Class Of Shape Optimization Problems Research Articles

Shape optimization for interface identification in nonlocal models

Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface–dependent kernels. We derive a novel shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques. The code for obtaining the results in this paper is published at (https://github.com/schustermatthias/nlshape).

Epsilon-regularity for the solutions of a free boundary system

This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions u and v , and a domain \Omega ; with u and v being both positive in \Omega , vanishing simultaneously on \partial\Omega , and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on \partial\Omega . Precisely, we consider solutions u, v \in C(B_1) of -\Delta u= f \ \text{ and } \ -\Delta v=g \quad\text{in }\Omega=\{u>0\}=\{v>0\}, \frac{\partial u}{\partial n}\frac{\partial v}{\partial n}=Q \quad\text{on }\partial\Omega\cap B_1. Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions \sqrt{uv} and \frac12(u+v) . Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies C^{1,\alpha} regularity.

Analysis of General Shape Optimization Problems in Nonlinear Acoustics

In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures. This work analyzes a class of shape optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theoretical framework, the Westervelt and Kuznetsov equations of nonlinear acoustics are obtained as particular cases. The quadratic gradient nonlinearity, specific to the Kuznetsov equation, requires special attention throughout. To prove the existence of the Eulerian shape derivative, we successively study the local well-posedness and regularity of the forward problem, uniformly with respect to shape variations, and prove that it does not degenerate under the hypothesis of small initial and boundary data. Additionally, we prove Hölder-continuity of the acoustic potential with respect to domain deformations. We then derive and analyze the corresponding adjoint problems for several different cost functionals of practical interest and conclude with the expressions of well-defined shape derivatives.

A class of shape optimization problems for some nonlocal operators

Abstract In this work we study a family of shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.

A Glimpse of Shape Optimization Problems

In this mini review, we give a glimpse of a branch of geometric analysis known as shape optimization problems. We introduce isoperimetric problems as a special class of shape optimization problems. We include a brief history of the isoperimetric problems and give a brief survey of the kind of shape optimization problems that we (with our collaborators) have worked on. We discuss the key ideas used in proving these results in the Euclidean case. Without getting into the technicalities, we mention how we generalized the results which were known in the Euclidean case to other geometric spaces. We also describe how we extended these results from the linear setting to a non-linear one. We describe briefly the difficulties faced in proving these generalized versions and how we overcame these difficulties.

Shape Optimization of Shell Structure Acoustics

This paper provides a rigorous framework for the numerical solution of shape optimization problems in shell structure acoustics using a reference-domain approach. The structure is modeled with Naghdi shell equations, fully coupled to boundary integral equations on a minimally regular surface, permitting the formulation of three-dimensional radiation and scattering problems on a two-dimensional set of reference coordinates. We prove well-posedness of this model, and Fréchet differentiability of the state with respect to the surface shape. For a class of shape optimization problems we prove existence of optimal solutions under slightly stronger surface regularity assumptions. Finally, adjoint equations are used to efficiently compute derivatives of the radiated field with respect to large numbers of shape parameters, which allows consideration of a rich space of shapes and, thus, of a broad range of design problems. A numerical example is presented to illustrate the applicability of our theoretical results.

Distributed shape derivativeviaaveraged adjoint method and applications

The structure theorem of Hadamard–Zolésio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. Actually the domain representation, also known as distributed shape derivative, is more general than the boundary expression as it is well-defined for shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithms. In this paper we describe several advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We identify a tensor representation of the distributed shape derivative, study its properties and show how it allows to recover the boundary expression directly. We use a novel Lagrangian approach, which is applicable to a large class of shape optimization problems, to compute the distributed shape derivative. We also apply the technique to retrieve the distributed shape derivative for electrical impedance tomography. Finally we explain how to adapt the level set method to the distributed shape derivative framework and present numerical results.

The method of fundamental solutions applied to boundary eigenvalue problems

We develop methods based on fundamental solutions to compute the Steklov, Wentzell and Laplace–Beltrami eigenvalues in the context of shape optimization. In the class of smooth simply connected two dimensional domains the numerical method is accurate and fast. A theoretical error bound is given along with comparisons with mesh-based methods. We illustrate the use of this method in the study of a wide class of shape optimization problems in two dimensions. We extend the method to the computation of the Laplace–Beltrami eigenvalues on surfaces and we investigate some spectral optimal partitioning problems.

Shape optimization problems with Robin conditions on the free boundary

We provide a free discontinuity approach to a class of shape optimization problems involving Robin conditions on the free boundary. More precisely, we identify a large family of domains on which such problems are well posed in a way that the extended problem can be considered a relaxed version of the corresponding one on regular domains, we prove existence of a solution and obtain some qualitative information on the optimal sets.

On Shape Stability of Incompressible Fluids Subject to Navier’s Slip Condition

The paper is concerned with the equations of motion for incompressible fluids that slip at the wall. Particular interest is in the domain dependence of weak solutions. We prove that the solutions depend continuously on the perturbation of the boundary provided that the latter remains in the class \({\mathcal {C}^{1, 1}}\) . The result is applicable to a wide class of shape optimization problems and is optimal in terms of boundary regularity.

Spectral optimization problems

In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered.

Level set method with topological derivatives in shape optimization

A class of shape optimization problems is solved numerically by the level set method combined with the topological derivatives for topology optimization. Actually, the topology variations are introduced on the basis of asymptotic analysis, by an evaluation of extremal points (local maxima for the specific problem) of the so-called topological derivatives introduced by Sokolowski and Zochowski [J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4) (1999), pp. 1251–1272] for elliptic boundary value problems. Topological derivatives are given for energy functionals of linear boundary value problems. We present results, including numerical examples, which confirm that the application of topological derivatives in the framework of the level set method really improves the efficiency of the method. Examples show that the level set method combined with the asymptotic analysis is robust for the shape optimization problems, and it allows us to identify the better solution compared to the pure level set method exclusively based on the boundary variation technique.

Second-order shape optimization using wavelet BEM

This present paper is concerned with second-order methods for a class of shape optimization problems. We employ a complete boundary integral representation of the shape Hessian which involves first- and second-order derivatives of the state and the adjoint state function, as well as normal derivatives of its local shape derivatives. We introduce a boundary integral formulation to compute these quantities. The derived boundary integral equations are solved efficiently by a wavelet Galerkin scheme. A numerical example validates that, in spite of the higher effort of the Newton method compared to first-order algorithms, we obtain more accurate solutions in less computational time.

Shape Optimization for Semi-Linear Elliptic Equations Based on an Embedding Domain Method

We study a class of shape optimization problems for semi-linear elliptic equations with Dirichlet boundary conditions in smooth domains in R2. A part of the boundary of the domain is variable as the graph of a smooth function. The problem is equivalently reformulated on a fixed domain. Continuity of the solution to the state equation with respect to domain variations is shown. This is used to obtain differentiability in the general case, and moreover a useful formula for the gradient of the cost functional in the case where the principal part of the differential operator is the Laplacian.

$C^2,\alpha$ Existence Result for a Class of Shape Optimization Problems

In this paper we give a sufficient condition for the existence of regular optimal domains for a class of shape optimization problems. This consists of finding a $C^{2,\alpha}$ domain minimizing locally a shape functional depending on the perimeter of the domain and on a general term, which in most PDE applications represents the energy associated with the state equation, under the constraint that the measure of the domain is given. The proof of this result is based on another existence result for $C^{2,\alpha}$ solutions for a class of free boundary problems that are critical domains for the shape functionals considered previously. A key point is the introduction of a special domain transformation, which has a separate term responsible for the domain translation and another which is basically only responsible for "pure" deformation of the domain. As an application, a typical example involving the Dirichlet problem in RN is considered.

Optimality Conditions for Simultaneous Topology and Shape Optimization

New optimality conditions are derived for a class of shape optimization problems. The conditions are established on the boundary by an application of the boundary variations technique and in the interior of an optimal domain by exploiting the topological derivative method. An example is provided for which the classical second order sufficient optimality conditions are verified for an optimal simply connected domain. However, the value of the cost can be improved by the topology variations, and therefore, the optimal solution can be substantially changed by applying the topology optimization.

On the Bernoulli free boundary problem and related shape optimization problems

This paper deals with the classical Bernoulli free boundary problem. We are interested in solving some shape optimization problems related to this free boundary problem. We prove the continuous dependence of the solution with respect to the data K, working with Hausdorff convergence. We can deduce an existence result for a large class of shape optimization problems. Finally, we give some ideas for a numerical method, based on the use of conformal mappings, to solve such problems in two dimensions.

An existence result for a class of shape optimization problems

Given a bounded open subset Ω of R n, we prove the existence of a minimum point for a functional F defined on the family A(Ω) of all “quasiopen” subsets of Ω, under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A(Ω) with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal k th eigenvalue (or with minimal capacity) with respect to a given elliptic operator.