Abstract: Gecko and many insects have adopted nanoscale fibrillar structures on their feet as adhesion devices. Here, we consider adhesion between a single fiber and a substrate by van der Waals or electrostatic interactions. For a given contact area A, the theoretical pull-off force of the fiber is σthA where σth is the theoretical strength of adhesion. We show that it is possible to design an optimal shape of the tip of the fiber to achieve the theoretical pull-off force. However, such design tends to be unreliable at the macroscopic scale because the pull-off force is sensitive to small variations in the tip shape. We find that a robust design of shape-insensitive optimal adhesion becomes possible only when the diameter of the fiber is reduced to length scales on the order of 100 nm. In general, optimal adhesion could be achieved by a combination of size reduction and shape optimization. The smaller the size, the less important the shape. At large contact sizes, optimal adhesion could still be achieved if the shape can be manufactured to a sufficiently high precision. The robust design of optimal adhesion at nanoscale provides a plausible explanation for the convergent evolution of hairy attachment systems in biology.

Abstract: ▪ AbstractThis paper is a short and nonexhaustive survey of some recent developments in optimal shape design (OSD) for fluids. OSD is an interesting field both mathematically and for industrial applications. Existence, sensitivity, and compatibility of discretizations are important theoretical issues. Efficient algorithmic implementations with low complexity are also critical. In this paper we discuss topological optimization, algorithmic differentiation, gradient smoothers, Computer Aided Design (CAD)-free platforms and shock differentiation; all these are applied to a multicriterion optimization for a supersonic business jet.

Abstract: This paper presents a free-form deformation technique suitable for aerodynamic shape optimization. Because the proposed technique is independent of grid topology, we can treat structured and unstructured computational fluid dynamics grids in the same manner. The proposed technique is an alternative shape parameterization technique to a trivariate volume technique. It retains the flexibility and freedom of trivariate volumes for CFD shape optimization, but it uses a bivariate surface representation. This reduces the number of design variables by an order of magnitude, and it provides much better control for surface shape changes. The proposed technique is simple, compact, and efficient. The analytical sensitivity derivatives are independent of the design variables and are easily computed for use in a gradient-based optimization. The paper includes the complete formulation and aerodynamics shape optimization results.

Abstract: The problem of segmentation of a given image using the active contour technique is considered. An abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented. The speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or Newton-type directions for the underlying optimization problem. The Newton-type speed function is found by solving an elliptic problem on the current active contour in every time step. Numerical experiments comparing the classical gradient method with Newton's method are presented.

Abstract: To establish an alternative analytical framework for the elastic-wave imaging of underground cavities, the focus of this study is an extension of the concept of topological derivative, rooted in elastostatics and shape optimization, to three-dimensional elastodynamics involving semi-infinite and infinite solids. The main result of the proposed boundary integral approach is a formula for topological derivative, explicit in terms of the elastodynamic fundamental solution, obtained by an asymptotic expansion of the misfit-type cost functional with respect to the creation of an infinitesimal hole in an otherwise intact (semi-infinite or infinite) elastic medium. Valid for an arbitrary shape of the infinitesimal cavity, the formula involves the solution of six canonical exterior elastostatic problems, and becomes fully explicit when the vanishing cavity is spherical. A set of numerical results is included to illustrate the potential of topological derivative as a computationally efficient tool for exposing an approximate cavity topology, location, and shape via a grid-type exploration of the host solid. For a comprehensive solution to three-dimensional inverse scattering problems involving elastic waves, the proposed approach can be used most effectively as a pre-conditioning tool for more refined, albeit computationally intensive minimization-based imaging algorithms. To the authors' knowledge, an application of topological derivative to inverse scattering problems has not been attempted before; the methodology proposed in this paper could also be extended to acoustic problems.

Abstract: An optimality criteria algorithm is presented for three-dimentional truss structure optimization with multiple constraints on its natural frequencies. Both nodal coordinates and element cross-sectional areas, which are quite different in their natures, are treated simultaneously in a unified design space for structural weight minimization. First the optimality criterion is developed for a single constraint based on differentiation of the Lagrangian function. It states that, at the optimum, all of the variables should have equal efficiencies. Then, the global sensitivity numbers are introduced to solve multiple constraints of frequencies, avoiding computation of the Lagrange multipliers. Finally, upon the sensitivity analysis, the most efficient variables are identified and modified in priority. The optimal solution is achieved gradually from the initial design with a minimum weight increment. Four typical trusses are used to demonstrate the feasibility and validity of the proposed method.

Abstract: The topological sensitivity analysis provides an asymptotic expansion of a shape function with respect to the insertion of a small hole or obstacle inside a domain. This expansion can then be used for shape optimization. In this paper, such an expansion is obtained for the Stokes equations with general shape functions and arbitrarily shaped holes. A numerical example illustrates the use of the topological sensitivity in a shape optimization problem.

Abstract: Topology optimization has become very popular in industrial applications, and most FEM codes have implemented certain capabilities of topology optimization. However, most codes do not allow simultaneous treatment of sizing and shape optimization during the topology optimization phase. This poses a limitation on the design space and therefore prevents finding possible better designs since the interaction of sizing and shape variables with topology modification is excluded. In this paper, an integrated approach is developed to provide the user with the freedom of combining sizing, shape, and topology optimization in a single process.

Abstract: This paper addresses the problem of structural shape and topology optimization. A level set method is adopted as an alternative approach to the popular homogenization based methods. The paper focuses on four areas of discussion: (1) The level-set model of the structure’s shape is characterized as a region and global representation; the shape boundary is embedded in a higher-dimensional scalar function as its “iso-surface.” Changes of the shape and topology are governed by a partial differential equation (PDE). (2) The velocity vector of the Hamilton-Jacobi PDE is shown to be naturally related to the shape derivative from the classical shape variational analysis. Thus, the level set method provides a natural setting to combine the rigorous shape variations into the optimization process. (3) Perimeter regularization is incorporated in the method to make the optimization problem well-posed. It also produces an effect of the geometric heat equation, regularizing and smoothing the geometric boundaries as an anisotropic filter. (4)We further describe numerical techniques for efficient and robust implementation of the method, by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain. This would separate the issues of accuracy in numerical calculations of the physical equation and in the level-set model propagation. Finally, the benefit and the advantages of the developed method are illustrated with several 2D examples that have been extensively used in the recent literature of topology optimization, especially in the homogenization based methods. keyword: Topology optimization, level set method, shape sensitivity, curvature flow 1 Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong. Tel.: +852-2609-8487; Fax: +852-2603-6002. E-mail: yuwang@acae.cuhk.edu.hk (M. Y. Wang). 2 School of Mechanical Engineering Dalian University of Technology Dalian 116024, China

Abstract: Efficient formulas for computing the probabilities of finding exactly ν electrons in an arbitrarily chosen volume Ω⊂ℝ3 for Hartree–Fock wavefunctions are presented. These formulas allow the use of shape optimization techniques, such as level set methods, for optimizing with respect to Ω various criteria involving such probabilities. The criterion defined as the difference between the Hartree–Fock and the independent-particle model probabilities of finding ν electrons in Ω stresses the quantum effects due to the Pauli principle. We have implemented a 2D level set method for optimizing this criterion in order to study spatial separation of electron pairs in linear molecules. The method is described and the illustrative example of the BH molecule is reported.

Abstract: This paper is concerned with the formulation of a variational r-adaption method for finite-deformation elastostatic problems. The distinguishing characteristic of the method is that the variational principle simultaneously supplies the solution, the optimal mesh and, in problems of shape optimization, the equilibrium shapes of the system. This is accomplished by minimizing the energy functional with respect to the nodal field values as well as with respect to the triangulation of the domain of analysis. Energy minimization with respect to the referential nodal positions has the effect of equilibrating the energetic or configurational forces acting on the nodes. We derive general expressions for the configuration forces for isoparametric elements and nonlinear, possibly anisotropic, materials under general loading. We illustrate the versatility and convergence characteristics of the method by way of selected numerical tests and applications, including the problem of a semi-infinite crack in linear and nonlinear elastic bodies; and the optimization of the shape of elastic inclusions.

Abstract: The problem of segmentation of a given gray scale image by minimization of the Mumford-Shah functional is considered. The minimization problem is formulated as a shape optimization problem where the contour which separates homogeneous regions is the (geometric) optimization variable. Expressions for first and second order shape sensitivities are derived using the speed method from classical shape sensitivity calculus. Second order information (the shape Hessian of the cost functional) is used to set up a Newton-type algorithm, where a preconditioning operator is applied to the gradient direction to obtain a better descent direction. The issue of positive definiteness of the shape Hessian is addressed in a heuristic way. It is suggested to use a positive definite approximation of the shape Hessian as a preconditioner for the gradient direction. The descent vector field is used as speed vector field in the level set formulation for the propagating contour. The implementation of the algorithm is discussed in some detail. Numerical experiments comparing gradient and Newton-type flows for different images are presented.

Abstract: An important problem in automotive industry is how to achieve better design concepts by considering structure performance and manufacturing cost in the early stages of product development. The topology design and optimisation provide an initial design concept for downstream applications, which leads to achieving better design by using computer-aided techniques. In this research, engine mount bracket design is presented to illustrate the application of topology optimisation approach for optimal design of vehicle components under dynamic loading conditions. The objective of the proposed research is to create an initial design concept, which has optimal structural layout. The effectiveness and verification of proposed approach are demonstrated with experimental results.

Abstract: The design optimization of a 7 × 7 pin-fin heat sink is performed numerically. To achieve higher thermal performance of the heat sink, the thermal resistance at the junction of the chip and the heat sink and the overall pressure drop in the heat sink have to be minimized simultaneously. The fin height (h), fin width (w), and fan-to-heat sink distance (c) are chosen as the design variables, and the pressure drop (ΔP) and thermal resistance (θ ja ) are adopted as the objective functions. To obtain the optimum design values, we used the finite-volume method for calculating the objective functions, the Broydon-Fletcher-Goldfarb-Shanno method for solving the unconstrained nonlinear optimization problem, and the weighting method for predicting the multiobjective problem. The results show that the optimum design variables for the weighting coefficient of 0.5 are as follows: w = 4.653 mm, h = 59.215 mm, and c = 2.669 mm. The objective functions corresponding to the optimal design are calculated as ΔP = 6.82 Pa an...

Abstract: The response surface method using a three-dimensional Navier-Stokes analysis to optimize the shape of forward-curved-blade centrifugal fan is described. For the numerical analysis, Reynolds-averaged Navier-Stokes equations with the standard κ-e turbulence model are discretized with finite volume approximations. The SIMPLEC algorithm is used as a velocity-pressure correction procedure. In order to reduce the huge computing time due to a large number of blades in forward-curved-blade centrifugal fan, the flow inside of the fan is regarded as steady flow by introducing the impeller force models. Four design variables, i.e., location of cutoff, radius of cutoff, expansion angle of scroll, and width of impeller were selected to optimize the shapes of scroll and blades. Data points for response evaluations were selected by D-optimal design, and a linear programming method was used for the optimization on the response surface

Abstract: A method is presented for the efficient solution of the inverse geometric problem of detection of subsurface cavities and flaws using thermographic techniques. Here, a superposition of clusters of sources/sinks with a boundary-element solution of the forward problem offers a numerical scheme that does not require remeshing of the interior geometry as the inverse problem is solved iteratively to detect the flaw or cavity. The approach offers tremendous advantage in reducing the computational burden involved in remeshing and presents a promising technique for three-dimensional applications.

Abstract: In this investigation the hybrid cellular automaton (HCA) method for structural synthesis is extended to facilitate simultaneous topology and shape optimization. The HCA methodology has been developed for application to continuum structures. The development of this methodology has been inspired by the biological process of bone remodeling. In bone remodeling, only those elements located on the surface of the mineralized structure can be modified. In the HCA methodology implemented in this research only surface elements are allowed to change density during the structural synthesis process. The HCA method combines local design rules based on the cellular automaton paradigm and finite element analysis. Closed-loop control is used to modify the mass distribution on the internal and external surfaces of the design domain to find an optimum structure. The local control maintains a balance between mass and rigidity. The new methodology effectively combines elements of topology optimization and shape optimization into a single tool. Three classes of test problems are used to illustrate the method’s efficacy.

Abstract: The optimum design of stiffened shell structures is investigated using a robust and efficient optimization algorithm where the total weight of the structure is to be minimized subject to behavioral constraints imposed by structural design codes. Evolutionary algorithms and more specifically the evolution strategies (ES) method specially tailored for this type of problems is implemented for the solution of the structural optimization problem. The discretization of the stiffened shell is performed by means of cost-effective and reliable shell and beam elements that incorporate the natural mode concept. Three types of design variables are considered: sizing, shape, and topology. A benchmark test example is examined where the efficiency and robustness of ES over other optimization methods is investigated. Two case studies of stiffened shells are subsequently presented, where a parametric study is undertaken to obtain the most efficient design compatible with the regulations suggested by design codes such as Eurocode. The important role of the stiffeners and how they can be optimally chosen to improve the performance of shell structures in terms of carrying capacity and economy is demonstrated.

Abstract: Electrical impedance tomography (EIT) is a diffuse imaging modality in which the resistivity distribution inside the object is estimated based on electrical measurements made on the boundary. Several applications can be found in geophysics, medicine and industry. Image reconstruction is an iterative procedure in which the norm between the computed and measured voltages is minimized. Also, an additional regularization term is included in the minimized functional due to the ill-posedness of the problem. In the reconstruction process, the geometry of the object is assumed to be known. Geometry is known in many cases but there are various situations in which the shape of the domain is unknown. For example, in medical applications the shape of the domain, a part of the human body on which the measurement electrodes are attached, is unknown unless some other imaging modality is used for receiving the shape. In industrial applications, such as in the imaging of a stirrer vessel for detecting air distribution or detecting large air bubbles in pipelines, the free surface between the liquid and air is unknown and should be estimated. Within the domain we may also have 'voids' having zero conductivity and we might be interested in detecting the shapes and locations of the voids. An example could be the detection of corrosion faults in metallic plates. In this paper, two approaches for shape and free surface estimation are proposed. The approaches taken here are based on the idea used in shape optimization problems. In the first approach, the unknown shape is parametrized using the mesh nodes as parameters. In the second approach, we define new 'design variables' which are used as parameters. These design variables are the coefficients of a Bezier curve that defines the shape of the surface. In this paper, we show results of the free surface estimation from both computer simulations and tank measurements. Also, results of simultaneous reconstruction of the resistivity distribution and the free surface are shown. Comparison of the results between these two approaches will be given. The comparison shows better performance in the Bezier curve approach.

Abstract: This paper considers an optimization technique in which the objective is attained via alterations to the physical geometry of the system. This optimization framework, to be considered in the context of electron guns, is known as optimal shape design. Optimal shape design has been used in a number of applications including wing design, magnetic tape design, and nozzle design, among others. In this investigation, we use the methods of shape optimization to design the cathode of an electron gun. The dynamical equations modeling the electron particle path as well as the generalized shape optimization problem will be presented. Illustrative examples of the technique on gun designs that were previously limited to spherical cathodes will be given.

Abstract: In the present work, an environment for the shape optimization of a labyrinth seal is described. A program for a parameterized, automated grid generation is coupled with a commercial Computational Fluid Dynamics (CFD) flow solver and an optimization algorithm. Standard optimization strategies, like gradient-based methods, mostly are trapped to local optima. Therefore, the simulated annealing method is applied. It allows the finding of global minima or maxima of arbitrary functions.

Abstract: Combining the vector level set model, the shape sensitivity analysis theory with the gradient projection technique, a level set method for topology optimization with multi-constraints and multi-materials is presented in this paper. The method implicitly describes structural material interfaces by the vector level set and achieves the optimal shape and topology through the continuous evolution of the material interfaces in the structure. In order to increase computational efficiency for a fast convergence, an appropriate nonlinear speed mapping is established in the tangential space of the active constraints. Meanwhile, in order to overcome the numerical instability of general topology optimization problems, the regularization with the mean curvature flow is utilized to maintain the interface smoothness during the optimization process. The numerical examples demonstrate that the approach possesses a good flexibility in handling topological changes and gives an interface representation in a high fidelity, compared with other methods based on explicit boundary variations in the literature.

Abstract: Evolution strategies (ES) are very robust and general techniques for finding global optima in optimisation problems As with all evolutionary algorithms, ES apply evolutionary operators and select the most fit from a set of possible solutions Unlike genetic algorithms, ES do not use binary coding of individuals, working instead with real variables Many recent studies have applied evolutionary algorithms to structural problems, particularly the optimisation of trusses This paper focuses on shape optimisation of continuum structures via ES Stress analysis is accomplished by using the fixed grid finite element method, which reduces the computing time while keeping track of the boundary representation of the structure This boundary is represented by b-spline functions, circles, and polylines, whose control points constitute the parameters that govern the shape of the structure Evolutionary operations are applied to each set of variables until a global optimum is reached Several numerical examples are presented to illustrate the performance of the method Finally, structures with multiple load cases are considered along with examples illustrating the results obtained

Abstract: We construct and justify the asymptotic expansion of a solution and the corresponding energy functional of the mixed boundary-value problem for the Poisson equation in a domain with a ligament, i.e., thin curvilinear strip connecting two small parts of the boundary outside the domain. Asymptotic analysis is required in the theory of shape optimization; therefore, in contrast to other publications, we use no simplifying assumptions of the flattening of the boundary near the junction zones.

Abstract: Optimization strategies for use within an automated optimization framework for PVC profile extrusion dies are presented. The methods are designed to work in an industrial environment and must therefore account for the specific design and manufacturing techniques for these profile dies. The complex three-dimensional die geometry will be represented through a series of two-dimensional so-called ‘die-slices’. The purpose of the presented optimization cycle is to find the optimal shape of the die cross-section of the die-slices. Each die-slice geometry is optimized by a computerized optimization loop using a finite element (FE) analysis of the polymer flow through the die. Simple data representation of the complex die geometry allows an efficient parameterization. Several optimization strategies are compared regarding the achieved quality of the optimization, the computational costs, the required user interaction and the robustness in industrial applications. The strategies are (i) a global scheme in which all design variables (DVs) are dealt with simultaneously, (ii) a sequential optimization in which DVs are addressed one after each other, (iii) employing a height approximation type method in which the new values for the DVs are found using assumptions of the flow between two parallel plates, and (iv) a global scheme in which the DVs are decoupled taking advantage of the particular FE analysis applied.