Showing papers on "Boundary value problem published in 1978"

Thin-layer approximation and algebraic model for separated turbulent flows

Abstract: An algebraic turbulence model for two- and three-dimensional separated flows is specified that avoids the necessity for finding the edge of the boundary layer. Properties of the model are determined and comparisons made with experiment for an incident shock on a flat plate, separated flow over a compression corner, and transonic flow over an airfoil. Separation and reattachment points from numerical Navier-Stokes solutions agree with experiment within one boundary-layer thickness. Use of law-of-the-wall boundary conditions does not alter the predictions significantly. Applications of the model to other cases are contained in companion papers.

Integral formulation for migration in two and three dimensions

Abstract: Computer migration of seismic data emerged in the late 1960s as a natural outgrowth of manual migration techniques based on wavefront charts and diffraction curves. Summation (integration) along a diffraction hyperbola was recognized as a way to automate the familiar point‐to‐point coordinate transformation performed by interpreters in mapping reflections from the x, t (traveltime) domain into the x, z (depth domain). We will discuss the mathematical formulation of migration as a solution to the scalar wave equation in which surface seismic observations are the known boundary values. Solution of this boundary value problem follows standard techniques, and the migrated image is expressed as a surface integral over the known seismic observations when areal or 3-D overage exists. If only 2-D seismic coverage is available, wave equation migration is still possible by assuming the subsurface and hence surface recorded data do not vary perpendicular to the seismic profile. With this assumption, the surface inte...

On Some Questions in Boundary Value Problems of Mathematical Physics

Abstract: Publisher Summary This chapter introduces some questions that arise in boundary value problems of mathematical physics. Some problems of the hydrodynamics of incompressible nonhomogeneous fluids are described in the chapter. The chapter describes the equations of flows of incompressible fluids that are nonhomogeneous in the sense of not having a constant density. The classical Navier–Stokes equations are described in the chapter. Except in some details of presentations, the chapter follows the notes of Antonzev and Kajikov. The problems studied in Antonzev and Kajikov include in particular the existence of strong solutions in the two dimensional case. Statement of existence theorem is presented and Galerkin's approximation is discussed in the chapter. A discussion is presented in the chapter on a linear equation arising in the kinetic theory of gases and containing some nonstandard aspects. An introduction to the method of homogenization for composite materials is also given in the chapter.

Implicit Finite-Difference Simulation of Flow about Arbitrary Two-Dimensional Geometries

Abstract: Finite-difference procedures are used to solve either the Euler equations or the "thin-layer" Navier-Stokes equations subject to arbitrary boundary conditions. An automatic grid generation program is employed, and because an implicit finite-difference algorithm for the flow equations is used, time steps are not severely limited when grid points are finely distributed. Computational efficiency and compatibility to vectorized computer processors is maintained by use of approximate factorization techniques. Computed results for both inviscid and viscous flow about airfoils are described and compared to viscous known solutions.

Development of Hydrodynamic Models Suitable for Air Pollution and Other Mesometerological Studies

Abstract: We describe the development of a general, predictive, hydrostatic meteorological model. The model is three-dimensional and is suitable for a wide variety of problems, ranging from the synoptic scale to the small end of the mesoscale. The model contains provisions for variable terrain, a moisture cycle, sensible heat addition at the earth's interface, and high- and low-resolution boundary layer physics. This paper presents the mathematical and numerical formulation used in the various options of the model. First we write the basic equations on a Lambert conformal projection. Then we describe the horizontal and vertical grid structure, the finite-difference equations, and the energetics of the three-dimensional model and its two-dimensional analog. We consider the role of the lateral boundary conditions for limited area forecasts, with emphasis on their effect on the mean motion over the domain. Two options for including the frictional and diabatic effects at the earth's surface are presented. Thes...

Theory of Thermoelasticity With Applications

Abstract: 1 Introduction.- 2 Mathematical groundwork.- 2.1 Tensor calculus.- 2.2 List of useful formulas.- 3 Fundamentals of thermodynamics.- 3.1 System. State. State parameters and functions.- 3.2 The laws of thermodynamics.- 3.3 Nonuniform systems.- 4 Thermodynamics of elastic deformations.- 5 Modes of heat transfer.- 5.1 Radiation.- 5.2 Convection.- 5.3 Conduction.- 6 Theory of heat conduction.- 6.1 Classical differential equation of heat conduction.- 6.2 Initial and boundary conditions.- 7 An hyperbolic equation of heat conduction.- 8 The linear thermoelastic solid.- 8.1 Anisotropy of materials.- 8.2 Certain types of thermoelastic coupling.- 9 The temperature field.- 9.1 Integral transforms.- 9.1a The Laplace transform.- 9.1b Fourier transforms.- 9.1c Hankel transforms.- 9.2 Separation of variables.- 9.3 Green's, or influence, functions.- 9.3a Steady states.- 9.3b Time-dependent states.- 9.4 Duhamel's superposition theorems.- 9.5 Solidification and melting.- 10 Stress and deformation fields.- 10.1 Goodier's thermoelastic potential.- 10.2 Method of biharmonic representations.- 10.3 Betti-Maysel reciprocal method.- 10.4 Thermoelastic-elastic correspondence principle.- 10.5 Method of Green's function.- 10.6 Method of a complex variable.- 10.6a General concepts and theorems.- 10.6b Series expansions.- 10.6c Conformai mapping.- 10.6d Applications to elasticity.- 10.6e Uniqueness of solution. Connectivity of regions.- 10.6f Cauchy integrals.- 10.7 The extended Boussinesq-Papkovich-Neuber solution.- 11 Uniqueness of solution. Stress-free thermoelastic fields.- 11.1 Uniqueness of solution.- 11.2 Stress-free thermoelastic fields.- 11.2a Three-dimensional regions.- 11.2b Two-dimensional regions.- 12 Anisotropic bodies.- 12.1 Correspondence principle for anisotropic bodies.- 12.2 Thermal stresses in an orthotropic hollow cylinder.- 12.3 Thermal stresses in a transversely isotropic half-space.- 13 Stresses due to solidification.- 14 Thermoelastic stresses in plates.- 14.1 General equations.- 14.2 Boundary conditions.- 14.3 Correspondence principle for isotropic plates.- 14.4 Two characteristic cases.- 14.5 Laminated composite plates.- 15 Thermoelastic stresses in shells.- 15.1 Deformation of shells of revolution under axisymmetric mechanical and thermal load.- 15.2 State of stress in shells of revolution deformed axisymmetrically.- 15.3 General theory of shells.- 15.4 Shells of revolution deformed arbitrarily.- 15.5 Donnell's theory of cylindrical shells.- 15.6 Boundary conditions.- 15.7 Equation of heat conduction for shells.- 16 Thermoelastic stresses in bars.- 16.1 Bars of solid cross-section.- 16.2 Bars of thin-walled open cross-section.- 16.3 Bars of thin-walled closed cross-section.- 16.4 Torsion of bars of thin-walled open cross-section.- 17 Thermoelastic stresses around cracks.- 18 Thermoelastic stability of bars and plates.- 18.1 Bars of solid and thin-walled closed cross-section.- 18.2 Bars of thin-walled open cross-section.- 18.3 Plates.- 18.4 Post-buckling behavior of plates.- 19 Moving and periodic fields.- 19.1 General remarks.- 19.2 Illustrative examples.- 20 Thermoelastic vibrations and waves.- 20.1 General concepts and equations.- 20.2 Thermoelastic harmonic waves in infinite media.- 20.3 Thermoelastic Rayleigh waves.- 20.4 Thermoelastic vibrations of a spinning disk.- 20.5 Wave discontinuities.- 21 Coupled thermoelasticity.- 22 Thermoelasticity of porous materials.- 23 Electromagnetic thermoelasticity.- 23.1 Basic concepts of electromagnetism.- 23.2 Maxwell's equations.- 23.3 Lorentz force. Maxwell stresses.- 23.4 Moving bodies.- 23.5 Electromagnetic energy.- 23.6 Electromagnetic thermoelastic equations.- 23.6a Thermoelasticity of dielectrics.- 23.6b Thermoelasticity of ferromagnetic bodies.- 23.6c Applications.- 24 Piezothermoelasticity.- 25 Random thermoelastic processes.- 25.1 General concepts and equations.- 25.1a Random variables.- 25.1b Random processes.- 25.2 Spectral density.- 26 Variational methods in thermoelasticity.- 26.1 General remarks.- 26.2 Virtual work.- 26.3 Principles of stationary energy of Hemp.- 26.4 Principle of Washizu.- 26.5 Principle of Biot.- Literature.- Author index.

Numerical Simulation of Hydrostatic Mountain Waves

Abstract: A numerical model is developed for simulating the flow of stably stratified nonrotating air over finite-amplitude, two-dimensional mountain ranges. Special attention is paid to accurate treatment of internal dissipation and to formulation of an upper boundary region and lateral boundary conditions which allow upward and lateral propagation of wave energy out of the model. The model is hydrostatic and uses potential temperature for the vertical coordinate. A local adjustment procedure is derived to parameterize low Richardson number instability. The model behavior is tested against analytic theory and then applied to a variety of idealized and real flow situations, leading to some new insights and new questions on the nature of large-amplitude mountain waves. The model proves to be effective in simulating the structure of two observed cases of strong mountain waves with very different characteristics.

A Mass-Consistent Model for Wind Fields over Complex Terrain.

Abstract: An adjustment model was developed to provide a pollutant transport model with input wind fields that are mass-consistent, three-dimensional, and also representative of the available meteorological measurements. Interpolated three-dimensional mean winds were adjusted in a weighted least-squares sense to satisfy the continuity equation within the volume specified. The upper and lateral boundaries above topography were assumed to be open air and thus allowed mass flow through the boundaries. The bottom boundary was determined by the topographic elevations of the area being studied. The topographic boundary was assumed to be solid.

Potential techniques for boundary value problems on C1-domains

Abstract: In troduct ion In this work we consider the Dirichlet and Neumann problems for Laplace's equation in a bounded domain, D, of R ' , n >~ 3. Assuming the boundary, ~D, to be of class C 1 and the boundary data in/2 '(~D), 1 < p < 0% we resolve the above problems in the form of classical double and single layer potentials respectively. More precisely, given g ELr(8D) we find a solution to the I)irichlet problem,

Frequency and Loss Factors of Sandwich Beams under Various Boundary Conditions

Abstract: A complete set of equations of motion and boundary conditions governing the vibration of sandwich beams are derived by using the energy approach. They are solved exactly for important boundary conditions. The computational difficulties that were encountered in previous attempts at the exact solution of these equations have been overcome by careful programming. These exact results are presented in the form of design graphs and formulae, and their usage is illustrated by examples.

Solution of the Smoluchowski equation with a Coulomb potential. I. General results

Abstract: The time‐dependent Smoluchowski equation with a Coulomb potential is solved analytically for a general boundary condition, and expressions for the distribution function, reaction rate, and survival probability are given. The expressions are evaluated numerically and the long‐time behavior is derived. The theoretical results apply to experiments involving ion recombination without an electric field (or where there is a time delay in the application of the field), to scavenging experiments, and to fluorescence quenching.

Mixed boundary value problems in mechanics

Abstract: Definitions in the case of multiple series equations and multiple integral equations are examined. In considering the solution of a given mixed boundary value problem perhaps the simplest technique is the direct application of the method of complex potentials provided the problem admits such potentials and the domain and the boundary conditions are suitable for such an application. The direct application of complex potentials is described with the aid of examples, taking into account a problem in potential theory, the case of periodic cuts, and an elasticity problem for a nonhomogeneous plane. The reduction to singular integral equations is discussed along with the numerical solution of singular integral equations of the first kind, integral equations with generalized Cauchy kernels, and singular integral equations of the second kind.

Characterizations of the ranges of some nonlinear operators and applications to boundary value problems

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Mixed boundary-value problems in mechanics

Abstract: Definitions in the case of multiple series equations and multiple integral equations are examined. In considering the solution of a given mixed boundary value problem perhaps the simplest technique is the direct application of the method of complex potentials provided the problem admits such potentials and the domain and the boundary conditions are suitable for such an application. The direct application of complex potentials is described with the aid of examples, taking into account a problem in potential theory, the case of periodic cuts, and an elasticity problem for a nonhomogeneous plane. The reduction to singular integral equations is discussed along with the numerical solution of singular integral equations of the first kind, integral equations with generalized Cauchy kernels, and singular integral equations of the second kind.

Perturbation theory for linear electroelastic equations for small fields superposed on a bias

Abstract: A perturbation formulation of the equations of linear piezoelectricity for small fields superposed on a bias is obtained from a Green’s function representation. It is shown that the resulting equation for the first perturbation of the eigenvalue may be obtained without the use of a Green’s tensor or a complete set of orthogonal eigensolutions. Since the bias enters the constitutive equations, the boundary conditions contain perturbation terms as well as the differential equations. The linear electroelastic equations for small fields superposed on a bias differ from the equations of linear piezoelectricity because the effective material constants of linear electroelasticity have less symmetry than the constants of linear piezoelectricity. Consequently, a perturbation formulation of the linear electroelastic equations for small fields superposed on a bias is presented. It is shown that the effective constants of linear electroelasticity have just the symmetry required for the condition of the orthogonality of linear electroelastic vibrations to hold.

Finite temperature and boundary effects in static space-times

Abstract: Expressions are derived for the free energy of a massless scalar gas confined to a spatial cavity in a static space-time at a finite temperature. A high temperature expansion is presented in terms of the Minakshisundaram coefficients. This gives curvature and boundary corrections to the Planckian form. The regularisation used is the zeta function one, and yields a finite total internal energy. However, it is known that the local energy density diverges in a non-integrable way as the boundary is approached. A 'surface energy' is suggested to reconcile these two facts. Explicit expressions for the total energy inside two infinite rectangular waveguides are obtained.

Boundary conditions for the numerical solution of wave propagation problems

Abstract: Many finite difference models in use for generating synthetic seismograms produce unwanted reflections from the edges of the model due to the use of Dirichlet or Neumann boundary conditions. In this paper we develop boundary conditions which greatly reduce this edge reflection. A reflection coefficient analysis is given which indicates that, for the specified boundary conditions, smaller reflection coefficients than those obtained for Dirichlet or Neumann boundary conditions are obtained. Numerical calculations support this conclusion.

Solitary Waves on a Two-Layer Fluid

Abstract: This paper deals with weakly nonlinear long gravity waves on a stably stratified two-layer fluid. By using the reductive perturbation method, it is found that the fast mode is always governed by a Korteweg-de Vries (K-dV) equation whose coefficients depend on the thickness ratio and the density ratio. On the other hand, the slow mode is also governed, in general, by another K-dV equation except near and at the critical thickness ratio. At the critical thickness ratio, however, the slow mode is shown to be governed by a modified K-dV equation with cubic nonlinearity and near the critical thickness ratio it is governed by an equation of a combined form of the K-dV and modified K-dV equation. Steady solitary wave solutions to these equations are investigated in detail. A special solution representing a dispersive bore or hydraulic jump is also found.

Uniqueness of Solutions to Hyperbolic Conservation Laws.

Abstract: : It is known that conservative systems of differential equations which result from continuum mechanics (e.g. the equations of shallow water waves, fluid dynamics, magneto-fluid dynamics and certain elasticity problems) do not have unique solutions. Thus the problem arises of proving that systems of this type have only one physically meaningful solution. This report shows that there exists at most one solution satisfying an entropy condition which generalizes the second law of thermodynamics.

Adaptive Mesh Selection Strategies for Solving Boundary Value Problems

Abstract: Various adaptive mesh selection strategies for solving two-point boundary value problems are brought together and a limited comparison is made. The mesh strategies are applied using collocation met...

Determination of constant α force-free solar magnetic fields from magnetograph data

Abstract: At first it is shown that a magnetic field being force-free, i.e. satisfying ▽ × B = αB, with α = constant (α ≠ 0) in the whole exterior of the Sun cannot have a finite energy content and cannot be determined uniquely from only one magnetic field component given at the photosphere. Then the boundary value problem for a semi-infinite column of arbitrary cross section is solved by a Green's function method.

Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations

Abstract: We consider the problem of finding the restrictions on the domain Ω⊂R n,n=2,3, under which the space of the solenoidal vector fields from coincides with the space , the closure in W 2 1 (Ω) of the set of all solenoidal vectors from . We give domains Ω⊂Rn, for which the factor space has a finite nonzero dimension. A similar problem is considered for the spaces of solenoidal vectors with a finite Dirichlet integral. Based on this, one compares two generalized formulations of boundary-value problems for the Stokes and Navier-Stokes systems. The following auxiliary problems are studied: , .

Incompletely parabolic problems in fluid dynamics

Abstract: Some partial differential equations encountered in physical applications are of incompletely parabolic type; the Navier–Stokes equations in fluid dynamics are a typical example. In this paper we analyze such systems; in particular we treat the mixed initial-boundary value problem. In many applications there is a small parameter $\varepsilon $ multiplying the coefficient for the highest derivative. The energy method is used to derive well-posed boundary conditions such that, when $\varepsilon $ tends to zero, the reduced problem is also well posed.