A new fictitious domain method in shape optimization

The present paper is concerned with investigating the capability of the smoothness preserving fictitious domain method from Mommer (IMA J. Numer. Anal. 26:503–524, 2006) to shape optimization problems. We consider the problem of maximizing the Dirichlet energy functional in the class of all simply connected domains with fixed volume, where the state equation involves an elliptic second order differential operator with non-constant coefficients. Numerical experiments in two dimensions validate that we arrive at a fast and robust algorithm for the solution of the considered class of problems. The proposed method can be applied to three dimensional shape optimization problems.     

A measure of 2D shape-of-object dissimilarity

We describe an approach for measuring 2D shape-of-object dissimilarity. The shape of the different objects to be compared is mapped to a representation invariant under translation, rotation, and area. Thus, the shapes will have the same amount of information to describe them (equal number of pixels). The measure of dissimilarity is based on the transformation of one shape into another. This transformation is performed by moving pixels. Thus, the shape difference could be ascertained by counting how many pixels we have to move and how far to change one shape into another. When the shape transformation is performed, the distribution of the shape difference is computed, which permits an improvement in shape comparison.     

A Transformation Approach in Shape Optimization: Existence and Regularity Results

In this article, we consider a model shape optimization problem. The state variable solves an elliptic equation on a star-shaped domain, where the radius is given via a control function. First, we reformulate the problem on a fixed reference domain, where we focus on the regularity needed to ensure the existence of an optimal solution. Second, we introduce the Lagrangian and use it to show that the optimal solution possesses a higher regularity, which allows for the explicit computation of the derivative of the reduced cost functional as a boundary integral. We finish the article with some second-order optimality conditions.     

Inverse sequences, rooted trees and their end spaces

In this paper we prove that if we consider the standard real metric on simplicial rooted trees then the category Tower-Set of inverse sequences can be described by means of the bounded coarse geometry of the naturally associated trees. Using this we give a geometrical characterization of Mittag-Leffler property in inverse sequences in terms of the metrically proper homotopy type of the corresponding tree and its maximal geodesically complete subtree. We also obtain some consequences in shape theory. In particular we describe some new representations of shape morphisms related to infinite branches in trees.     

Shape Methods for the Transmission Problem with a Single Measurement

In the current work, we consider the inverse conductivity problem of recovering inclusion with one measurement. First, we use conformal mapping techniques for determining the location of the anomaly and estimating its size. We then get a good initial guess for quasi-Newton type method. The inverse problem is treated from the shape optimization point of view. We give a rigorous proof for the existence of the derivative of the state function and of shape functionals. We consider both least squares fitting and Kohn and Vogelius functionals. For the numerical implementation, we use a parameterization of shapes coupled with a boundary element method. Several numerical examples indicate the superiority of the Kohn and Vogelius functional over least squares fitting.     

Advances in concrete arch dams shape optimization

This paper presents an efficient methodology to find the optimum shape of arch dams. In order to create the geometry of arch dams a new algorithm based on Hermit Splines is proposed. A finite element based shape sensitivity analysis for design-dependent loadings involving body force, hydrostatic pressure and earthquake loadings is implemented. The sensitivity analysis is performed using the concept of mesh design velocity. In order to consider the practical requirements in the optimization model such as construction stages, many geometrical and behavioral constrains are included in the model in comparison with previous researches. The optimization problem is solved via the sequential quadratic programming (SQP) method. The proposed methods are applied successfully to an Iranian arch dam, and good results are achieved. By using such methodology, efficient software for shape optimization of concrete arch dams for practical and reliable design now is available.     

Attractors for a Three-Dimensional Thermo-Mechanical Model of Shape Memory Alloys

Abstract In this note, we consider a Frémond model of shape memory alloys. Let us imagine a piece of a shape memory alloy which is fixed on one part of its boundary, and assume that forcing terms, e.g., heat sources and external stress on the remaining part of its boundary, converge to some time-independent functions, in appropriate senses, as time goes to infinity. Under the above assumption, we shall discuss the asymptotic stability for the dynamical system from the viewpoint of the global attractor. More precisely, we generalize the paper [12] dealing with the one-dimensional case. First, we show the existence of the global attractor for the limiting autonomous dynamical system; then we characterize the asymptotic stability for the non-autonomous case by the limiting global attractor. * Project supported by the MIUR-COFIN 2004 research program on “Mathematical Modelling and Analysis of Free Boundary Problems”.     

Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the characteristic functions of the subsets of R² and their distance functions. The L² norm of the difference of characteristic functions, the L and the W^1,2 norms of the difference of distance functions define interesting topologies, in particular the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with image shapes defined on finite grids of pixels, we restrict our attention to subsets of ² of positive reach in the sense of Federer [16], with smooth boundaries of bounded curvature. For this particular set of shapes we show that the three previous topologies are equivalent. The next problem we consider is that of warping a shape onto another by infinitesimal gradient descent, minimizing the corresponding distance. Because the distance function involves an inf, it is not differentiable with respect to the shape. We propose a family of smooth approximations of the distance function which are continuous with respect to the Hausdorff topology, and hence with respect to the other two topologies. We compute the corresponding Gâteaux derivatives. They define deformation flows that can be used to warp a shape onto another by solving an initial value problem.We show several examples of this warping and prove properties of our approximations that relate to the existence of local minima. We then use this tool to produce computational definitions of the empirical mean and covariance of a set of shape examples. They yield an analog of the notion of principal modes of variation. We illustrate them on a variety of examples.     

Compact gradient tracking in shape optimization

In the present paper we consider the minimization of gradient tracking functionals defined on a compact and fixed subdomain of the domain of interest. The underlying state is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that, in contrast to the situation of gradient tracking on the whole domain, the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the shape problem under consideration. Shape functional and gradient require only knowledge of the Cauchy data of the state and its adjoint on the boundaries of the domain and the subdomain. These data can be computed by means of boundary integral equations when reformulating the underlying differential equations as transmission problems. Thanks to fast boundary element techniques, we derive an efficient algorithm to solve the problem under consideration.     

On the quartic curve of Han

The quartic curve of Han [X. Han, Piecewise quartic polynomial curves with shape parameter, Journal of Computational and Applied Mathematics 195 (2006) 34–45] can be considered as the generalization of the cubic B-spline curve incorporating shape parameters into the polynomial basis functions. We show that this curve can be considered as the linear blending of the original cubic B-spline curve and a fixed quartic curve. Moreover, we present the Bézier form of the curve, which is useful in terms of incorporating the curve into existing CAD systems. Geometric effects of the alteration of shape parameters is also discussed, including design oriented computational methods for constrained shape control of the curve.     

Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

<正>In this work we consider the problem of shape reconstruction from an unorganized data set which has many important applications in medical imaging,scientific computing,reverse engineering and geometric modelling.The reconstructed surface is obtained by continuously deforming an initial surface following the Partial Differential Equation(PDE)-based diffusion model derived by a minimal volume-like variational formulation.The evolution is driven both by the distance from the data set and by the curvature analytically computed by it.The distance function is computed by implicit local interpolants defined in terms of radial basis functions.Space discretization of the PDE model is obtained by finite co-volume schemes and semi-implicit approach is used in time/scale.The use of a level set method for the numerical computation of the surface reconstruction allows us to handle complex geometry and even changing topology, without the need of user-interaction.Numerical examples demonstrate the ability of the proposed method to produce high quality reconstructions.Moreover,we show the effectiveness of the new approach to solve hole filling problems and Boolean operations between different data sets.     

Shape optimization for Stokes flow

This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations. The structures of continuous shape gradients with respect to the shape of the variable domain for some given cost functionals are established by introducing the Piola transformation and then deriving the state derivative and its associated adjoint state. Finally we give the finite element approximation of the problem and a gradient type algorithm is effectively used for our problem.     

Shape sensitivity analysis of an elastic contact problem: Convergence of the Nitsche based finite element approximation

In a recent work, we introduced a finite element approximation for the shape optimization of an elastic structure in sliding contact with a rigid foundation where the contact condition (Signorini’s condition) is approximated by Nitsche’s method and the shape gradient is obtained via the adjoint state method. The motivation of this work is to propose an a priori convergence analysis of the numerical approximation of the variables of the shape gradient (displacement and adjoint state) and to show some numerical results in agreement with the theoretical ones. The main difficulty comes from the non-differentiability of the contact condition in the classical sense which requires the notion of conical differentiability.     

Shape inverse problem for Stokes–Brinkmann equations

In this paper, we propose an imaging technique for the detection of porous inclusions in a stationary flow governed by Stokes–Brinkmann equations. We introduce the velocity method to perform the shape deformation, and derive the structure of shape gradient for the cost functional based on the continuous adjoint method and the function space parametrization technique. Moreover, we present a gradient-type algorithm to the shape inverse problem. The numerical results demonstrate the proposed algorithm is feasible and effective for the quite high Reynolds numbers problems.     

On the selection of a good value of shape parameter in solving time-dependent partial differential equations using RBF approximation method

Radial basis function method is an effective tool for solving differential equations in engineering and sciences. Many radial basis functions contain a shape parameter c which is directly connected to the accuracy of the method. Rippa [1] proposed an algorithm for selecting good value of shape parameter c in RBF-interpolation. Based on this idea, we extended the proposed algorithm for selecting a good value of shape parameter c in solving time-dependent partial differential equations.     

On Distributed $H^1$ Shape Gradient Flows in Optimal Shape Design of Stokes Flows: Convergence Analysis and Numerical Applications

We consider optimal shape design in Stokes flow using $H^1$ shape gradient flows based on the distributed Eulerian derivatives. MINI element is used for discretizations of Stokes equation and Galerkin finite element is used for discretizations of distributed and boundary $H^1$ shape gradient flows. Convergence analysis with a priori error estimates is provided under general and different regularity assumptions. We investigate the performances of shape gradient descent algorithms for energy dissipation minimization and obstacle flow. Numerical comparisons in 2D and 3D show that the distributed $H^1$ shape gradient flow is more accurate than the popular boundary type. The corresponding distributed shape gradient algorithm is more effective.     

Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u‖² _H ¹ _(Ω) ²/‖u‖² _L ² _(∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes. Mathematics Subject Classification (2000) 35P15, 49K20, 49M25, 49Q10     

Preconditioning the Pressure Tracking in Fluid Dynamics by Shape Hessian Information

Potential flow pressure matching is a classical inverse design aerodynamic problem. The resulting loss of regularity during the optimization poses challenges for shape optimization with normal perturbation of the surface mesh nodes. Smoothness is not enforced by the parameterization but by a proper choice of the scalar product based on the shape Hessian, which is derived in local coordinates for starshaped domains. Significant parts of the Hessian are identified and combined with an aerodynamic panel solver. The resulting shape Hessian preconditioner is shown to lead to superior convergence properties of the resulting optimization method. Additionally, preconditioning gives the potential for level independent convergence.     

Shape analysis of cubic trigonometric Bézier curves with a shape parameter

For the cubic trigonometric polynomial curves with a shape parameter (TB curves, for short), the effects of the shape parameter on the TB curve are made clear, the shape features of the TB curve are analyzed. The necessary and sufficient conditions are derived for these curves having single or double inflection points, a loop or a cusp, or be locally or globally convex. The results are summarized in a shape diagram of TB curves, which is useful when using TB curves for curve and surface modeling. Furthermore the influences of shape parameter on the shape diagram and the ability for adjusting the shape of the curve are shown by graph examples, respectively.