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     

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.     

On a Kohn-Vogelius like formulation of free boundary problems

The present paper is concerned with the solution of a Bernoulli type free boundary problem by means of shape optimization. Two state functions are introduced, namely one which satisfies the mixed boundary value problem, whereas the second one satisfies the pure Dirichlet problem. The shape problem under consideration is the minimization of the L ²-distance of the gradients of the state functions. We compute the corresponding shape gradient and Hessian. By the investigation of sufficient second order conditions we prove algebraic ill-posedness of the present formulation. Our theoretical findings are supported by numerical experiments.     

Tracking Dirichlet Data in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> is an Ill-Posed Problem

A stationary free boundary problem is solved by tracking the Dirichlet data at the free boundary. The shape gradient and Hessian of the tracking functional under consideration are computed. By analyzing the shape Hessian in case of matching Dirichlet data, it is shown that this shape optimization problem is algebraically ill-posed. Numerical experiments are carried out to validate and quantify the results.     

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. Online publication 23 January 2004.     

Solving a free boundary problem with nonconstant coefficients

The present article is concerned with the numerical solution of a free boundary problem for an elliptic state equation with nonconstant coefficients. We maximize the Dirichlet energy functional over all domains of fixed volume. The domain under consideration is represented by a level set function, which is driven by the objective's shape gradient. The state is computed by the finite element method where the underlying triangulation is constructed by means of a marching cubes algorithm. We show that the combination of these tools lead to an efficient solver for general shape optimization problems.     

Legendre-Laguerre coupled spectral element methods for second- and fourth-order equations on the half line

Some Legendre spectral element/Laguerre spectral coupled methods are proposed to numerically solve second- and fourth-order equations on the half line. The proposed methods are based on splitting the infinite domain into two parts, then using the Legendre spectral element method in the finite subdomain and Laguerre method in the infinite subdomain. C⁰ or C¹-continuity, according to the problem under consideration, is imposed to couple the two methods. Rigorous error analysis is carried out to establish the convergence of the method. More importantly, an efficient computational process is introduced to solve the discrete system. Several numerical examples are provided to confirm the theoretical results and the efficiency of the method.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

A Green’s function formulation of nonlocal finite-difference schemes for reaction-diffusion equations

Reaction-diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. The aim of this work is to show that a Green’s function formulation of reaction-diffusion PDEs is a suitable framework to derive FD schemes incorporating both O(h²) accuracy and nonlocal approximations in the whole domain (including boundary nodes). By doing so, the approach departs from a Green’s function formulation of the boundary-value problem to pose an approximation problem based on a domain decomposition. Within each subdomain, the corresponding integral equation is forced to have zero residual at given grid points. Different FD schemes are obtained depending on the numerical scheme used for computing the Green’s integral over each subdomain. Dirichlet and Neumann boundary conditions are considered, showing that the FD scheme based on the Green’s function formulation incorporates, in a natural way, the effects of boundary nodes in the discretization approximation.     

Convergence of euler-stokes splitting of the navier-stokes equations

We consider approximation by partial time steps of a smooth solution of the Navier-Stokes equations in a smooth domain in two or three space dimensions with no-slip boundary condition. For small k > 0, we alternate the solution for time k of the inviscid Euler equations, with tangential boundary condition, and the solution of the linear Stokes equations for time k, with the no-slip condition imposed. We show that this approximation remains bounded in H^2,p and is accurate to order k in L^p for p > ∞. The principal difficulty is that the initial state for each Stokes step has tangential velocity at the boundary generated during the Euler step, and thus does not satisfy the boundary condition for the Stokes step. The validity of such a fractional step method or splitting is an underlying principle for some computational methods. © 1994 John Wiley & Sons, Inc.     

On the convergence of the conjugate gradient method for singular capacitance matrix equations from the Neumann problem of the Poisson equation

Summary It is shown analytically in this work that the conjugate gradient method is an efficient means of solving the singular capacitance matrix equations arising from the Neumann problem of the Poisson equation. The total operation count of the algorithm does not exceed constant timesN ²logN(N=1/h) for any bounded domain with sufficiently smooth boundary.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under Grant No. MCS75-17385. Also partially supported by the Energy Research and Development Administration     

Sampling of Discontinuous Dirac Systems

Homogenization and error analysis of an optimal interior control problem in the framework of Stokes’ system, on a domain with rapidly oscillating boundary, are the subject matters of this article. We consider a three dimensional domain constituted of a parallelepiped with a large number of rectangular cylinders at the top of it. An interior control is applied in a proper subdomain of the parallelepiped, away from the oscillating volume. We consider two types of functionals, namely a functional involving the L ²-norm of the state variable and another one involving its H ¹-norm. The asymptotic analysis of optimality systems for both cases, when the cross sectional area of the rectangular cylinders tends to zero, is done here. Our major contribution is to derive error estimates for the state, the co-state and the associated pressures, in appropriate functional spaces.     

Energy models where the equations are defined on different domains

We investigate a stationary energy model for semiconductor devices and respect the more realistic assumption that the continuity equations for electrons and holes have to be investigated only in a subdomain Ω₀ of the domain of definition Ω of the energy balance equation and of the Poisson equation. This nonlinear system of model equations is strongly coupled and has to be considered in heterostructures and with mixed boundary conditions. We obtain a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a W^1,p -regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodic solutions of the Navier�CStokes equations with the inhomogeneous time-dependent boundary data under the general flux condition

We consider the initial boundary value problem to the Navier–Stokes equations in a bounded domain with the inhomogeneous time-dependent data b(t) ? H^1/2(?W){\beta(t) \in H^{1/2}(\partial\Omega)} under the general flux condition. We establish a reproductive property for weak solutions of the Navier–Stokes equations. Here, the reproductive property is regarded as the generalization of the time periodicity. As an application, we can prove the existence of periodic weak solutions.     

Finite element approximation of a shape optimization problem for von karman system

An optimal shape design problem of an elastic body described by a system of two nonlinear elliptic equations is considered. The problem is to find the boundary of the domain occupied by the body in such a way that the stiffnes of the system in the equilibrium state is minimized.

It is assumed that the volume of the body is constant. Moreover, the function describing the boundary of the domain and its gradient are bounded.     

Finite element solutions for three‐dimensional elliptic boundary value problems on unbounded domains

The finite element (FE) solutions of a general elliptic equation ?div([a_ij] ??u) + u = f in an exterior domain Ω, which is the complement of a bounded subset of R ³, is considered. The most common approach to deal with exterior domain problems is truncating an unbounded subdomain Ω_∞, so that the remaining part Ω_B = Ω\Ω _∞ is bounded, and imposing an artificial boundary condition on the resulted artificial boundary Γ_a = Ω _∞ ∩ Ω _B. In this article, instead of discarding an unbounded subdomain Ω_∞ and introducing an artificial boundary condition, the unbounded domain is mapped to a unit ball by an auxiliary mapping. Then, a similar technique to the method of auxiliary mapping, introduced by Babu?ka and Oh for handling the domain singularities, is applied to obtain an accurate FE solution of this problem at low cost. This method thus does have neither artificial boundary nor any restrictions on f. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A priori estimates for a solution to the first initial-boundary-value problem for a system of fully nonlinear parabolic equations

A priori estimates for a solution to a system of fully nonlinear parabolic equations are obtained in a bounded domain under the condition that the solution vanishes on the boundary of the domain. The method of obtaining a priori estimates is based on the possibility of reducing the problem under consideration to the Cauchy problem for a scalar equation on a manifold without boundary in some linear space. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 46–71.     

On the cost of controlling unstable systems: The case of boundary controls

We discuss the cost of controlling parabolic equations of the formy _t + δ² y +kδy = 0 in a bounded smooth domain Ώ ofℝ ^d by means of a boundary control. More precisely, we are interested in the cost of controlling from zero initial state to a given final state (in a suitable approximate sense) at timeT > 0 and in the behavior of this cost ask → ∞. We introduce finite-dimensional Galerkin approximations and prove that they are exactly controllable. Moreover, we also prove that the cost of controlling converges exponentially to zero ask → ∞. This proves, roughly speaking, that when the system becomes more unstable it is easier to control. The system under consideration does not admit a variational formulation. Thus, in order to introduce its Galerkin approximation, we first approximate it by means of a singular perturbation. We also develop a method for the construction of special Galerkin bases well adapted to the control problem. Dedicated to John E. Lagnese on his 60th Birthday. Supported by project PB93-1203 of the DGICYT (Spain) and grant CHRX-CT94-0471 of the European Union.     

Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions

In this Note, we discuss the numerical solution of a system of Eikonal equations with Dirichlet boundary conditions. Since the problem under consideration has infinitely many solutions, we look for those solutions which are nonnegative and of maximal (or nearly maximal) L¹-norm. The computational methodology combines penalty, biharmonic regularization, operator splitting, and finite element approximations. Its practical implementation requires essentially the solution of cubic equations in one variable and of discrete linear elliptic problems of the Poisson and Helmholtz type. As expected, when the spatial domain is a square whose sides are parallel to the coordinate axes, and when the Dirichlet data vanishes at the boundary, the computed solutions show a fractal behavior near the boundary, and particularly, close to the corners. To cite this article: B. Dacorogna et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Persistence of bounded solutions of parabolic equations under domain perturbation

The aim of this paper is to study the behavior of bounded solutions of parabolic equations on the whole real line under perturbation of the underlying domain. We give the convergence of bounded solutions of linear parabolic equations in the L ² and the L ^p -settings. For the L ^p -theory, we also prove the H?lder regularity of bounded solutions with respect to time. In addition, we study the persistence of a class of bounded solutions which decay to zero at t → ±∞ of semilinear parabolic equations under domain perturbation.