Showing papers on "Boundary value problem published in 1980"

Non-Homogeneous Media and Vibration Theory

Abstract: Distributions and Sobolev spaces.- Operators in Banach spaces.- Examples of boundary value problems.- Semigroups and laplace transform.- Homogenization of second order equations.- Homogenization in elasticity and electromagnetism.- Fluid flow in porous media.- Vibration of mixtures of solids and fluids.- Examples of perturbations for elliptic problems.- The Trotter-Kato theorem and related topics.- Spectral perturbation. Case of isolated eigenvalues.- Perturbation of spectral families and applications to selfadjoint eigenvalue problems.- Stiff problems in constant and varialbe domains.- Averaging and two-scale methods.- Generalities and potential method.- Functional methods.- Scattering problems depending on a parameter.

Nonlinear Differential Equations

Abstract: In the foregoing Chapter V, we considered a 1inear partial differential operator g of order 2k and constructed a suitable weighted Sobolev space Wk,2(Ω;S) in which some boundary value problem (mainly, the Dirichlet problem) for g was uniquely weakly solvable. The collection S of weight functions wα∈ W(Ω) was determined by the operator or, more precisely, by its coefficients aαs.

Radiation boundary conditions for wave-like equations

Abstract: In the numerical computation of hyperbolic equations it is not practical to use infinite domains; instead, the domain is truncated with an artificial boundary. In the present study, a sequence of radiating boundary conditions is constructed for wave-like equations. It is proved that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r exp -m-1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature of the boundary conditions.

Transient Response of Multiconductor Transmission Lines Excited by a Nonuniform Electromagnetic Field

Abstract: The time-domain transmission-line equations for uniform multiconductor transmission lines in a conductive, homogeneous medium excited by a transient, nonuniform electromagnetic (EM) field, are derived from Maxwell's equations. Depending on how the line voltage is defined, two formulations are possible. One of these formulations is considerably more convenient to apply than the other. The assumptions made in the derivation of the transmission-line equations and the boundary conditions at the terminations are discussed. For numerical calculations, the transmission -line equations are represented by finite-difference techniques, and numerical examples are included.

Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces

Abstract: Simple and efficient numerical methods are developed for treating electromagnetic problems of scattering and radiation from surfaces. Special consideration is given to the treatment of edges so that rather arbitrary geometrical configurations may be handled. For the conducting body problems considered, an electric field integral formulation is used, and the method of moments is applied using pulse expansions to represent both the current and the charge. It is demonstrated that proper placement of the current and charge subdomains relative to edges not only is important in treating edges but also yields a convenient numerical procedure. A simple testing scheme is used which is almost as efficient as point-matching. Numerical results indicate that the approach is free of anomalies in the behavior of current near edges and of other previously observed numerical instabilities. Problems considered include conducting strips (both TM and TE), a bent rectangular plate, and both material and conducting bodies of revolution.

Differentiation with Respect to the Domain in Boundary Value Problems

Abstract: (1980). Differentiation with Respect to the Domain in Boundary Value Problems. Numerical Functional Analysis and Optimization: Vol. 2, No. 7-8, pp. 649-687.

Direct Control of the Grid Point Distribution in Meshes Generated by Elliptic Equations

Abstract: The generation of computational grids suitable for obtaining accurate numerical solutions to the three-dimensional Navier-Stokes equations is the subject of intensive research. For a wide class of nozzle configurations, a three-dimensional grid can be constructed by a sequence of two-dimensional grids in successive cross-sectional planes. The present paper is concerned with numerical generation of two-dimensional grids. An effective method of interior grid control is presented based on a modified elliptic system containing free parameters. For a simply connected region, the free parameters are computed from the Dirichlet boundary values. The resulting interior grid point distribution is controlled entirely by a priori selection of the grid point distribution along the boundaries of the section.

A Nonlinear Singular Boundary Value Problem in the Theory of Pseudoplastic Fluids

Abstract: The boundary layer equations for the class of non-Newtonian fluids termed pseudoplastic are examined under the classical conditions of uniform flow past a semi-infinite flat plate. The adoption of Crocco variables results in a nonlinear, singular boundary value problem for the shear function which is an interesting and natural generalization of the well known Crocco equation arising from the standard Newtonian fluid case. The uniqueness, existence and analyticity of the solution are established and subsequently an explicit power series solution is exhibited.

On the boundary value problem of the biharmonic operator on domains with angular corners

Abstract: The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Karman model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.

An improved slender-body theory for Stokes flow

Abstract: The present study examines the flow past slender bodies possessing finite centre-line curvature in a viscous, incompressible fluid without any appreciable inertia effects. We consider slender bodies having arbitrary centre-line configurations, circular transverse cross-sections, and longitudinal cross-sections which are approximately elliptic close to the body ends (i.e. prolate-spheroidal body ends). The no-slip boundary condition on the body surface is satisfied, using a convenient stepwise procedure, to higher orders in the slenderness parameter (e) than has previously been possible. In fact, the boundary condition is satisfied up to an error term of O(e2) by distributing appropriate stokeslets, potential doublets, rotlets, sources, stresslets and quadrupoles on the body centre-line. The methods used here produce an integral equation valid along the entire body length, including the ends, whose solution determines the stokeslet strength or equivalently the force per unit length up to a term of O(e2). The O(e2) correction to the stokeslet strength is also found. The theory is used to examine the motion of a partial torus and a helix of finite length. For helical bodies comparisons are made between the present theory and the resistive-force theory using the force coefficients of Gray & Hancock and Lighthill. For the motion considered the Gray & Hancock force coefficients generally underestimate the force per unit length, whereas Lighthill's coefficients provide good agreement except in the vicinity of the body ends.

Dynamics in nonglobally hyperbolic, static space-times

Abstract: Ordinary Cauchy evolution determines a solution of a partial differential equation only within the domain of dependence of the initial data surface. Hence, in a nonglobally hyperbolic space‐time, one does not have fully deterministic dynamics. We show here that for the case of a Klein–Gordon scalar field propagating in an arbitrary static space‐time, a physically sensible, fully deterministic dynamical evolution prescription can be given. If the cosmic censor hypothesis should be overthrown, a prescription of this sort could rescue deterministic physics.

A local compactness theorem for Maxwell's equations

Abstract: The paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: Let G be a bounded domain, β be a tensor-valued function on G satisfying certain restrictions, and let {n} be a sequence of vector-valued functions on G where the L2-norms of {n}, {curl n}, and {div(β n)} are bounded, and where all n either satisfy x n = 0 or (β Fn) = 0 at the boundary ∂G of G ( = normal to ∂G): then {n} has a L2-convergent subsequence The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ∂G if β is interpreted as electric dielectricity ϵ or as magnetic permeability μ, respectively These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for Maxwell's equations

A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface

Abstract: The problem of the diffraction of an arbitrary ray optical electromagnetic field by a smooth perfectly conducting convex surface is investigated. A pure ray optical solution to this problem has been developed by Keller within the framework of his geometrical theory of diffraction (GTD). However, the original GTD solution fails in the transition region adjacent to the shadow boundary where the diffracted field plays a significant role. A uniform GTD solution is developed which remains valid within the shadow boundary transition region, and which reduces to the GTD solution outside this transition region where the latter solution is valid. The construction of this uniform solution is based on an asymptotic solution obtained previously for a simpler canonical problem. The present uniform GTD solution can be conveniently and efficiently applied to many practical problems. Numerical results based on this uniform GTD solution are shown to agree very well with experiments.

On Implementing a Numeric Huygen's Source Scheme in a Finite Difference Program to Illuminate Scattering Bodies

Abstract: A numerical procedure is described that will simplify the analysis of the EMP response of structures with dielectric or poorly conducting segments.

A justification of the von Kármán equations

Abstract: The method of asymptotic expansions, with the thickness as the parameter, is applied to the nonlinear, three-dimensional, equations for the equilibrium of a special class of elastic plates under suitable loads. It is shown that the leading term of the expansion is the solution of a system of equations equivalent to those of von Karman. The existence of solutions of this system is established. It is also shown that the displacement and stress corresponding to the leading term of the expansion have the specific form generally assumed in the usual derivations of the von Karman equations; in particular, the displacement field is of Kirchhoff-Love type. This approach also clarifies the nature of admissible boundary conditions for both the von Karman equations and the three-dimensional model from which these equations are obtained. A careful discussion of the limitations of this approach is given in the conclusion.

Error estimates for mixed methods

Abstract: : This paper presents abstract error estimates for mixed methods for the approximate solution of elliptic boundary value problems. These estimates are then applied to obtain quasi-optimal error estimates in the usual Sobolev norms for four examples: three mixed methods for the biharmonic problem and a mixed method for 2nd order elliptic problems. (Author)

Charged macromolecules in external fields. I. The sphere

Abstract: The response of rigid macroions in aqueous solution to an external electric field is considered. The external field, which may be steady or oscillating, induces perturbations in the distribution of coions and counterions, and also in the electrical and solvent velocity fields. The basic equations that describe the various fields are reviewed, and then specialized to the special problem of thin double layers. Boundary conditions are derived which differ from those used in previous work. Previous treatments of the counterion flux into the double layer have either omitted the flux, or taken it to be in phase with the applied field. We find that this flux is large, and has a significant component out of phase. The new boundary conditions make substantial changes in the complex dielectric response. The dominant slow process that controls relaxation is the slow diffusion of neutral salt in the environment of the macromolecule. Large perturbations are induced in the neutral salt concentration by charge distortio...

Transient response of multiconductor transmission lines excited by a nonuniform electromagnetic field

Abstract: The time-domain transmission-line equations for uniform multiconductor transmission lines in a conductive, homogeneous medium excited by a transient, nonuniform electromagnetic (EM) field, are derived from Maxwell's equations. Depending on how the line voltage is defined, two formulations are possible. One of these formulations is considerably more convenient to apply than the other. The assumptions made in the derivation of the transmission-line equations and the boundary conditions at the terminations are discussed. For numerical calculations, the transmission -line equations are represented by finite-difference techniques, and numerical examples are included.

Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows

Abstract: A spectral method for numerical computation of 3-D time-dependent incompressible flows between two plane parallel plates is presented. Fourier expansions in the coordinates parallel to the walls and expansions in Chebyshev polynomials in the normal coordinate are used. The time coordinate is discretized with second order finite differences, treating the viscous terms implicitly. An efficient direct solution procedure for the implicit equations is developed which reduces the 3-D problem to a set of essentially tridiagonal linear equations in one space coordinate. Boundary and continuity conditions are satisfied exactly, apart from round-off errors.

Analysis of mixed methods using mesh dependent norms

Abstract: : This paper presents a new approach to the analysis of mixed methods for the approximate solution of 4th order elliptic boundary value problems. In this approach one introduces a pair of mesh dependent norms and proves the approximation method is stable with respect to these norms. The error estimates then follow in a direct manner. In a mixed method, one introduces an auxiliary variable, usually representing another physically important quantity, and writes the differential equation as a lower order system. One then considers Ritz-Galerkin approximation schemes based on a variational formulation of this lower order system, thereby obtaining direct approximations to both the original and auxiliary variables. Three particular mixed methods for the approximate solution of the biharmonic problem are examined in detail.