Showing papers on "Free boundary problem published in 1988"

Boundary Element Methods in Elastodynamics

Abstract: Governing equations of motions boundary integral formulation in elastodynamics numerical treatment of boundary equations other boundary methods wave propagation analysis soil structure interaction vibrations of structures other linear material models computer implementation aspects.

Inverse boundary value problems at the boundary—continuous dependence

Abstract: We use the methods of microlocal analysis to give a new proof of a theorem of Kohn and Vogelius, showing that the boundary values of a continuous isotropic conductivity can be recovered from voltage and current measurements at the boundary. Moreover, we prove sharp estimates to establish the continuous dependence of the boundary values of the conductivity on the voltage to current maps.

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Abstract: This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.

On the modeling of brine transport in porous media

Abstract: The problem of concentrated brine transport arises in the study of transport of pollutants released from a repository in a rock salt formation. An important characteristic of brine, as compared to other solutions normally encountered in groundwater problems, is that it contains a high concentration of solutes. This factor requires special attention in the development of mathematical models for brine transport problems. In this work we discuss certain important physical and mathematical differences between low- and high-concentration situations. In particular, we consider three primary aspects of a model: basic equations, boundary conditions, and numerical techniques. Recognizing the fact that in high-concentration situations, the fluid motion is not independent of the solutes movement, a new formulation of Darcy's and Fick's law are proposed. The basic equations comprise a set of two nonlinear coupled partial differential equations to be solved for the pressure p and the solute mass fraction ω. These equations have to be solved by means of iterative methods. Various possibilities involving finite difference methods have been studied. In one case, after discretizing the equations in a fully implicit way, the Newton-Raphson method has been employed to solve the system of nonlinear difference equations simultaneously. In another case, after removing part of the nonlinearity by a transformation of the dependent variable ω, a procedure of sequential solution of the two equations by successive substitution is employed. It turns out that the latter method is considerably faster than the former one as a result of the quasi-linearization. Finally, considering boundary conditions, it is shown that often they are also nonlinear and coupled. Appropriate conditions for a rock salt boundary and an outflow boundary are developed and their significance in high-concentration situations are discussed. In particular, a nonlinear time-dependent boundary condition at a rock salt boundary is developed which takes into account the process of salt dissolution and cap rock formation.

Absorbing boundary conditions for the vector wave equation

Abstract: Approximate absorbing boundary conditions for the vector wave equation are developed. These permit a truncation of the computational domain required to accurately model an open-region electromagnetic scattering problem using the finite element method. Details are provided to illustrate the manner in which the boundary conditions can be coupled to a weak form of the vector wave equation.

On a boundary layer problem for the nonlinear Boltzmann equation

Abstract: This article deals with a boundary-layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero. The model considered here is the stationary, nonlinear Boltzmann equation in one dimension with a slightly perturbed reflection boundary condition. We restrict our attention to the case of hard spheres collisions, with Grad's cutoff assumption. Existence, uniqueness and asymptotic behavior are derived by means of energy estimates.

Explicit formula for scalar non-linear conservation laws with boundary condition

Abstract: We prove an uniqueness and existence theorem for the entropy weak solution of non-linear hyperbolic conservation laws of the form , with initial data and boundary condition. The scalar function u = u(x, t), x > 0, t > 0, is the unknown; the function f = f(u) is assumed to be strictly convex. We also study the weighted Burgers' equation: α ϵ ℝ . We give an explicit formula, which generalizes a result of Lax. In particular, a free boundary problem for the flux f(u(.,.)) at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to Keyfitz.

Boundary integral equation method for shape optimization of elastic structures

Abstract: A general method for shape design sensitivity analysis as applied to plane elasticity problems is developed with a direct boundary integral equation formulation, using the material derivative concept and adjoint variable method. The problem formulation is very general and a complete consideration is given to describing the boundary variation by including the tangential component of the velocity field. The method is then applied to obtain the sensitivity formula for a general stress constraint imposed over a small part of the boundary. The accuracy of the design sensitivity analysis is studied with a fillet and an elastic ring design problem. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary have shown the best accuracy. A smooth characteristic function is found to be better than a plateau function for localization of the stress constraint. Optimal shapes for the two problems are presented to show numerical applications.

SomeL 1 existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions

Abstract: In this paper we study questions of existence, uniqueness, and continuous dependence for semilinear elliptic equations with nonlinear boundary conditions. In particular, we obtain results concerning the continuous dependence of the solutions on the nonlinearities in the problem, which in turn implies analogous results for a related parabolic problem. Such questions arise naturally in the study of potential theory, flow through porous media, and obstacle problems.

A weakly nonlinear stability analysis of the reactive infiltration interface

Abstract: This paper investigates the infiltration flow of a reactive fluid in a porous medium. The reaction-induced porosity changes couple in a nonlinear manner with the transport and may lead to fingering instabilities. A mathematical model for this phenomenon is given in the form of a moving free boundary problem. The morphological instability of a planar dissolution front is demonstrated using a weakly nonlinear stability analysis. The technique used here at each order is to solve the equations uniquely and explicitly, obtaining the resulting restrictions by substituting into the evolution equation for the dissolution interface, as opposed to the more traditional method of integrating against the solution of the homogeneous adjoint problem.

Non‐linear heat conduction in composite bodies: A boundary element formulation

Abstract: The present paper discusses the numerical solution of steady-state non-linear heat conduction problems in composite bodies by using the boundary element method. Two kinds of non-linearities are considered: the temperature dependence of the thermal conductivity and boundary conditions of the radiative type. By introducing the integral of conductivity as a new variable the governing equation of the problem becomes linear in the transform space. Transformed boundary conditions of the Dirichlet and Neumann types are also linear but convective boundary conditions become non-linear. Also, discontinuities arise in the value of the integral of conductivity across the interface between materials with different properties since continuity of temperature is imposed. The problem is numerically solved by discretizing the external and interface boundaries of the region under consideration with constant boundary elements and applying an iterative scheme of the Newton–Raphson type.

A two point boundary value problem with a rapidly oscillating solution

Abstract: Some numerical methods are developed for a two point boundary value problem with a rapidly oscillating solution. The two point boundary value problem is chosen to model some of the difficulties that may be expected to occur in solving the reduced wave equation at moderately high frequencies.

On a nonlinear moving boundary problem with heterogeneity: application of a reciprocal transformation

Abstract: A nonlinear problem of Stefan-type is analysed. An exact solution is obtained by means of a reciprocal transformation.

Unsteady motion of a finite mass of fluid, bounded by a free surface

Abstract: It is proved that the problem with a free boundary for the Navier-Stokes equations, describing the motion of a finite mass of viscous, incompressible capillary fluid, has a unique solution for all t > 0 if the domain occupied by the fluid is nearly a ball and the velocity vector field is small at the initial moment.

On the boundary element method for some nonlinear boundary value problems

Abstract: Here we analyse the boundary element Galerkin method for two-dimensional nonlinear boundary value problems governed by the Laplacian in an interior (or exterior) domain and by highly nonlinear boundary conditions. The underlying boundary integral operator here can be decomposed into the sum of a monotoneous Hammerstein operator and a compact mapping. We show stability and convergence by using Leray-Schauder fixed-point arguments due to Petryshyn and Necas.

Solution of the advection‐dispersion equation with free exit boundary

Abstract: I investigated the exit boundary condition for the advection-dispersion equation and found that in numerical solutions of this equation, using Galekin finite elements, a free exit boundary condition requiring no a priori information is possible, provided the advective component in the numerical equations is of sufficient magnitude relative to the dispersive component. Since the relationship between these two components is controlled by the spatial discretization through the grid Peclet number, the free exit boundary condition can in fact be applied whenever there is a non-zero advective component. The numerical solution in a finite domain with free exit boundary, using a correctly proportioned spatial discrezation, behaves like an infinite-domain solution.

Singular boundary-value problems for ordinary second-order differential equations

Abstract: This article gives an exposition of the fundamental results of the theory of boundary-value problems for ordinary second-order differential equations having singularities with respect to the independent variable or one of the phase variables. In particular criteria are given for solvability and unique solvability of two-point boundary-value problems and problems concerning bounded and monotonic solutions. Several specific problems are considered which arise in applications (atomic physics, field theory, boundary-layer theory of a viscous incompressible fluid, etc.)

A super-absorbing boundary algorithm for solving electromagnetic problems by time-domain finite-difference method

Abstract: An algorithm is presented to treat the absorbing boundaries for the time-domain finite-difference method. With this approach, the leading order error of the conventional local absorbing boundary conditions can be cancelled by a simple algorithm, so that the absorbing quality of the boundary conditions can be greatly improved. The authors concentrate on the local boundary conditions which represent the field values on boundary nodes by several field values inside (and possibly on) the boundary. >

Boundary control of distributed parabolic system with boundary condition involving a time-varying lag

Abstract: An optimal boundary control problem for a distributed-parameter system with a boundary condition involving a time-varying lag is solved. The time horizon is fixed. Making use of the Lions scheme, necessary and sufficient conditions of optimality for the Neumann problem with the quadratic performance functional and constrained control are derived.

Plates on biparametric elastic foundation by bdie method

Abstract: An efficient boundary differential integral equation (BDIE) method is presented for the analysis of thin elastic plates with free boundaries of any shape resting on biparametric elastic foundation. The plate, which may have holes, is subjected to concentrated loads, line loads, or distributed surface loads. The solution is achieved by converting the governing boundary value problem to an equivalent problem consisting of five coupled boundary equations, two of which are differential and three of which are integral. The boundary differential equations are derived from the boundary conditions, while the boundary integral equations are derived from the integral representations for the deflections of the plate and of the foundation region. A numerical technique based on the discretization of the boundary is developed for the solution of the boundary equations. The computational efficiency of the method is increased by converting the domain integrals attributable to loading into boundary line integrals. Numeric...

Isospectral sets for boundary value problems on the unit interval

Abstract: We analyse isospectral sets of potentials associated to a given ‘generalized periodic’ boundary condition in SL(2, R ) for the Sturm-Liouville equation on the unit interval. This is done by first studying the larger manifold M of all pairs of boundary conditions and potentials with a given spectrum and characterizing the critical points of the map from M to the trace a + d Isospectral sets appear as slices of M whose geometry is determined by the critical point structure of the trace function. This paper completes the classification of isospectral sets for all real self-adjoint boundary conditions.

The First Boundary Value Problem for Nondivergence Second Order Parabolic Equations with Discontinuous Coefficients

Abstract: This paper is concerned with the questions of solvability and smoothness of weak solutions of the first boundary value problem for a parabolic equation of the form Bibliography: 18 titles.

Control of free boundary problem with hysteresis

Abstract: We consider the problem of controlling the free boundary of the two-phase Stefan problem by means of boundary hysteresis control based on the Preisach model. It is proved that for each control $\mu $ there is a corresponding solution of the Stefan problem and that there exists an optimal control.

Two new moving boundary problems for scalar conservation laws

Abstract: The application of the theory of scalar conservation laws to semiconductor device fabrication is described. This application is the source of a Stefan problem and another moving boundary problem for a class of such equations. The analogue of the Riemann problem for these problems is analyzed and solved. Conditions on the boundary values that characterize physically correct solutions are derived.

Constriction resistance of circular contacts on coated surfaces - Effect of boundary conditions

Abstract: The thermal constriction resistance of a circular contact spot on a coated half-space is developed for both heat flux and temperature-specified boundary conditions on the contact. Solutions are obtained with the Hankel transform method for flux-specified contacts and with a novel technique of linear superposition for the mixed boundary value problem created by an isothermal contact. A comparison of the results obtained shows that the thermal constriction resistance, which is based on average contact temperature, is insensitive to the contact boundary condition for most practical purposes.