Showing papers on "Boundary value problem published in 1973"

Acoustic Fields and Waves in Solids

Abstract: This work, part of a two-volume set, applies the material developed in the Volume One to various boundary value problems (reflection and refraction at plane surfaces, composite media, waveguides and resonators). The text also covers topics such as perturbation and variational methods.

The finite element method with Lagrangian multipliers

Abstract: The Dirichlet problem for second order differential equations is chosen as a model problem to show how the finite element method may be implemented to avoid difficulty in fulfilling essential (stable) boundary conditions. The implementation is based on the application of Lagrangian multiplier. The rate of convergence is proved.

Method for Solving the Sine-Gordon Equation

Abstract: The initial value problem for the sine-Gordon equation is solved by the inverse-scattering method.

Second-Order Equations With Nonnegative Characteristic Form

Abstract: I. The First Boundary Value Problem.- 1. Notation. Auxiliary results. Formulation of the first boundary value problem.- 2. A priori estimates in the spaces Lp (?).- 3. Existence of a solution of the first boundary value problem in the spaces Lp (?).- 4. Existence of a weak solution of the first boundary value problem in Hilbert space.- 5. Solution of the first boundary value problem by the method of elliptic regularization.- 6. Uniqueness theorems for weak solutions of the first boundary value problem.- 7. A lemma on nonnegative quadratic forms.- 8. On smoothness of weak solutions of the first boundary value problem. Conditions for existence of solutions with bounded derivatives.- 9. On conditions for the existence of a solution of the first boundary value problem in the spaces of S. L. Sobolev.- II. On the Local Smoothness of Weak Solutions and Hypoellipticity of Second Order Differential Equations.- 1. The spaces Hs.- 2. Some properties of pseudodifferential operators.- 3. A necessary condition for hypoellipticity.- 4. Sufficient conditions for local smoothness of weak solutions and hypoellipticity of differential operators.- 5. A priori estimates and hypoellipticity theorems for the operators of Hormander.- 6. A priori estimates and hypoellipticity theorems for general second order differential equations.- 7. On the solution of the first boundary value problem in nonsmooth domains. The method of M. V. Keldys.- 8. On hypoellipticity of second order differential operators with analytic coefficients.- III. Additional Topics.- 1. Qualitative properties of solutions of second order equations with non- negative characteristic form.- 2. The Cauchy problem for degenerating second order hyperbolic equations.- 3. Necessary conditions for correctness of the Cauchy problem for second order equations.

Existence Theorems in Elasticity

Abstract: The subject to be developed in this article covers a very large field of existence theory for linear and nonlinear partial differential equations. Indeed, problems of static elasticity, of the propagation of waves in elastic media, and of the thermodynamics of continua require existence theorems for elliptic, hyperbolic and parabolic equations both linear and nonlinear. Even if one restricts oneself to linear elasticity, there are several kinds of partial differential equations to be considered. In static problems we encounter second order systems, either with constant or with variable coefficients (homogeneous and non-homogeneous bodies), scalar second order equations (for instance either in the St. Venant torsion problems or in the membrane theory), fourth order equations (equilibrium of thin plates), eighth order equations (equilibrium of shells). Each case must be considered with several kinds of boundary conditions, corresponding to different physical situations. On the other hand, to every problem of static elasticity corresponds a dynamical one, connected with the study of vibrations in the elastic system under consideration. Moreover, problems of thermodynamics require the study of certain diffusion problems of parabolic type. In addition to that, the study of materials with memory requires existence theorems for certain integro-differential equations, first considered by Volterra.

The Finite Element Method with Penalty

Abstract: An application of the penalty method to the finite element method is analyzed For a model Poisson equation with homogeneous Dirichlet boundary conditions, a varia- tional principle with penalty is discussed This principle leads to the solution of the Poisson equation by using functions that do not satisfy the boundary condition The rate of con- vergence is discussed 1 Introduction The finite element method in all of its versions has become the subject of current practical and theoretical study A particular problem associated with the finite element method has recently attracted considerable interest Specifically, this problem is the application of variational principles to spaces of functions in which the boundary conditions need not be satisfied See for example references (1) to (7) In references (5) and (6), this author has studied the penalty method approach to this problem This approach consists in the use of a "penalty" parameter which depends on the smoothness of the original problem The selection of the penalty parameter is, in some sense, arbitrary Moreover, the solution of the original problem may be quite sensitive to this parameter This paper studies the model Poisson problem - Au = f with homogeneous boundary conditions of Dirichlet type A variational principle for this model problem on spaces of functions not satisfying the boundary conditions is studied and, based on this principle, a variant of the finite element method is given This new scheme has a rate of convergence that is arbitrarily close to the optimal rate found by using the usual finite element method with elements satisfying the boundary conditions The analysis also shows that the finite element method with penalty is not overly sensitive to the choice of the penalty parameter 2 Some Principal Notions Let Rn be an n-dimensional Euclidian space For x = (x,, , x) C R, we define IIxH2 = Jn= x2 and dx = dx dxn Let Q be a bounded domain in Rn with boundary IF C C' Let Hm(Rn), Hm(Q) and Hm(F), m > 0 m not necessarily an integer, be the fractional Sobolev spaces of order m on Rn, Q and F, respectively We will designate the respec- tive norms of these Sobolev spaces by | | I- (RnX| ( ) and |HH (r) Recall that Hm(Q) and Hm(T) are sometimes also denoted by W'(Q) and W'(F), respectively, and that H1(Q) = L2(Q) and H0(F) = L2(F) Let the spaces Ho'(Q) be the closure in

Spectral-Domain Approach for Calculating the Dispersion Characteristics of Microstrip Lines (Short Papers)

Abstract: The boundary value problem associated with the open microstrip line structure is formulated in terms of a rigorous, hybird-mode representation The resulting equations are subsequently transformed, via the application of Galerkin's method in the spectral domain, to yield a characteristic equation for the dispersion properties of the open microstrip line Numerical results are included for several different structural parameters These are compared with other available data and with some experimental measurements

On the no-slip boundary condition

Abstract: It has been argued that the no-slip boundary condition, applicable when a viscous fluid flows over a solid surface, may be an inevitable consequence of the fact that all such surfaces are, in practice, rough on a microscopic scale: the energy lost through viscous dissipation as a fluid passes over and around these irregularities is sufficient to ensure that it is effectively brought to rest. The present paper analyses the flow over a particularly simple model of such a rough wall to support these physical ideas.

Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries

Abstract: The general study of random walks on a lattice is developed further with emphasis on continuous-time walks with an asymmetric bias. Continuous time walks are characterized by random pauses between jumps, with a common pausing time distributionψ(t). An analytic solution in the form of an inverse Laplace transform for P(l, t), the probability of a walker being atl at timet if it started atlo att=0, is obtained in the presence of completely absorbing boundaries. Numerical results for P(l, t) are presented for characteristically different ψ(t), including one which leads to a non-Gaussian behavior for P(l, t) even for larget. Asymptotic results are obtained for the number of surviving walkers and the mean 〈l〉 showing the effect of the absorption at the boundary.

Nonlinear electroelastic equations for small fields superposed on a bias

Abstract: The nonlinear differential equations and boundary conditions in small‐field variables, for small fields superposed on large static biasing states, are obtained from general rotationally invariant nonlinear electroelastic equations derived previously. The small‐field equations are directly applicable in the consistent description of parametric effects in high‐coupling piezoelectric materials in terms of the fundamental material parameters. The application of the equations to homogeneously polarized ferroelectrics reveals that in the linear limit the electroelastic equations are identical with the equations of linear piezoelectricity for the symmetry of the polarized state. The influence of a thickness directed homogeneous biasing electric field on the thickness vibrations of a piezoelectric plate, to second order in the biasing field, has been determined. To first order in the biasing field the results indicate that the effective fifth‐rank tensor assumed in earlier quasilinear work on the subject did not have correct symmetry properties because the influence of the homogeneous static deformation under the biasing field was ignored.

Low Reynolds number flow past a porous spherical shell

Abstract: This paper is concerned with the flow of viscous fluids around and through porous bodies. Previous boundary conditions that have been used are discussed and a generalization of a boundary condition adopted by Beavers and Joseph, for plane boundaries, is proposed for curved surfaces. Using this condition the problem of slow viscous flow past a spherical shell is solved and several special limiting cases are considered.

A survey of the mean turbulent field closure models.

Abstract: C solutions to the differential boundary-layer equations have for some years now been applied to turbulent boundary layers where relief from the difficulty of solution permits more concern for the physical elements of models which purport to simulate some statistical features of turbulent flowfields. A first step has been accomplished; that is, accurate, quite versatile, and practically/useful computer programs have been combined with rather simple and successful empirical statements" which allow one to estimate the Reynolds shear stress in the equations for the mean velocity field. We call this Mean Velocity Field (MVF) closure since it predicts only the mean velocity field in addition to the mean shear stress. For boundarylayer flows a set of empirical constants must be selected. However, it is then possible to accurately predict flows with wall transpiration, heat transfer, and a variety of other boundary conditions, and, remarkably, with no adjustment in the constants. However, the constants must be adjusted for, say, pipe or channel flow or free shear flows. By moving on to the more complicated Mean Turbulent Field (MTF) closure there is some hope of discovering increasingly universal models and a greater range of predictability. There are two other incentives: First, it is rather comforting actually to compute the turbulent kinetic energy; it is, after all, the premier property that distinguishes turbulent from laminar flow. Second, it appears possible to include body forcelike effects such as curvature, buoyancy, and Coriolis effects with no further empiricism. The latter is a line of thought that is not new,' and it has occupied the present authors' interest for some time. However, in this paper we avoid these topics in order to simplify discussion of an already complicated field. We also assume that the fluid is incompressible. Extension to high Mach number flows does not, however, seem to be a major problem. A basis for MTF calculations began appropriately enough with the semiheuristic models of Kolmogoroff and Prandtl in the early 1940's; they include the turbulent kinetic energy transport equation, a turbulent-energy-related eddy viscosity, and either a prescribed length scale function or a differential equation for a length scale. We wish to call this Mean Turbulent Energy (MTE) closure which together with Mean Reynolds Stress (MRS) closure forms two subsets of MTF closure.* MRS closure implies a closed set of equations which include equations for all nonzero components of the Reynolds stress tensor. Chou' seems to be the first to initiate a study of the full set of equations with an eye towards closure. However, it was Rotta in 1951 who laid the foundation for almost all of the current models. In the Reynolds stress tensor equation (the tensor equation for the single-point, double-velocity correlations, the trace of which is the kinetic energy equation) there appear pressure-velocity gradient correlations, (pdujdxj), which Rotta called the energy redistribution terms and which he argued should be proportional to the deviation from isotropy — dtj(uky/3. On the whole, the assumption seems physically correct, but of further importance is the fact that it provides a unity that was lacking, say, in the 1940's.f Thus, the Reynolds shear stress is now determined as a part of the whole; MTE closure can be obtained as an analytic simplification of MRS closure, and, furthermore, MVF closure (that is, eddy viscosity or mixing length concepts) can be viewed as a further simplification. We shall follow this process of simplification in this paper. Despite the unity of thought provided by Rotta's basic assumption, it is, of course, an approximation to nature and is subject to modification in the hands of investigators eager to achieve agreement with data. Furthermore, there are other terms in the Reynolds stress equations, such as the dissipation and diffusion terms, which are modeled differently by different investigators and represent some impass to a consensus theory such as is the near state of MVF closure. In the present development we have attempted to present the basic ideas and a core model for MRS and MTE closure and

The Rayleigh hypothesis and a related least‐squares solution to scattering problems for periodic surfaces and other scatterers

Abstract: The Rayleigh hypothesis is reviewed in relation to scattering by periodic surfaces, aperiodic surfaces, and bounded, two-dimensional bodies. Conditions for its validity are described, and explicit results are quoted for a sinusoidal grating. Some methods to solve scattering problems for periodic surfaces are outlined. One particular procedure for periodic surfaces and bounded scatterers is examined in detail. This involves an expansion for the scattered field in terms of the same sets of elementary wavefunctions that occur in connection with the Rayleigh hypothesis. The coefficients are determined by satisfying the boundary condition in the least-squares sense. It is shown that this solution converges uniformly to the scattered field at all points exterior to the boundary of the scatterer. Necessary completeness properties of the sets of wavefunctions are established in the appendices.

Fluctuating hydrodynamics and Brownian motion

Abstract: The Langevin equation describing Brownian motion is considered as a contraction from the more fundamental, but still phenomenological, description of an incompressible fluid governed by fluctuating hydrodynamics in which a Brownian particle with stick boundary condition is immersed. First, the derivation of fluctuating hydrodynamics is reconsidered to clarify certain ambiguities as to the treatment of boundaries. Subsequently the contraction is carried out. Since Brownian particles of arbitrary shape are considered, rotations and translations are in general coupled. The symmetry of the 6×6 friction tensorγ ij (t) is proved for arbitrary shape without appeal to microscopic arguments. This symmetry is then used to prove that the fluctuation-dissipation theorem on the contracted level (nonwhite noise in general) follows from the corresponding statement on the level of fluctuating hydrodynamics (white noise). The condition under which the contracted description reduces to the classical Langevin equation is given, and the connection between our theory and related work is discussed.

$L^2 $-Estimates for Galerkin Methods for Second Order Hyperbolic Equations

Abstract: A priori error estimates are established in the $L^2 $-norm for Galerkin approximations to the solution of a generalized wave equation. Optimal rates of convergence are established for several boundary conditions using both continuous and discrete Galerkin procedures.

Curved Elements in the Finite Element Method. I

Abstract: Curved elements, introduced by the author in [13] and [14], which are suitable for solving boundary value problems of the second order in plane domains with an arbitrary boundary are discussed. An approximation theorem is proved, the Dirichlet problem for a ${\mathop W\limits^{\circ}} _2^{(1)} $-elliptic equation is considered as a model problem and error bounds are derived.

A unitarity bound on the evolution of nonstationary states

Abstract: A version of an equation derived by Ersak is studied to obtain rigorous bounds on the rate of change of the integrity of a normalized state.

Theory of space‐charge polarization and electrode‐discharge effects

Abstract: A combined, general treatment of intrinsic and extrinsic conduction in a liquid or solid is presented. Positive and negative species of mobile charge of arbitary valences and mobilities are assumed present, together with homogeneous immobile charge in the extrinsic case. The general equations are specialized to a one‐dimensional situation and then to that where both position‐dependent static and much smaller sinusoidally time‐varying components of charge, field, and current are simultaneously present. Sufficiently general boundary conditions are used that any condition from complete blocking to free discharge of positive and negative mobile carriers separately can occur at the electrodes. For the flat‐band condition (zero static field; in the binary electrolyte case, coincidence of the zero charge potential and the equilibrium potential) exact equivalent circuits and an exact expression for the small‐signal impedance are obtained. Relatively simple, closed‐form expressions for the zero‐frequency limiting ...

Rate of deposition of Brownian particles under the action of London and double-layer forces

Abstract: The rate of collection of Brownian particles under the influence of interaction forces between the collector surface and the particles is calculated by (a) incorporating the interaction forces in the rate constant of a virtual, first order, chemical reaction taking place on the surface of the collector, and by (b) solving the convective diffusion equation subject to that chemical reaction as a boundary condition. Several geometries (sphere, cylinder, rotating disc) are considered for the collector.An Arrhenius type equation is obtained for the apparent reaction rate constant. Equations for the apparent activation energy and for the frequency factor are established as functions of Hamaker's constant, ionic strength, surface potentials and particle radius.

Uni-moment method of solving antenna and scattering problems

Abstract: It has been shown by this investigator and numerous others [6], [7], [8] that exterior boundary value problems involving localized inhomogeneous media are most conveniently solved using finite difference or finite element techniques together with integral equations or harmonic expansions, which satisfy the radiation conditions. The methods result in large matrices that are partly full and partly sparse; and methods to solve them, such as iteration or banded matrix methods are not very satisfactory. The unimoment method alleviates the difficulties by decoupling exterior problems from the interior boundary value problems. This is done by solving the interior problem many times so that N linearly independent solutions are generated. The continuity conditions are then enforced by a linear combination of the N independent solutions, which may be done by solving much smaller matrices. Methods of generating solutions of the interior problems are discussed.

Boundary Value Problems of Elasticity with Unilateral Constraints

Abstract: In the preceding article “Existence Theorems in Elasticity”, which henceforth will be cited as E.T.E.1, I have treated boundary value problems of Elasticity in the case when the side conditions to be associated with the differential equations of equilibrium correspond to bilateral constraints imposed upon the elastic body. In this article I will treat the analytical problems which arise when unilateral constraints are imposed.