A smart repair embedded memetic algorithm for 2D shape matching problems

Shape representation plays a major role in any shape optimization exercise. The ability to identify a shape with good performance is dependent on both the flexibility of the shape representation scheme and the efficiency of the optimization algorithm. In this article, a memetic algorithm is presented for 2D shape matching problems. The shape is represented using B-splines, in which the control points representing the shape are repaired and subsequently evolved within the optimization framework. The underlying memetic algorithm is a multi-feature hybrid that combines the strength of a real coded genetic algorithm, differential evolution and a local search. The efficiency of the proposed algorithm is illustrated using three test problems, wherein the shapes were identified using a mere 5000 function evaluations. Extension of the approach to deal with problems of unknown shape complexity is also presented in the article.     

A POD-selective inverse distance weighting method for fast parametrized shape morphing

Efficient shape morphing techniques play a crucial role in the approximation of partial differential equations defined in parametrized domains, such as for fluid-structure interaction or shape optimization problems. In this paper, we focus on inverse distance weighting (IDW) interpolation techniques, where a reference domain is morphed into a deformed one via the displacement of a set of control points. We aim at reducing the computational burden characterizing a standard IDW approach without significantly compromising the accuracy. To this aim, first we propose an improvement of IDW based on a geometric criterion that automatically selects a subset of the original set of control points. Then, we combine this new approach with a dimensionality reduction technique based on a proper orthogonal decomposition of the set of admissible displacements. This choice further reduces computational costs. We verify the performances of the new IDW techniques on several tests by investigating the trade-off reached in terms of accuracy and efficiency.     

CFD shape optimization using an incomplete‐gradient adjoint formulation

A design methodology based on the adjoint approach for flow problems governed by the incompressible Euler equations is presented. The main feature of the algorithm is that it avoids solving the adjoint equations, which saves an important amount of CPU time. Furthermore, the methodology is general in the sense it does not depend on the geometry representation. All the grid points on the surface to be optimized can be chosen as design parameters. In addition, the methodology can be applied to any type of mesh. The partial derivatives of the flow equations with respect to the design parameters are computed by finite differences. In this way, this computation is independent of the numerical scheme employed to obtain the flow solution. Once the design parameters have been updated, the new solid surface is obtained with a pseudo‐shell approach in such a way that local singularities, which can degrade or inhibit the convergence to the optimal solution, are avoided. Some 2D and 3D numerical examples are shown to demonstrate the proposed methodology. Copyright © 2001 John Wiley & Sons, Ltd.     

Structural shape and topology optimization based on level-set modelling and the element-propagating method

This article introduces the element-propagating method to structural shape and topology optimization. Structural optimization based on the conventional level-set method needs to solve several partial differential equations. By the insertion and deletion of basic material elements around the geometric boundary, the element-propagating method can avoid solving the partial differential equations and realize the dynamic updating of the material region. This approach also places no restrictions on the signed distance function and the Courant–Friedrichs–Lewy condition for numerical stability. At the same time, in order to suppress the dependence on the design initialization for the 2D structural optimization problem, the strain energy density is taken as a criterion to generate new holes in the material region. The coupled algorithm of the element-propagating method and the method for generating new holes makes the structural optimization more robust. Numerical examples demonstrate that the proposed approach greatly improves numerical efficiency, compared with the conventional level-set method for structural topology optimization.     

Sensitivity computation and shape optimization for a non-linear arch model with limit-points instabilities

A shape optimization method for geometrically non-linear structural mechanics based on a sensitivity gradient is proposed. This gradient is computed by means of an adjoint state equation and the structure is analysed with a total Lagrangian formulation. This classical method is well understood for regular cases, but standard equations have to be modified for limit points and simple bifurcation points. These modifications introduce numerical problems which occur at limit points. Numerical systems are very stiff and the quadratic convergence of Newton–Raphson algorithm vanishes, then higher-order derivatives have to be computed with respect to state variables. A geometrically non-linear curved arch is implemented with a finite element method via a formal calculus approach. Thickness and/or shape for differentiable costs under linear and non-linear constraints are optimized. Numerical results are given for linear and non-linear examples and are compared with analytic solutions. © 1998 John Wiley & Sons, Ltd.     

3D preform shape optimization in forging using reduced basis techniques

Three-dimensional preform shape optimization of complex forgings with a weighted summation of multiple basis shapes is presented in this article. Currently, 2D preform shape optimization is well developed; however, in cases in which the parts are neither axisymmetric nor plane strain, 2D assumptions do not hold well. The number of design variables required to define the 3D preform shape is high, making most iterative design methods impractical for shape optimization. The goal here is to make design optimization practical and efficient by developing reduced-order modeling techniques for 3D preform shape optimization. The preform shape is treated as a linear combination of various billet shapes, called basis shapes, with the weights for each basis shape used as design variables, thereby reducing the number of design variables. It is very difficult to obtain the necessary gradient information for 3D forging simulations, so a non-gradient method is used to build the surrogate model on which optimization is performed. The optimization problem is formulated to minimize strain variance while placing constraints on underfill. Representative problems are used to demonstrate the effectiveness of the approach.     

Efficient handling of stability problems in shell optimization by asymmetric ‘worst‐case’ shape imperfection

The paper presents an approach to shape optimization of proportionally loaded elastic shell structures under stability constraints. To reduce the stability‐related problems, a special technique is utilized, by which the response analysis is always terminated before the first critical point is reached. In this way, the optimization is always related to a precritical structural state. The necessary load‐carrying capability of the optimal structure is assured by extending the usual formulation of the optimization problem by a constraint on an estimated critical load factor. Since limit points are easier to handle, the possible presence of bifurcation points is avoided by introducing imperfection parameters. They are related to an asymmetric shape perturbation of the structure. During the optimization, the imperfection parameters are updated to get automatically the ‘worst‐case’ pattern and amplitude of the imperfection. Both, the imperfection parameters and the design variables are related to the structural shape via the design element technique. A gradient‐based optimizer is employed to solve the optimization problem. Three examples illustrate the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Shape optimization with topological changes and parametric control

Recent advances in shape optimization rely on free-form implicit representations, such as level sets, to support boundary deformations and topological changes. By contrast, parametric shape optimization is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. We propose a novel approach to shape optimization that combines and retains the advantages of the earlier optimization techniques. The shapes in the design space are represented implicitly as level sets of a higher-dimensional function that is constructed using B-splines (to allow free-form deformations), and parameterized primitives combined with R-functions (to support desired parametric changes). Our approach to shape design and optimization offers great flexibility because it provides explicit parametric control of geometry and topology within a large space of free-form shapes. The resulting method is also general in that it subsumes most other types of shape optimization as special cases. We describe an implementation of the proposed technique with attractive numerical properties. The explicit construction of an implicit representation supports straightforward sensitivity analysis that can be used with most gradient-based optimization methods. Furthermore, our implementation does not require any error-prone polygonization or approximation of level sets (isocurves and isosurfaces). The effectiveness of the method is demonstrated by several numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

Velocity field level-set method for topological shape optimization using freely distributed design variables

The velocity field level-set topological shape optimization method combines the implicit representation in the standard level-set method and the capabilities of general mathematical programming algorithms in handling multiple constraints and additional design variables. The key concept is to construct the normal velocity field using basis functions and the velocity design variables at specified points (referred to as velocity knots) in the entire design domain. In this study, the velocity design variables are decoupled from the level-set grid points. Making use of this property, we can adaptively change the arrangement of the velocity knots as the structural boundary evolves. This provides more design freedom in the optimization and allows for a significant reduction in the number of design variables. Several numerical examples in two- and three-dimensional design domains are presented to demonstrate the robustness and efficiency of the proposed method. We also show that changing the number of velocity knots may implicitly exert certain control on topological complexity and length scale.     

A velocity field level set method for shape and topology optimization

In this paper, we propose a new implementation of the level set shape and topology optimization, the velocity field level set method. Therein, the normal velocity field is constructed with specified basis functions and velocity design variables defined on a given set of points that are independent of the finite element mesh. A general mathematical programming algorithm can be employed to find the optimal normal velocities on the basis of the sensitivity analysis. As compared with conventional level set methods, mapping the variational boundary shape optimization problem into a finite‐dimensional design space and the use of a general optimizer makes it more efficient and straightforward to handle multiple constraints and additional design variables. Moreover, the level set function is updated by the Hamilton‐Jacobi equation using the normal velocity field; thus, the inherent merits of the implicit representation is retained. Therefore, this method combines the merits of both the general mathematical programming and conventional level set methods. Integrated topology optimization of structures with embedded components of designable geometries is considered to show the capability of this method to deal with general design variables. Several numerical examples in 2D or 3D design domains illustrate the robustness and efficiency of the method using different basis functions.     

Lie group method: A new approach to image matching with arbitrary orientations

In this article, we develop a new method for image matching of any two images with arbitrary orientations. The idea comes from the workpiece localization in machining industry. We first describe an image as a 3D point set other than the common 2D function f(x, y), then, making the sets corresponding to the compared images form solid surfaces, we equivalently translate the matching problem into an optimization problem on the Lie group SE(3). Through developing a kind of steepest descent algorithms on a general Lie group, we present an practical algorithm for matching problem. Simulations of eye detection and face detection are presented to show the feasibility and efficiency of the proposed algorithm. © 2010 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 20, 245–252, 2010.     

A global optimization algorithm for simulation-based problems via the extended DIRECT scheme

This article presents a global optimization algorithm via the extension of the DIviding RECTangles (DIRECT) scheme to handle problems with computationally expensive simulations efficiently. The new optimization strategy improves the regular partition scheme of DIRECT to a flexible irregular partition scheme in order to utilize information from irregular points. The metamodelling technique is introduced to work with the flexible partition scheme to speed up the convergence, which is meaningful for simulation-based problems. Comparative results on eight representative benchmark problems and an engineering application with some existing global optimization algorithms indicate that the proposed global optimization strategy is promising for simulation-based problems in terms of efficiency and accuracy.     

Optimized stereo reconstruction of free-form space curves based on a nonuniform rational B-spline model

Analytical reconstruction of 3D curves from their stereo images is an important issue in computer vision. We present an optimization framework for such a problem based on a nonuniform rational B-spline (NURBS) curve model that converts reconstruction of a 3D curve into reconstruction of control points and weights of a NURBS representation of the curve, accordingly bypassing the error-prone point-to-point correspondence matching. Perspective invariance of NURBS curves and constraints deduced on stereo NURBS curves are employed to formulate the 3D curve reconstruction problem into a constrained nonlinear optimization. A parallel rectification technique is then adopted to simplify the constraints, and the Levenberg-Marquardt algorithm is applied to search for the optimal solution of the simplified problem. The results from our experiments show that the proposed framework works stably in the presence of different data samplings, randomly posed noise, and partial loss of data and is potentially suitable for real scenes.     

A comparative study of controlled random search algorithms with application to inverse aerofoil design

This article presents a comparative study of some versions of the controlled random search algorithm (CRSA) in global optimization problems. The basic CRSA, originally proposed by Price in 1977 and improved by Ali et al. in 1997, is taken as a starting point. Then, some new modifications are proposed to improve the efficiency and reliability of this global optimization technique. The performance of the algorithms is assessed using traditional benchmark test problems commonly invoked in the literature. This comparative study points out the key features of the modified algorithm. Finally, a comparison is also made in a practical engineering application, namely the inverse aerofoil shape design.     

Genetic algorithms for high-precision reconstructions of three-dimensional topographies using stereo fractographs

A software is developed which enables reconstruction of the three-dimensional (3D) shape of fracture surfaces without human assistance. It is based upon computer image processing and pattern recognition techniques by using a stereo-pair of scanning electron micrographs. The processing consists of two subprocesses: searching the matching points between two images; and computation of heights using the relative shift of the matching points. By using the previously developed system, some mismatches were inevitable in the search process, in particular, for low-contrast SEM images, e.g. striations, intergranular facets, etc. In order to improve the accuracy of the search, a genetic algorithm (GA) was implemented into the developed system. By using the GA method, the 3D shapes of a wide variety of fracture surfaces including cleavage failures, intergranular cracking, dimples and fatigue striations, were successfully reconstructed with sufficient accuracy. The searching processes by the GA method and the previously developed two-step algorithm of coarse and close searching were compared. These proved that the GA method has both the advantage of accuracy in the searching process and a short run-time. A detailed 3D shape, of more than a 120 × 120 reconstructed point-sized shape, was thus obtained with sufficient accuracy and with a relatively short run-time.     

Robust shape and topology optimization considering geometric uncertainties with stochastic level set perturbation

When geometric uncertainties arising from manufacturing errors are comparable with the characteristic length or the product responses are sensitive to such uncertainties, the products of deterministic design cannot perform robustly. This paper presents a new level set‐based framework for robust shape and topology optimization against geometric uncertainties. We first propose a stochastic level set perturbation model of uncertain topology/shape to characterize manufacturing errors in conjunction with Karhunen–Loève (K–L) expansion. We then utilize polynomial chaos expansion to implement the stochastic response analysis. In this context, the mathematical formulation of the considered robust shape and topology optimization problem is developed, and the adjoint‐variable shape sensitivity scheme is derived. An advantage of this method is that relatively large shape variations and even topological changes can be accounted for with desired accuracy and efficiency. Numerical examples are given to demonstrate the validity of the present formulation and numerical techniques. In particular, this method is justified by the observations in minimum compliance problems, where slender bars vanish when the manufacturing errors become comparable with the characteristic length of the structures. Copyright © 2016 John Wiley & Sons, Ltd.     

A shape optimization approach for simulating contact of elastic membranes with rigid obstacles

The obstacle problem consists in computing equilibrium shapes of elastic membranes in contact with rigid obstacles. In addition to the displacement u of the membrane, the interface Γ on the membrane demarcating the region in contact with the obstacle is also an unknown and plays the role of a free boundary. Numerical methods that simulate obstacle problems as variational inequalities share the unifying feature of first computing membrane displacements and then deducing the location of the free boundary a posteriori. We present a shape optimization-based approach here that inverts this paradigm by considering the free boundary to be the primary unknown and compute it as the minimizer of a certain shape functional using a gradient descent algorithm. In a nutshell, we compute Γ then u, and not u then Γ. Our approach proffers clear algorithmic advantages. Unilateral contact constraints on displacements, which render traditional approaches into expensive quadratic programs, appear only as Dirichlet boundary conditions along the free boundary. Displacements of the membrane need to be approximated only over the noncoincidence set, thereby rendering smaller discrete problems to be resolved. The issue of suboptimal convergence of finite element solutions stemming from the reduced regularity of displacements across the free boundary is naturally circumvented. Most importantly perhaps, our numerical experiments reveal that the free boundary can be approximated to within distances that are two orders of magnitude smaller than the mesh size used for spatial discretization. The success of the proposed algorithm relies on a confluence of factors- choosing a suitable shape functional, representing free boundary iterates with smooth implicit functions, an ansatz for the velocity of the free boundary that helps realize a gradient descent scheme and triangulating evolving domains with universal meshes. We discuss these aspects in detail and present numerous examples examining the performance of the algorithm.     

Dual-mode nested search method for categorical uncertain multi-objective optimization

Categorical multi-objective optimization is an important issue involved in many matching design problems. Non-numerical variables and their uncertainty are the major challenges of such optimizations. Therefore, this article proposes a dual-mode nested search (DMNS) method. In the outer layer, kriging metamodels are established using standard regular simplex mapping (SRSM) from categorical candidates to numerical values. Assisted by the metamodels, a k-cluster-based intelligent sampling strategy is developed to search Pareto frontier points. The inner layer uses an interval number method to model the uncertainty of categorical candidates. To improve the efficiency, a multi-feature convergent optimization via most-promising-area stochastic search (MFCOMPASS) is proposed to determine the bounds of objectives. Finally, typical numerical examples are employed to demonstrate the effectiveness of the proposed DMNS method.     

Structural shape optimization using equivalent static loads transformed from dynamic loads

In structural optimization, static loads are generally utilized although real external forces are dynamic. Dynamic loads have been considered only in small‐scale problems. Recently, an algorithm for dynamic response optimization using transformation of dynamic loads into equivalent static loads has been proposed. The transformation is conducted to match the displacement fields from dynamic and static analyses. This algorithm can be applied to large‐scale problems. However, the application has been limited to size optimization. The present study applies the algorithm to shape optimization. Because the number of degrees of freedom of finite element models is usually very large in shape optimization, it is difficult to conduct dynamic response optimization with conventional methods that directly treat dynamic response in the time domain. The optimization process is carried out by interfacing an optimization system and an analysis system for structural dynamics. Various examples are solved to verify the algorithm. The results are compared to the results from static loads. It is found that the algorithm using static loads transformed from dynamic loads based on displacement is valid for very large‐scale shape optimization problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Shape design sensitivity analysis via material derivative-adjoint variable technique for 3-D and 2-D curved boundary elements

A general approach to shape design sensitivity analysis of three- and two-dimensional elastic solid objects is developed using the material derivative-adjoint variable technique and boundary element method. The formulation of the problem is general and first-order sensitivities in the form of boundary integrals for the effect of boundary shape variations are derived for an arbitrary performance functional. Second-order quadrilateral surface elements (for 3-D problems) and quadratic boundary elements (for 2-D problems) are employed in the solution of primary and adjoint systems and discretization of the boundary integral expressions for sensitivities. The accuracy of sensitivity information is studied for selected global performance functionals and also for boundary state fields at discrete points. Numerical results are presented to demonstrate the accuracy and efficiency of this approach.