Efficient treatment of stationary free boundary problems

In the present paper we consider the efficient treatment of free boundary problems by shape optimization. We reformulate the free boundary problem as shape optimization problem. A second order shape calculus enables us to analyze the shape problem under consideration and to prove convergence of a Ritz–Galerkin approximation of the shape. We show that Newton's method requires only access to the underlying state function on the boundary of the domain. We compute these data by boundary integral equations which are numerically solved by a fast wavelet Galerkin scheme. Numerical results prove that we succeeded in finding a fast and robust algorithm for solving the considered class of problems.     

Shape Optimization in 2D Contact Problems with Given Friction and a Solution-Dependent Coefficient of Friction

The paper deals with shape optimization of elastic bodies in unilateral contact. The aim is to extend the existing results to the case of contact problems, where the coefficient of friction depends on the solution. We consider the two-dimensional Signorini problem, coupled with the physically less accurate model of given friction, but assume a solution-dependent coefficient of friction. First, we investigate the shape optimization problem in the continuous, infinite-dimensional setting, followed by a suitable finite-dimensional approximation based on the finite-element method. Convergence analysis is presented as well. Next, an algebraic form of the state problem is studied, which is obtained from the discretized problem by further approximating the frictional term by a quadrature rule. It is shown that if the coefficient of friction is Lipschitz continuous with a sufficiently small modulus, then the algebraic state problem is uniquely solvable and its solution is a Lipschitz continuous function of the control variable, describing the shape of the elastic body. For the purpose of numerical solution of the shape optimization problem via the so-called implicit programming approach we perform sensitivity analysis by using the tools from the generalized differential calculus of Mordukhovich. The paper is concluded first order optimality conditions.     

2D shape optimization under proximity constraints by CFD and response surface methodology

In this paper we consider two-dimensional CFD-based shape optimization in the presence of obstacles, which introduce nontrivial proximity constraints to the optimization problem. Built on Gregory’s piecewise rational cubic splines, the main contribution of this paper is the introduction of such parametric deformations to a nominal shape that are guaranteed to satisfy the proximity constraints. These deformed shape candidates are then used in the identification of a multivariate polynomial response surface; proximity-constrained shape optimization thus reduces to parametric optimization on this polynomial model, with simple interval bounds on the design variables. We illustrate the proposed approach by carrying out lift and/or drag optimization for the NACA 0012 airfoil containing a rectangular fuel tank: By identifying polynomial response surfaces using a large batch of 1800 design candidates, we conclude that the lift coefficient can be optimized by a linear model, whereas the drag coefficient can be optimized by using a quadratic model. Higher order polynomial models yield no improvement in the optimization.     

Design sensitivity for arch structures with respect to midsurface shape under static loading

We consider in this paper an example of structural optimization in which the structure is a loaded arch and the design variable is the shape of the arch. We concentrate on differentiability of static response with respect to shape changes. After recalling the arch equation with its functional spaces and the optimization problem, we state a differentiability theorem and provide a detailed proof. Numerical use of this result is finally discussed.     

Shape optimization for the Maxwell equations under weaker regularity of the data

We consider a shape optimization problem for Maxwell's equations with a strictly dissipative boundary condition. In order to characterize the shape derivative as a solution to a boundary value problem, sharp regularity of the boundary traces is critical. This Note establishes the Fréchet differentiability of a shape functional.     

The optimal shape of a dendrite sealed at both ends

In this paper, we are interested in the geometric structures which appear in nature. We consider the example of a nerve fiber and we suppose that shapes in nature arise in order to optimize some criterion. Then, we try to solve the problem consisting in searching the shape of a nerve fiber for a given criterion. The first considered criterion represents the attenuation in space of the electrical message troughout the fiber and seems to be relevant. Our second criterion represents the attenuation in time of the electrical message and doesn't provide a realistic shape. We prove that the associated optimization problem has no solution.     

A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.     

Nonsmooth domain optimization for elliptic equations with unilateral conditions

In the paper we consider elliptic boundary problems in domains having cuts (cracks). The non-penetration condition of inequality type is prescribed at the crack faces. A dependence of the derivative of the energy functional with respect to variations of crack shape is investigated. This shape derivative can be associated with the crack propagation criterion in the elasticity theory. We analyze an optimization problem of finding the crack shape which provides a minimum of the energy functional derivative with respect to a perturbation parameter and prove a solution existence to this problem.     

Non-linear eigenvalue problems for <Emphasis Type="Italic">p</Emphasis>-Laplacian with variable domain

In this work we consider some eigenvalue problems for p-Laplacian with variable domain. Eigenvalues of this operator are taken as a functional of the domain. We calculate the first variation of this functional, using the obtained formula investigate behavior of the eigenvalues when the domain varies. Then we consider one shape optimization problem for the first eigenvalue, prove the necessary condition of optimality relatively domain, offer an algorithm for the numerical solution of this problem.     

Existence and approximation results for shape optimization problems in rotordynamics

We consider a shape optimization problem in rotordynamics where the mass of a rotor is minimized subject to constraints on the natural frequencies. Our analysis is based on a class of rotors described by a Rayleigh beam model including effects of rotary inertia and gyroscopic moments. The solution of the equation of motion leads to a generalized eigenvalue problem. The governing operators are non-symmetric due to the gyroscopic terms. We prove the existence of solutions for the optimization problem by using the theory of compact operators. For the numerical treatment of the problem a finite element discretization based on a variational formulation is considered. Applying results on spectral approximation of linear operators we prove that the solution of the discretized optimization problem converges towards the solution of the continuous problem if the discretization parameter tends to zero. Finally, a priori estimates for the convergence order of the eigenvalues are presented and illustrated by a numerical example.     

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.     

Optimal design of a fin in steady-state

A fin serves as an extended surface to enhance the heat transfer from a larger heated mass to which it is attached. We consider a fin in steady-state and find its shape that maximizes the efficiency. Our approach is based on a piecewise linear approximation of design. We develop the corresponding algorithm and conduct numerical experiments. Our optimization problem is similar but not identical to the problem studied by L. Hanin. We discuss the applicability of Schmidt’s hypothesis for our model. This paper is an extension of the previous authors’ paper where we considered the shape of the fin that minimizes the cooling time of a heated mass.     

Variational sensitivity analysis and changes in the material configuration

This contribution is concerned with some aspects of variational design sensitivity analysis in the physical and material configuration. Sensitivity analysis is a branch of structural optimization, e.g. shape or topology optimization. In these disciplines we consider variations of the material configuration and we are interested in the change of the state variables and the objective functional due to these variations. In the context of structural optimization this is termed as design sensitivity analysis. The sensitivities are required in order to solve the corresponding Lagrangian equation within standard nonlinear programming algorithms. In many engineering applications, the energy functional of the problem is used as the objective functional. In this paper, we consider variations of the energy with respect to the state and the design and we investigate sensitivity relations for the physical and material problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Controls of Systems Governed by Semilinear Fractional Differential Equations with Not Instantaneous Impulses

In this note, we consider a control theory problem involving a strictly convex energy functional, which is not Gâteaux differentiable. The functional came up in the study of a shape optimization problem, and here we focus on the minimization of this functional. We relax the problem in two different ways and show that the relaxed variants can be solved by applying some recent results on two-phase obstacle-like problems of free boundary type. We derive an important qualitative property of the solutions, i.e., we prove that the minimizers are three-valued, a result which significantly reduces the search space for the relevant numerical algorithms.     

Coupling of FEM and BEM in Shape Optimization

In the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown shape, especially L ²-tracking type functionals. The underlying state equation is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that the shape Hessian is not strictly H ^1/2-coercive at the optimal domain which implies ill-posedness of the optimization problem under consideration. Since the adjoint state depends directly on the state, we propose a coupling of finite element methods (FEM) and boundary element methods (BEM) to realize an efficient first order shape optimization algorithm. FEM is applied in the compact region while the rest is treated by BEM. The coupling of FEM and BEM essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations.This research has been carried out when the second author stayed at the Department of Mathematics, Utrecht University, The Netherlands, supported by the EU-IHP project Nonlinear Approximation and Adaptivity: Breaking Complexity in Numerical Modelling and Data Representation     

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.     

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.     

On Solution to an Optimal Shape Design Problem in 3-Dimensional Linear Magnetostatics

In this paper we present theoretical, computational, and practical aspects concerning 3-dimensional shape optimization governed by linear magnetostatics. The state solution is approximated by the finite element method using Nédélec elements on tetrahedra. Concerning optimization, the shape controls the interface between the air and the ferromagnetic parts while the whole domain is fixed. We prove the existence of an optimal shape. Then we state a finite element approximation to the optimization problem and prove the convergence of the approximated solutions. In the end, we solve the problem for the optimal shape of an electromagnet that arises in the research on magnetooptic effects and that was manufactured afterwards.     

Optimization in constrained crack problems

Following the optimization approach to brittle fracture, we consider the evolution of a crack in a domain as solution of the global problem of constrained minimization of the total potential energy with respect to both state variables (the displacement vector) and shape variables (parameters of the crack shape). This formulation describes the initiation of a crack in a homogeneous body and its stable as well as unstable propagation, which allows also a kink (topology change) of the crack during the evolution process. By this we assume that contact can occurs between the opposite crack faces. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fast Kriging-based strategy for the optimization of electrical device

The primary goal of a shape optimization problem is to find the best design w.r.t. a given criterion. A however non less important goal is to achieve this task within a reasonable time. In realistic situations, a reliable but slow method might not always provide its best capacity if the computational resources are worked out before convergence. The expensive simulation-based optimization of a physical system is one of these situations. In order to tackle this issue, we consider a fast but low-fidelity model given by a Kriging method. We propose a simple adaptive scheme in order to enhance the reliability of the optimization procedure and thus avoid a premature convergence. Numerical experiments are conducted for the electrostatic design of a realistic device. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)