Showing papers on "Boundary value problem published in 1975"
Book•
01 Jan 1975
TL;DR: The Finite Element Method as discussed by the authors is a method to meet the Finite Elements Method of Linear Elasticity Theory (LETI) and is used in many of the problems of mesh generation.
Abstract: PART I. Meet the Finite Element Method. The Direct Approach: A Physical Interpretation. The Mathematical Approach: A Variational Interpretation. The Mathematical Approach: A Generalized Interpretation. Elements and Interpolation Functions. PART II. Elasticity Problems. General Field Problems. Heat Transfer Problems. Fluid Mechanics Problems. Boundary Conditions, Mesh Generation, and Other Practical Considerations. Appendix A: Matrices. Appendix B: Variational Calculus. Appendix C: Basic Equations from Linear Elasticity Theory. Appendix D: Basic Equations from Fluid Mechanics. Appendix E: Basic Equations from Heat Transfer. References. Index.
1,497 citations
TL;DR: A review of the models of magnetic field line merging is given in this paper, where the authors describe the process whereby plasma flows across a surface which separates regions including topologically different magnetic field lines.
Abstract: A review is presented of the models of magnetic field line merging defined as the process whereby plasma flows across a surface which separates regions including topologically different magnetic field lines. The models examined are characterized by uniform and antiparallel external magnetic fields. An attempt is made to simplify the presentation of the models, to clarify some doubtful mathematical points, or to extend the results to a different range of physical parameters. The models are described from a hydromagnetic point of view, with the configuration in any given case being determined by the boundary conditions. It is shown that the models developed by Sweet (1958), Parker (1957, 1963), Petschek (1964), Sonnerup (1970), and by Yeh and Axford (1970) are basically consistent, describing different aspects of the same problem; however, there is not a single model that would account for all the cases considered. The singular models and the compressible similarity models are physically not feasible.
1,310 citations
TL;DR: In this article, the general dynamical system with constraints is quantized, and the S-matrix is constructed in the most general class of gauges including relativistic ones, and a new type of additional diagrams arises securing unitarity of the theory: the fourfermion interaction of ghost fields.
846 citations
TL;DR: In this article, conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time were investigated and sharp results were derived for certain classes of nonlinearities.
Abstract: A number of authors have investigated conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time. For certain classes of nonlinearities sharp results are derived in this paper. Extensions to parabolic and to abstract operator equations are also given.
700 citations
TL;DR: In this article, the equations of fluid motion have been formulated in a generalized noncartesian, nonorthogonal coordinate system and a particular coordinate transformation which transforms a domain with an irregular lower boundary into a cube has been constructed.
645 citations
TL;DR: In this article, it was shown that the problem of locating complex poles can be reduced to solving a polynomial with real roots whose order is one half that of the original all-pole transfer function.
Abstract: It has been known that the linear predictor coefficients (LPC) of speech signals can be transformed into a “pseudo” vocal‐tract area function whose boundary conditions are (a) a complete opening at the lips and (b) a matching resistance termination at the glottis. If the boundary condition at the glottis is replaced by a complete opening or a complete closure, all the poles of the resulting system function will move onto the unit circle in z plane. Using this fact it is possible to describe the original LPCs by two sets of pole frequencies corresponding to the two new boundary conditions at the glottis, or a set of frequency‐residue pairs corresponding to either set of poles. These representations have several important features: (1) If an original pole is narrow band, the new pole is close to the original pole; (2) the two sets of pole frequencies alternate and are ordered on the frequency axis; and (3) the problem of locating complex poles can be reduced to solving a polynomial with real roots whose order is one‐half that of the original all‐pole transfer function.
630 citations
TL;DR: In this article, a solution of electromagnetic scattering by a homogeneous prolate (or oblate) spheroidal particle with an arbitrary size and refractive index is obtained for any angle of incidence by solving Maxwell's equations under given boundary conditions.
Abstract: The solution of electromagnetic scattering by a homogeneous prolate (or oblate) spheroidal particle with an arbitrary size and refractive index is obtained for any angle of incidence by solving Maxwell's equations under given boundary conditions. The method used is that of separating the vector wave equations in the spheroidal coordinates and expanding them in terms of the spheroidal wavefunctions. The unknown coefficients for the expansion are determined by a system of equations derived from the boundary conditions regarding the continuity of tangential components of the electric and magnetic vectors across the surface of the spheroid. The solutions both in the prolate and oblate spheroidal coordinate systems result in a same form, and the equations for the oblate spheroidal system can be obtained from those for the prolate one by replacing the prolate spheroidal wavefunctions with the oblate ones and vice versa. For an oblique incidence, the polarized incident wave is resolved into two components, the TM mode for which the magnetic vector vibrates perpendicularly to the incident plane and the TE mode for which the electric vector vibrates perpendicularly to this plane. For the incidence along the rotation axis the resultant equations are given in the form similar to the one for a sphere given by the Mie theory. The physical parameters involved are the following five quantities: the size parameter defined by the product of the semifocal distance of the spheroid and the propagation constant of the incident wave, the eccentricity, the refractive index of the spheroid relative to the surrounding medium, the incident angle between the direction of the incident wave and the rotation axis, and the angles that specify the direction of the scattered wave.
622 citations
TL;DR: The differential scattering characteristics of closed three-dimensional dielectric objects are theoretically investigated and the method developed here appears to be most applicable to objects whose physical size is on the order of the wavelength of the incident radiation.
Abstract: The differential scattering characteristics of closed three-dimensional dielectric objects are theoretically investigated. The scattering problem is solved in a spherical basis by the Extended Boundary Condition Method (EBCM) which results in a system of linear equations for the expansion coefficients of the scattered field in terms of the incident field coefficients. The equations are solved numerically for dielectric spheres, spheroids, and finite cylinders to study the dependence of the differential scattering on the size, shape, and index of refraction of the scattering object. The method developed here appears to be most applicable to objects whose physical size is on the order of the wavelength of the incident radiation.
558 citations
TL;DR: The quasi-normal modes of a black hole represent solutions of the relevant perturbation equations which satisfy the boundary conditions appropriate for purely outgoing (gravitational) waves at infinity and purely ingoing waves at the horizon as discussed by the authors.
Abstract: The quasi-normal modes of a black hole represent solutions of the relevant perturbation equations which satisfy the boundary conditions appropriate for purely outgoing (gravitational) waves at infinity and purely ingoing waves at the horizon. For the Schwarzschild black hole the problem reduces to one of finding such solutions for a one-dimensional wave equation (Zerilli's equation) for a potential which is positive everywhere and is of short-range. The notion of quasi-normal modes of such one-dimensional potential barriers is examined with two illustrative examples; and numerical solutions for Zerilli's potential are obtained by integrating the associated Riccati equation.
508 citations
TL;DR: The convergence rate for difference approximations to mixed initial boundary value problems has been shown to be linear in the convergence rate of the difference approximation as mentioned in this paper, which is the best known convergence rate.
Abstract: The convergence rate for difference approximations to mixed initial boundary value problems
TL;DR: In this article, the boundary conditions for the discrete wave equation are constructed using projection operators, and the separation of waves into ingoing and outgoing waves inherent in these boundary conditions greatly facilitates diagnostics.
TL;DR: In this article, the authors derived the terminal characteristic parameters for a uniform coupled-line four-port (4P) IC for inhomogeneous systems in terms of two independent modes that propagate in two uniformly coupled propagating systems.
Abstract: Terminal characteristic parameters for a uniform coupled-line four-port for the general case of an asymmetric, inhomogeneous system are derived in this paper. The parameters (impedance, admittance, etc.) are derived in terms of two independent modes that propagate in two uniformly coupled propagating systems. The four-port parameters derived are of the same form as those obtained for the symmetric case resulting in similar port equivalent circuits for various circuit configurations considered by Zysman and Johnson. The results obtained should be quite useful in designing asymmetric coupled-line circuits in an inhomogeneous medium for various known applications.
TL;DR: Ishii as mentioned in this paper sets up the rigorous theoretical foundations for describing such diverse two phase flow probems as are associated with systems connected with power generation, heat transfer, transport, lubrication, information, environmental control, etc.
Abstract: M Ishii Paris: Eyrolles 1975 pp xxix + 248 price 83F (paperback) This book sets up the rigorous theoretical foundations for describing such diverse two phase flow probems as are associated with systems connected with power generation, heat transfer, transport, lubrication, information, environmental control, etc. The complete set of local laws and appropriate boundary conditions are clearly expounded, and one third of the work deals with the practical importance and unifying power of the time-average operator in making contact with macroscopic models.
TL;DR: In this article, the potential, scattering, and spectral theory associated with boundary value problems for the Laplacian on domains which are perturbed in very irregular fashions are studied.
TL;DR: In this paper, the authors identify the conditions necessary for the validity of a gradient transport model in a simple mean-free-path type of random transport process and present a collection of conditions that may be necessary and sufficient for the applicability of simple gradient transport models in turbulence.
Abstract: Publisher Summary This chapter identifies the conditions necessary for the validity of a gradient transport model in a simple mean-free-path type of random transport process. A principal shortcoming of the telegraph equation as a turbulent diffusion model is that it possesses no Eulerian statistical properties, whereas Eulerian coordinates are those in which most experimental investigations and theoretical analysis of turbulent motion are conveniently carried out. By qualitative analogy with random walk transport, analyzed without rigor, the chapter presents a collection of conditions that may be necessary for the applicability of simple gradient transport models in turbulence. These conditions stipulate degrees of homogeneity and stationarity of the mean field being transported, and of the turbulence properties central to the transport mechanism. One or more of these conditions appear to be violated in each of the traditional turbulent flow boundary value problems, such as boundary layer and jet. There are many alternative approaches to turbulent transport, which are related to neither a gradient transport approximation, nor to a hypothetical, long-path, radiative transport analogy. Closure schemes for hierarchies of turbulence moment equations have been based on a variety of truncated expansion procedures, which often invoke no explicit physical or phenomenological images, but are more or less ad hoc, based sometimes on the intuitive belief that higher order correlation coefficients tend to be very much smaller than lower order ones. It is possible that one of these more formal closure methods will eventually succeed.
01 Jan 1975
TL;DR: In this article, a numerical method for solving the compressible form of the unsteady Navier-Stokes equations is described, with emphasis on the choice of a computational mesh for high Reynolds number flows, finitedifference approximations for mixed partial derivatives, extension of the Courant-Friedrichs-Lewy stability condition for viscous flows, mesh boundary conditions, and numerical smoothing for strong shock-wave calculations.
Abstract: A numerical method for solving the compressible form of the unsteady Navier-Stokes equations is described. This method was originally presented in 1970 and has since been modified during the development of computer programs at Ames for implementing models that account for the effects of turbulence in shock-induced separated flows. Although this paper does not describe the turbulence models themselves, a complete description of the basic numerical method is given with emphasis on the choice of a computational mesh for high Reynolds number flows, finite-difference approximations for mixed partial derivatives, extension of the Courant-Friedrichs-Lewy stability condition for viscous flows, mesh boundary conditions, and numerical smoothing for strong shock-wave calculations.
TL;DR: In this paper, the authors employed a more realistic shape for benzene and found a difference of 10%, which is within the limits of the numerical and experimental uncertainty, for a benzene molecule rotating normal to its axis of symmetry.
Abstract: Friction coefficients are calculated numerically for ellipsoids rotating about their principal axes and for a benzene molecule rotating normal to its axis of symmetry under conditions of creeping flow and slip boundary conditions. It has been shown previously, that if a benzene molecule is approximated by an oblate spheroid, the predicted and experimental friction coefficients differ by more than 40%. The present study employs a more realistic shape for benzene and finds a difference of 10%, which is within the limits of the numerical and experimental uncertainty.
TL;DR: In this paper, a form of quantum electrodynamics is developed which allows us to treat a number of problems involving dielectric and conducting surfaces, the presence of which leads to new observable effects.
Abstract: A form of quantum electrodynamics is developed which allows us to treat a number of problems involving dielectric and conducting surfaces, the presence of which leads to a number of new observable effects. A number of suitably defined response functions play a basic role in the present approach, as these in conjunction with the fluctuation-dissipation theorem lead to electromagnetic field correlation functions, which describe physical effects such as lifetimes, frequency shifts of the excited states, dispersion forces, etc. The quantization of the electromagnetic field is only implicitly used. A large part of the present paper is devoted to the calculation of the response functions involving different geometries and various types of dielectrics. Both spatially dispersive and spatially nondispersive dielectrics are considered. The response functions are calculated using Maxwell's equations and the usual boundary conditions at the interface adjoining the two mediums. As a first application of the present approach, the black-body fluctuations in finite geometries and the influence of surfaces on its temporal and spatial coherence are studied. An interesting theorem is also proved which enables us to calculate the normally ordered (antinormally ordered) correlation functions from the symmetrized correlation functions.
TL;DR: In this paper, values of local flow properties, obtained by solving appropriate conservation equations in finite-difference form and with boundary conditions corresponding to four furnace arrangements, are presented and compared with measurements.
01 Nov 1975
TL;DR: In this article, the elastic analysis of structures with "unilateral contact" and "friction" boundary conditions is considered and it is proved that the considered inequality boundary value problems can be formulated equivalently as variational inequalities, which permit the derivation of theorems of minimum potential and complementary energy, to account for this type of boundary conditions.
Abstract: In the present paper the elastic analysis of structures with “unilateral contact” — and “friction” — boundary conditions is considered. It is proved that the considered inequality boundary value problems can be formulated equivalently as variational inequalities, which permit the derivation of the theorems of minimum potential and complementary energy, to account for this type of boundary conditions. These minimum theorems are used to formulate the analysis as a nonlinear programming problem. Numerical examples on structures with coupled unilateral contact- and friction-boundary conditions illustrate the theory.
TL;DR: A theorem on the unique solvability of the Cauchy problem for a certain nonlinear differential equation in Hilbert space is first proved in this article, which is a very simple abstract analogue of the indicated mixed problem for the quasilinear hyperbolic equation.
Abstract: In the bounded cylinder with arbitrary fixed 0$ SRC=http://ej.iop.org/images/0025-5734/25/1/A09/tex_sm_2203_img2.gif/> the mixed problem with Dirichlet boundary conditions is considered for the quasilinear hyperbolic equation A particular class of functions is introduced in which there is an existence and uniqueness theorem for solutions of this problem.A theorem on the unique solvability of the Cauchy problem for a certain nonlinear differential equation in Hilbert space is first proved. This problem is a very simple abstract analogue of the indicated mixed problem for the quasilinear hyperbolic equation.Bibliography: 2 items.
TL;DR: In this paper, the indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force is formulated as a mixed boundary value problem under the assumption of Coulomb friction with coefficient μ in the region of contact.
Abstract: The indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force is formulated as a mixed boundary value problem under the assumption of Coulomb friction with coefficient μ in the region of contact. Within an inner circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. The slip boundary between the two regions depends on μ and the Poisson ratio v, and is found uniquely as an eigenvalue of a certain integral equation.
TL;DR: The combination of automatic variable order (via deferred corrections) and automatic (adaptive) mesh selection produces, as in the case of initial value problem solvers, a versatile, robust, and efficient algorithm.
Abstract: A variable order variable step finite difference algorithm for approximately solving m-dimensional systems of the form y'' = f(t,y), t $\in$ [a,b] subject to the nonlinear boundary conditions g(y(a),y(b)) = 0 is presented. A program, PASVAR, implementing these ideas has been written and the results on several test runs are presented together with comparisons with other methods. The main features of the new procedure are: a) Its ability to produce very precise global error estimates, which in turn allow a very fine control between desired tolerance and actual output precision. b) Non-uniform meshes allow an economical and accurate treatment of boundary layers and other sharp changes in the solutions. c) The combination of automatic variable order (via deferred corrections) and automatic (adaptive) mesh selection produces, as in the case of initial value problem solvers, a versatile, robust, and efficient algorithm.
TL;DR: In this article, exact matrix equations for surface fields, as well as transmitted and reflected wave amplitudes are obtained for an interface having periodic height variation, in conjunction with the extended boundary condition, in which propagating waves are coupled with evanescent modes.
Abstract: Reflection behavior is examined for an interface having periodic height variation. Upon employing the Bloch theorem, in conjunction with the extended boundary condition, exact matrix equations are obtained for surface fields, as well as transmitted and reflected wave amplitudes. Tunneling considerations then lead to new energy constraints for problems of this type, in which propagating waves are coupled with evanescent modes. In the limit of surface corrugations shallow compared with impinging wavelength, analytic results are obtained confirming both the energy constraints and early computations by Rayleigh. Numerical results demonstrate the efficiency of the method.Subject Classification: 20.30.
TL;DR: In this paper, a cubic spline approximation is used for the solution of several problems in fluid mechanics, including the Burgers' equation, the diffusion equation and the vorticity-stream function system describing viscous flow in a driven cavity.
TL;DR: In this paper, the finite amplitude behavior of global magnetic fields and the large-scale flows induced by them in rotating systems is investigated, where viscous and ohmic dissipative mechanisms both play a role in determining the amplitude and structure of the flows and magnetic fields.
Abstract: Past study of the large-scale consequences of forced small-scale motions in electrically conducting fluids has led to the ‘α-effect’ dynamos. Various linear kinematic aspects of these dynamos have been explored, suggesting their value in the interpretation of observed planetary and stellar magnetic fields. However, large-scale magnetic fields with global boundary conditions can not be force free and in general will cause large-scale motions as they grow. I n this paper the finite amplitude behaviour of global magnetic fields and the large-scale flows induced by them in rotating systems is investigated. In general, viscous and ohmic dissipative mechanisms both play a role in determining the amplitude and structure of the flows and magnetic fields which evolve. In circumstances where ohmic loss is the principal dissipation, it is found that determination of a geo- strophic flow is an essential part of the solution of the basic stability problem. Nonlinear aspects of the theory include flow amplitudes which are independent of the rotation and a total magnetic energy which is directly proportional to the rotation. Constant a is the simplest example exhibiting the various dynamic balances of this stabilizing mechanism for planetary dynamos. A detailed analysis is made for this case to determine the initial equilibrium of fields and flows in a rotating sphere.
TL;DR: In this article, the authors treat some nonlinear two-point boundary value problems for certain ordinary differential equations having singular coefficients and compare traditional finite differences, patch bases and collocation as computational methods.
Abstract: The authors treat some nonlinear two-point boundary value problems for certain ordinary differential equations having singular coefficients. Typically these problems arise when reducing partial to ordinary differential equations by physical symmetry. Traditional finite differences, patch bases and collocation are all developed and compared as computational methods.
TL;DR: In this paper, the large amplitude vibrations of a thin-walled cylindrical shell are analyzed using the Donnell's shallow-shell equations and a perturbation method is applied to reduce the nonlinear partial differential equations into a system of linear PDEs.
Abstract: The large amplitude vibrations of a thin-walled cylindrical shell are analyzed using the Donnell's shallow-shell equations. A perturbation method is applied to reduce the nonlinear partial differential equations into a system of linear partial differential equations. The simply-supported boundary condition and the circumferential periodicity condition are satisfied. The resulting solution indicates that in addition to the fundamental modes, the response contains asymmetric modes as well as axisymmetric modes with the frequency twice that of the fundamental modes. In the previous investigations in which the Galerkins procedure was applied, only the additional axisymrnetric modes were assumed.
Vibrations involving a single driven mode response are investigated. The results indicate that the nonlinearity is either softening or hardening depending on the mode. The vibrations involving both a driven mode and a companion mode are also investigated. The region where the companion mode participates in the vibration is obtained and the effects due to the participation of the companion mode are studied.
An experimental investigation is also conducted. The
results are generally in agreement with the theory. "Non-stationary4 response is detected at some frequencies for large amplitude response where the amplitude drifts from one value to another. Various nonlinear phenomena are observed and quantitative comparisons with the theoretical results are made.