, which suggests the steepest descent direction. A geometric energy term is introduced to ensure smooth structural boundary. Augmented Lagrangian multiplier method is adopted to enforce volume constraint. Numerical examples are provided for 2D cases, considering design independent temperature distribution.

Abstract: We present an algorithm for shape optimization under stochastic loading and representative numerical results. Our strategy builds upon a combination of techniques from two-stage stochastic programming and level-set-based shape optimization. In particular, usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces, which often is much smaller than the number of different realizations of the stochastic forces. Numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions.

Abstract: A numerical investigation of 3D fluid flow and heat transfer in a rectangular micro-channel has been carried out using water as a cooling fluid in a silicon substrate. Navier-Stokes and energy equations for laminar flow and conjugate heat transfer are solved using a finite volume solver. Solutions are first carefully validated with available analytical and experimental results; the shape of the micro-channel is then optimized using surrogate methods. Ratios of the width of the micro-channel to the depth and the width of the fin to the depth are selected as design variables. Design points are selected through a four-level full factorial design. A single objective function thermal resistance, formulated using pumping power as a constraint, is optimized. Mass flow rate is adjusted by the constant pumping power constraint. Response surface approximation, kriging, and radial basis neural network methods are applied to construct surrogates and the optimum point is searched by sequential quadratic programming.

Abstract: A variational approach to shape feature control in topology optimization is presented in this paper. The method is based on a new class of surface energies known as higher-order energies as opposed to the conventional energies for problem regularization, which are linear. In employing a quadratic energy functional in the objective of the topology optimization, non-trivial interactions between different points on the structural boundary are introduced, thus favoring a family of shapes with strip-like (or beam) features. In addition, the quadratic energy functional can be seamlessly integrated into the level set framework that represents the geometry of the structure implicitly. The shape gradient of the quadratic energy functional is fully derived in the paper, and it is incorporated in the level set approach for topology optimization. The approach is demonstrated with benchmark examples of structure optimization and compliant mechanism design. The results presented show that this method is capable of generating strip-like (or beam) designs with specified feature width, which have highly desirable characteristics and practical benefits and uniquely distinguish the proposed method.

Abstract: A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems and the Navier-Stokes equations are given.

Abstract: Shape optimization is a problem which arises in numerous computer vision problems such as image segmentation and multiview reconstruction. In this paper, we focus on a certain class of binary labeling problems which can be globally optimized both in a spatially discrete setting and in a spatially continuous setting. The main contribution of this paper is to present a quantitative comparison of the reconstruction accuracy and computation times which allows to assess some of the strengths and limitations of both approaches. We also present a novel method to approximate length regularity in a graph cut based framework: Instead of using pairwise terms we introduce higher order terms. These allow to represent a more accurate discretization of the L 2 -norm in the length term.

Abstract: The purpose of this paper was to study the layout design of the components and their supporting structures in a finite packing space. A coupled shape and topology optimization (CSTO) technique is proposed. On one hand, by defining the location and orientation of each component as geometric design variables, shape optimization is carried out to find the optimal layout of these components and a finite-circle method (FCM) is used to avoid the overlap between the components. On the other hand, the material configuration of the supporting structures that interconnect components is optimized simultaneously based on topology optimization method. As the FE mesh discretizing the packing space, i.e., design domain, has to be updated itertively to accommodate the layout variation of involved components, topology design variables, i.e., density variables assigned to density points that are distributed regularly in the entire design domain will be introduced in this paper instead of using traditional pseudo-density variables associated with finite elements as in standard topology optimization procedures. These points will thus dominate the pseudo-densities of the surrounding elements. Besides, in the CSTO, the technique of embedded mesh is used to save the computing time of the remeshing procedure, and design sensitivities are calculated w.r.t both geometric variables and density variables. In this paper, several design problems maximizing structural stiffness are considered subject to the material volume constraint. Reasonable designs of components layout and supporting structures are obtained numerically.

Abstract: A parameterization level set method is presented for structural shape and topology optimization of compliant mechanisms involving large displacements. A level set model is established mathematically as the Hamilton-Jacobi equation to capture the motion of the free boundary of a continuum structure. The structural design boundary is thus described implicitly as the zero level set of a level set scalar function of higher dimension. The radial basis function with compact support is then applied to interpolate the level set function, leading to a relaxation and separation of the temporal and spatial discretizations related to the original partial differential equation. In doing so, the more difficult shape and topology optimization problem is now fully parameterized into a relatively easier size optimization of generalized expansion coefficients. As a result, the optimization is changed into a numerical process of implementing a series of motions of the implicit level set function via an existing efficient convex programming method. With the concept of the shape derivative, the geometrical non-linearity is included in the rigorous design sensitivity analysis to appropriately capture the large displacements of compliant mechanisms. Several numerical benchmark examples illustrate the effectiveness of the present level set method, in particular, its capability of generating new holes inside the material domain. The proposed method not only retains the favorable features of the implicit free boundary representation but also overcomes several unfavorable numerical considerations relevant to the explicit scheme, the reinitialization procedure, and the velocity extension algorithm in the conventional level set method.

Abstract: A level set based method is proposed for the simultaneous optimization of the material properties and the topology of functionally graded structures. The objective of the present study is to determine the optimal material properties (via the material volume fractions) and the structural topology to maximize the performance of the structure in a given application. In the proposed method, the volume fraction and the structural boundary are considered as the design variables, with the former being discretized as a scalar field and the latter being implicitly represented by the level set method. To perform simultaneous optimization, the two design variables are integrated into a common objective functional. Sensitivity analysis is conducted to obtain the descent directions. The optimization process is then expressed as the solution to a coupled Hamilton-Jacobi equation and diffusion partial differential equation. Numerical results are provided for the problem of mean compliance optimization in two dimensions.

Abstract: Topology optimization can be seen as optimizing a distribution of small topological elements within a domain with respect to given specifications. A numerical topology gradient (TG) algorithm is applied in the context of electromagnetism for optimizing microwave devices, computing the sensitivity on adding or removing small metallic elements. This method leads to an optimum topology with very little initial information in acceptable time consumption. The method is applied to the design of a microstrip component in which the topology gradient is directly used as a direction of descent. However, in some ill-behavior problems, topology gradient is not sufficient to converge to the global optimum. In the latter case, the basic TG is coupled with a genetic algorithm (GA) to make a more suitable algorithm for solving local optima problems. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.

Abstract: In aerodynamic shape optimization, gradient-based methods often rely on the adjoint approach, which is capable of computing the objective function sensitivities with respect to the design variables. In the literature adjoint approaches are proved to outperform other relevant methods, such as the direct sensitivity analysis, finite differences or the complex variable approach. They appear in two different formulations, namely the continuous and the discrete one, which are both discussed in this chapter.

Abstract: Adjoint-based shape optimization methods have proven to be computationally efficient for aerodynamic problems. The majority of the studies on adjoint methods have used structured grids to discretize the computational domain. Because of the potential advantages of unstructured grids for complex configurations, in this study we have developed and validated a continuous adjoint formulation for unstructured grids. The hurdles posed in the computation of the gradient for unstructured grids are resolved by using a reduced gradient formulation. The methods to impose thickness constraints on unstructured grids are also discussed. The results for two- and three-dimensional simulations of airfoils and wings in inviscid transonic flow are used to validate the design procedure. Finally, the design procedure is applied to redesign the shape of a transonic business jet configuration; we were able to reduce the inviscid drag of the aircraft from 235 to 216 counts resulting in a shock-free wing. Although the Euler equations are the focus of the study in this paper of the adjoint-based approach, the solution of the adjoint system and gradient formulation can be conceptually extended to viscous flows. The approach presented in this study has been successfully used by the first and third authors for viscous flows using structured grids. However, particular aspects of the design process, such as the robustness of the mesh deformation process for unstructured grids, need more attention for viscous flows and are therefore the subject of ongoing research.

Abstract: A conformable airfoil is proposed as an alternative to trailing-edge flaps used for active helicopter vibration reduction through high-frequency changes in camber. The design consists of several compliant mechanisms of predetermined topology that are placed serially within the airfoil along the chord, aft of the leading-edge spar. A shape optimization approach is used to design the compliant mechanisms, in which the objective is to maximize trailing-edge deflection while minimizing airfoil deflections due to aerodynamic loads. Solutions were obtained using a sequential linear programming method coupled with a finite element analysis. An optimized shape is predicted to achieve a trailing-edge deflection of ±6.0 mm or a ±4.6- deg equivalent flap deflection angle using the tip deflection objective. Results indicate that the deflection is dependent on the amount of passive material allowed and the objective function used. The aerodynamic loads are found to cause only small deformations in comparison with those caused by the actuation. Prototype fabrication and bench-top tests demonstrated that rotor airfoil camber is controllable using the proposed concept.

Abstract: This paper is devoted to the analysis of a second order method for recovering the a priori unknown shape of an inclusion $\omega$ inside a body $\Omega$ from boundary measurement. This inverse problem—known as electrical impedance tomography—has many important practical applications and hence has been the focus of much attention during the past few years. However, to the best of our knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: We investigate the existence of second order derivative of the state $u$ with respect to perturbations of the shape of the interface $\partial\omega$. Then we choose a cost function in order to recover the geometry of $\partial \omega$ and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer.

Abstract: A new topology optimization using adaptive inner-front level set method is presented. In the conventional level set-based topology optimization, the optimum topology strongly depends on the initial level set due to the incapability of inner-front creation during the optimization process. In the present work, in this regard, an algorithm for inner-front creation is proposed in which the sizes, the positions, and the number of new inner-fronts during the optimization process can be globally and consistently identified. In the algorithm, the criterion of inner-front creation for compliance minimization problems of linear elastic structures is chosen as the strain energy density along with volumetric constraint. To facilitate the inner-front creation process, the inner-front creation map is constructed and used to define new level set function. In the implementation of inner-front creation algorithm, to suppress the numerical oscillation of solutions due to the sharp edges in the level set function, domain regularization is carried out by solving the edge smoothing partial differential equation (smoothing PDE). To update the level set function during the optimization process, the least-squares finite element method (LSFEM) is adopted. Through the LSFEM, a symmetric positive definite system matrix is constructed, and non-diffused and non-oscillatory solution for the hyperbolic PDE such as level set equation can be obtained. As applications, three-dimensional topology optimization of shell structures is treated. From the numerical examples, it is shown that the present method brings in much needed flexibility in topologies during the level set-based topology optimization process.

Abstract: The Direct Numerical Optimization (DNO) approach for airfoil shape design requires the integration of modules: a) A geometrical shape function; b) Computational flow solver and; c) Search model for shape optimization. These modules operate iteratively until convergence based on defined objectives and constraints. The DNO architecture is to be validated to ensure efficient optimization simulations and is the focus of this paper. The PARSEC airfoil shape function is first validated by observing the effect of design coefficients on airfoil geometry and aerodynamics. The design variables provide independent one-to-one control over airfoil geometry, for imposing shape constraints. The aerodynamic performance of PARSEC airfoils through variable perturbations, conform to established aerodynamic principles. It confirms the design flexibility of the shape function in providing direct control over airfoil geometry. The Particle Swarm Optimization (PSO) algorithm is introduced as the search agent. A PSO simulation requires userinputs to define the search pattern. A methodology is presented to validate these parameters on pre-defined benchmark mathematical functions. Self Organizing Maps (SOM) are applied to illustrate trade-offs between PSO search variables. An Adaptive Inertia Weight (APSO) scheme that dynamically alters the search path of the swarm by monitoring the position of the particles, provides an acceptable convergence. Validation tests indicated the maximum velocity of the particles is less than 1% of computational domain size for convergence. The DNO approach is computationally inefficient, thus a surrogate model to address this issue is presented. An Artificial Neural Network (ANN) model with a training dataset of 3000 airfoils is applied to develop a model that applies the PARSEC airfoil geometry variables as inputs and the equating aerodynamic coefficient as output. System validation with 1000 randomly generated airfoils indicated 70% of the simulated solutions were within 10% of actual solver run. Future research will involve reducing the percentage error of the surrogate model against the theoretical solution.

Abstract: This paper deals with the numerical solution of structural optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with a given friction is described by an elliptic inequality of the second order governing a displacement field. The optimization problem consists in finding, in a contact region, such topology and shape of the boundary of the domain occupied by the body that the normal contact stress is minimized. Level set methods [3], [4] are numerically efficient and robust procedures for the tracking of interfaces, which allows domain boundary shape changes in the course of iteration. The evolution of the level set function is governed by the Hamilton Jacobi equation. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative [2] of an appropriately defined cost functional with respect to the free boundary.

Abstract: The paper continues along the work initiated by the authors in taking into account stress constraints in topology optimization of continuum structures. Revisiting some of their last developments in this field, the authors point out the im- portance of considering stress constraints as soon as the preliminary design phase, that is, to include stress constraints in the topology optimization problem in order to get the most appropriate structural layout. Numerical applications that can be solved using these new developments make possible to exhibit interesting results related to the specific nature of strength based structural layout for maximum strength compared to maximum stiffness. This particular character of stress design is clearly demonstrated in two kinds of situations: once several load cases are considered and when unequal stress limits in tension and compression are involved.