Showing papers on "Free boundary problem published in 1992"

Boundary Integral and Singularity Methods for Linearized Viscous Flow

Abstract: 1. Preliminaries 2. Green's Functions and the Boundary Integral Equation 3. Green's Functions in domains bounded by a solid surface 4. Generalized boundary integral methods 5. Interfacial motion 6. Boundary integral methods based on the Stream Function 7. Discrete representation of a boundary 8. Numerical solution of the integral equations.

A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests

Abstract: A new strategy based on the stabilized space-time finite element formulation is proposed for computations involving moving boundaries and interfaces. In the deforming-spatial-domain/space-time (DSD/ST) procedure the variational formulation of a problem is written over its space-time domain, and therefore the deformation of the spatial domain with respect to time is taken into account automatically. Because the space-time mesh is generated over the space-time domain of the problem, within each time step, the boundary (or interface) nodes move with the boundary (or interface). Whether the motion of the boundary is specified or not, the strategy is nearly the same. If the motion of the boundary is unknown, then the boundary nodes move as defined by the other unknowns at the boundary (such as the velocity or the displacement). At the end of each time step a new spatial mesh covers the new spatial domain. For computational feasibility, the finite element interpolation functions are chosen to be discontinuous in time, and the fully discretized equations are solved one space-time slab at a time.

A new outflow boundary condition

Abstract: SUMMARY Boundary conditions come from Nature. Therefore these conditions exist at natural boundaries. Often, owing to limitations in computing power and means, large domains are truncated and confined between artificial synthetic boundaries. Then the required boundary conditions there cannot be provided naturally and there is a need to fabricate them by intuition, experience, asymptotic behaviour and numerical experimentation. In this work several kinds of outflow boundary conditions, including essential, natural and free boundar conditions, are evaluated for two flow and heat transfer model problems. A new outflow boundary condition, called hereafter the f?ee boundary condition, is introduced and tested. This free boundary condition is equivalent to extending the validity of the weak form of the governing equations to the synthetic outflow instead of replacing them there with unknown essential or natural boundary conditions. In the limit of zero Reynolds number the free boundary condition minimizes the energy functional among all possible choices of outflow boundary conditions. A review of results from applications of the same boundary conditions to several other flow situations is also presented and discussed.

Solution of a boundary value problem for the Helmholtz equation via variation of the boundary into the complex domain

Abstract: In this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials. This corresponds to a two-dimensional quasi-periodic boundary value problem for the Helmholtz equation. We prove that solutions behave analytically with respect to variations of the interface. The interest of this result is both theoretical – the legitimacy of power series expansions in the parameters of the problem has indeed been questioned – and, perhaps more importantly, practical: we have found that the solution can be computed on the basis of this observation. The simple algorithm that results from such boundary variations is described. To establish the property of analyticity of the solution for the grating with respect to the height δ, we present a holomorphic formulation of the problem using surface potentials. We show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary, or, in other words, that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables ( x , y , δ). This permits us to conclude, in particular, that the partial derivatives of u with respect to δ at δ = 0 satisfy certain boundary value problems for the Helmholtz equation, in regions with plane boundaries, which can be solved in a closed form.

Partial differential equations and complex analysis

Abstract: The Dirichlet Problem in the Complex Plane Review of Fourier Analysis Pseudodifferential Operators Elliptic Operators Elliptic Boundary Value Problems A Degenerate Elliptic Boundary Value Problem The ?- Neumann Problem Applications of the ?- Neumann Problem The Local Solvability Issue and a Look Back.

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

Abstract: In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.

Laminar flow analysis

Abstract: Preface 1. Derivation of the Navier-Stokes equations 2. Exact solutions of the Navier-Stokes equations 3. Boundary layer theory 4. Thermal boundary layers - forced convection 5. Thermal boundary layers - free convection 6. Compressible boundary layers Appendices.

The conductive boundary condition for Maxwell's equations

Abstract: First, the conductive boundary value problem is derived for the quasi-stationary Maxwell equations that arise in the study of magnetotellurics. Then the boundary integral equation method is used to prove the existence and uniqueness of solutions of the problem. The final section is devoted to a study of the set of far field patterns for scattering problems with plane wave incidence.

The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its application

Abstract: The exterior boundary value problems of linear elastic equations are considered. A sequence of approximations to the exact boundary conditions at an artificial boundary is given. Then the original problem is reduced to a boundary value problem on a bounded domain. Furthermore, a finite element approximation of this problem and optimal error estimates are obtained. Finally, a numerical example shows the effectiveness of this method.

The Blasius problem formulated as a free boundary value problem

Abstract: In the present paper we point out that the correct way to solve the Blasius problem by numerical means is to reformulate it as free boundary value problem. In the new formulation the truncated boundary (instead of infinity) is the unknown free boundary and it has to be determined as part of the numerical solution. Taking into account the “partial” inavariance of the mathematical model at hand with respect to a stretching group we define a non-iterative transformation method. Further, we compare the improved numerical results, obtained by the method in point, with analytical and numerical ones. Moreover, the numerical results confirm that the question of accuracy depends on the value of the free boundary. Therefore, this indicates that boundary value problems with a boundary condition at infinity should be numerically reformulated as free boundary value problems.

Boundary-conforming mapping applied to computations of highly deformed solidification interfaces

Abstract: A new boundary-conforming mapping is developed for the calculation of highly deformed cellular solidification interfaces in a model of directional solidification of a binary alloy. The mapping is derived through a variational formulation that is designed so that the grid penetrates the grooves between cells along the interface without causing a loss of ellipticity of the mapping equations. A finite element/Newton method is presented for simultaneous solution of the free boundary problem described by the solutal model of directional solidification and the mapping equations. Results are compared to previous calculations and demonstrate the importance of accurate representation of the interface shape for understanding the solution structure.

Boundary conditions for approximate differential equations

Abstract: A large number of mathematical models are expressed as differential equations. Such models are often derived through a slowly-varying approximation under the assumption that the domain of interest is arbitrarily large; however, typical solutions and the physical problem of interest possess finite domains. The issue is: what are the correct boundary conditions to be used at the edge of the domain for such model equations? Centre manifold theory [24] and its generalisations may be used to derive these sorts of approximations, and higher-order refinements, in an appealing and systematic fashion. Furthermore, the centre manifold approach permits the derivation of appropriate initial conditions and forcing for the models [25, 7]. Here I show how to derive asymptotically-correct boundary conditions for models which are based on the slowly-varying approximation. The dominant terms in the boundary conditions typically agree with those obtained through physical arguments. However, refined models of higher order require subtle corrections to the previously-deduced boundary conditions, and also require the provision of additional boundary conditions to form a complete model.

BEM approach to inverse heat conduction problems

Abstract: In the paper the inverse problem of the estimation of the temperature and heat flux on the surface of a heat conducting body is considered. Since the problem belongs to the ill-posed, the method of solving the boundary probelem as well as the method of stabilizing the results of calculations are required. The boundary element method is applied to solve the boundary problem whereas combined ‘future steps’ and the regularization method is applied to obtain stable results. A numerical example is included.

Least Squares Method

Abstract: This chapter analyzes the procedure and yield of least square method. It is applicable to ordinary differential equations and partial differential equations. It yields an approximation to the solution. A variational principle is created for a given differential equation, and then an approximation to the solution with some free parameters is proposed. By use of the variational principle, the free parameters are determined. The solution of the two point boundary value problem is approximated. This approximation has been chosen in such a way that the boundary conditions for u ( x ) are satisfied.

Diffusion approximation for GI/G /1 controlled queues

Abstract: A queueing model is considered in which a controller can increase the service rate. There is a holding cost represented by functionh and the service cost proportional to the increased rate with coefficientl. The objective is to minimize the total expected discounted cost. Whenh andl are small and the system operates in heavy traffic, the control problem can be approximated by a singular stochastic control problem for the Brownian motion, namely, the so-called “reflected follower” problem. The optimal policy in this problem is characterized by a single numberz * so that the optimal process is a reflected diffusion in [0,z *]. To obtainz * one needs to solve a free boundary problem for the second order ordinary differential equation. For the original problem the policy which increases to the maximum the service rate when the normalized queue-length exceedsz * is approximately optimal.

Global uniform decay rates for the solutions to wave equation with nonlinear boundary conditions

Abstract: Wave equation with nonlinear boundary conditions of Neumann type is considered. It is shown that the boundary damping produces a uniform global stability of the corresponding solutions.

Local high‐order radiation boundary conditions for the two‐dimensional time‐dependent structural acoustics problem

Abstract: The time‐dependent structural acoustics problem involving solution of the coupled wave equation over an infinite fluid domain is posed as a coupled problem over a finite fluid domain with local time‐dependent radiation boundary conditions applied to the fluid truncation boundary. The proposed radiation boundary conditions are based on an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a nondimensional wave number. A sequence of differential operators that match the leading terms of the asymptotic expansion provide boundary conditions that are of progressively higher order and increasing accuracy. Time‐dependent boundary conditions are obtained through an inverse Fourier transform. The relationship of these approximate local operators to the exact nonlocal Dirichlet‐to‐Neumann map is examined. To illustrate their effectiveness, the boundary conditions are employed in a finite element formulation for the time‐dependent structural acoustics problem. In contrast to nonlocal boundary conditions based on the Dirichlet‐to‐Neumann map or retarded potential integral formulations, the proposed local boundary conditions preserve the data structure of the standard finite element method and do not require storage of a large pool of historical data during the solution process. Numerical results illustrate the accuracy of the proposed boundary conditions as effected by the operator order, acoustic wave number, radiation directionality, and distance from the acoustic source.

On the use of time-maps for the solvability of nonlinear boundary value problems

Abstract: (T > 0 is a fixed positive constant). The study of the periodic problem for equation (1.1) (or for some of its generalizations) represents a central subject in the qualitative theory of ordinary differential equations and it has been widely developed by the introduction of powerful tools from nonlinear functional analysis. See e.g. [22, 11, 9, 8, 16] and the references therein, for a source of various different techniques which can be used for this purpose. A classical method to deal with problem (1.1)-(1.2) consists into the search of fixed points of the translation operator (Poincar~-Andronov map) ~ : (x o, Yo) ~ (x (T; Xo, Yo), y (T; xo, Yo)) associated to the equivalent planar system

On the principle of smooth fit for a class of singular stochastic control problems for diffusions

Abstract: This paper considers the principle of smooth fit for a class of one-dimensional singular stochastic control problems allowing the system to be of nonlinear diffusion type The existence and the uniqueness of a convex $C^2 $-solution to the corresponding variational inequality are obtained It is proved that this solution gives the value function of the control problem, and the optimal control process is constructed As an example of the degenerate case, it is proved that the conclusion is also true for linear systems, and the explicit formula for the smooth fit points is derived