Showing papers on "Boundary value problem published in 1974"
TL;DR: In this paper, an extended form of the correspondence principle is employed to determine directly the quasi-static deformation of viscoelastic earth models by mass loads applied to the surface.
Abstract: An extended form of the correspondence principle is employed to determine directly the quasi-static deformation of viscoelastic earth models by mass loads applied to the surface. The stress-strain relation employed is that appropriate to a Maxwell medium. Most emphasis is placed on the discussion of spherically stratified self-gravitating earth models, although some consideration is given to the uniform elastic half space and to the uniform viscous sphere, since they determine certain limiting behaviors that are useful for interpretation and proper normalization of the general problem. Laplace transform domain solutions are obtained in the form of ‘s spectra’ of a set of viscoelastic Love numbers. These Love numbers are defined in analogy with the equivalent elastic problem. An efficient technique is described for the inversion of these s spectra, and this technique is employed to produce sets of time dependent Love numbers for a series of illustrative earth models. These sets of time dependent Love numbers are combined to produce Green functions for the surface mass load boundary value problem. Through these impulse response functions, which are obtained for radial displacement, gravity anomaly, and tilt, a brief discussion is given of the approach to isostatic equilibrium. The response of the earth to an arbitrary quasi-static surface loading may be determined by evaluating a space-time convolution integral over the loaded region using these response functions.
788 citations
TL;DR: In this article, the friction coefficients for the uniform rotation of spheroids in a viscous fluid with the slipping boundary condition were computed numerically and reported in tabular form.
Abstract: Friction coefficients for the uniform rotation of prolate and oblate spheroids in a viscous fluid, with the slipping boundary condition, are computed numerically and reported in tabular form.
674 citations
TL;DR: In this article, the authors present a general approach to transient heat conduction problems with non-linear physical properties and boundary conditions using an unconditionally stable central algorithm which does not require iteration.
Abstract: The paper presents a generally applicable approach to transient heat conduction problems with non-linear physical properties and boundary conditions. An unconditionally stable central algorithm is used which does not require iteration.
Several examples involving phase change (where latent heat effects are incorporated as heat capacity variations) and non-linear radiation boundary conditions are given which show very good accuracy.
Simple triangular elements are used throughout but the formulation is generally valid and not restricted to any single type of element.
506 citations
TL;DR: In this paper, the modified (underrelaxed, damped) Newton method is extended in such a way as to apply to the solution of ill-conditioned systems of nonlinear equations, i.e. systems having a "nearly singular" Jacobian at some iterate.
Abstract: In this paper the well-known modified (underrelaxed, damped) Newton method is extended in such a way as to apply to the solution of ill-conditioned systems of nonlinear equations, i.e. systems having a "nearly singular" Jacobian at some iterate. A special technique also derived herein may be useful, if only bad initial guesses of the solution point are available. Difficulties that arose previously in the numerical solution of nonlinear two-point boundary value problems by multiple shooting techniques can be removed by means of the results presented below.
453 citations
TL;DR: In this paper, an inhomogeneous linear system is derived whose solution yields the K matrix for the electron-molecule interaction, which is then used to obtain expressions for elastic scattering and photoionization differential cross sections.
Abstract: We adapt the multiple‐scattering method to treat unbound electronic states of molecules in the independent electron approximation. An inhomogeneous linear system is derived whose solution yields the K matrix for the electron‐molecule interaction. Using the K matrix, we derive continuum electronic wavefunctions by imposing boundary conditions corresponding to electron‐molecule scattering and molecular photoionization, i.e., the wavefunctions satisfy the so‐called outgoing‐wave and incoming‐wave normalization, respectively. These wavefunctions are then used to obtain expressions for elastic scattering and photoionization differential cross sections.
441 citations
TL;DR: In this article, the authors considered the problem of estimating the dimensions and paths of a crack in a rectilinear cut with respect to the number of cracks in the cut.
Abstract: One of the problems of fracture mechanics is the prediction of the propagation of cracks in solids. The present paper deals mainly with linear fracture mechanics which owes its origin to the works of A. A. Griffith [1, 2] and studies the development of cracks under sufficiently low loads when the behaviour of the material within a region sufficiently remote from the edges of cracks may be regarded as linearly elastic, At present, linear fracture mechanics [3] is restricted mainly to special kinds of loading geometry, with the crack extending rectilinearly (in a plane case) or in its plane (in a three-dimensional case). The main problem here is to establish a relationship between the dimensions of cracks and the loads applied. Within the framework of linear fracture mechanics the fracture itself and other non-linear phenomena that precede it are assumed to take place only within local regions which are small compared to the dimensions of cracks. The possibility that such a situation exists is associated with the fact that when the crack dimensions are sufficiently large the characteristic dimension of the end region is fully determined by a certain intrinsic dimension of the material structure. Therefore, if the material does not exhibit time dependency, the state of the end region at the moment of rupture becomes fully independent of the loads applied and the geometry of the solid, i.e. autonomous. The notion of autonomy [4] leads to the formulation of this theory as one of limit equilibrium. If the conditions of rectilinear extension of the crack (or those of the crack extension in its plane) are disturbed, there arises a problem of determining not only the dimensions of the crack, but also the path of the crack extension under such conditions of loading that a slow, quasi-static crack development is possible. This problem can be actually subdivided into two: (1) Criteria for the determination of the dimensions and paths of the crack extension, and (2) Expressions for the characteristics of the stress-strain state which are constituents of these criteria through the geometry of solid with cracks and the loads applied. As regards (1), there have been many assertions, and the connections between them are not quite clear at present. The first of the suggested criteria, namely that of local symmetry for the plane problem formulated by Barenblatt and Cherepanov [5, 6] and by Erdogan and Sih [7] can be within certain limits substantiated and generalized for the three-dimensional case. The guiding principle here is the treatment of the theory of cracks from the standpoint of the method of inner and outer expansions or that of singular perturbations [8]. The concept of the stress intensity factor which is basic in linear fracture mechanics is decisive in matching inner and outer expansions to find the main term of the asymptotic solution of the complete problem. Actually the construction of the theory of equilibrium cracks [4] implicitly employs this technique for a certain specific model. More explicit indications are given in Ref. 9. In the treatment of the problem of plastic zones in the vicinity of notches, the idea of the boundary layer is employed in Ref. 10. The problem of fracture of a solid is analysed from this standpoint in Ref. 11. As regards (2), progress has been hampered by the lack of efficient techniques for fording the stress-strain state of a solid having non-rectilinear cuts. A number of investigations have been carried out for cuts of a particular kind an arc of a circumference [12, 13], an arc of a parabola [l4], and a three-link broken line which is close to a straight line to such an extent that the boundary conditions are assumed referable to the direction of the middle portion [15, 16]. The problem of a semi-infinite curvilinear cut slightly deviating from a rectilinear one by expanding complex elastic potentials in the magnitude of deviation of the cut from the rectilinear axis tangent to the line of cut at its end is considered in Ref. 17. An exact solution of the problem of a semi-infinite cut having the form of a two-link broken line is given in Ref. 18. The present paper is devoted to the investigation of the development of cracks under arbitrary loading conditions. In Section 1 the criterion of local symmetry is substantiated and generalized for the three-dimensional case. In Section 2 an effective procedure of finding stress intensity factors for the plane case is given, in terms of which the criterion is formulated. Closed first approximation formulas for these magnitudes are presented in the case of a slightly curved crack, numerical calculations showing the applicability of the latter with an error not exceeding 10 to 15 with the angles of deviation of the crack from the straight line coming to 20°. In Section 3 equations of extension of curvilinear cracks are derived on the basis of the first approximation formulas and criterion of local symmetry. In Section 4 some examples are considered.
433 citations
TL;DR: In this article, the authors studied the Variational boundary value problem for quasilinear elliptic systems in divergence form and derived several existence theorems which generalize the basic results of Browder and Leray-Lions.
Abstract: Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined. © 1974 American Mathematical Society.
405 citations
TL;DR: In this article, a method for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations is presented, which consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter e, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration.
Abstract: A method is presented for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations. It consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter e, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration. The density ψ is expressed as the sum of an interior part ψi, a boundary layer part ψb, and an initial layer part ψ0. Then ψi is sought as a power series in e, while ψb decays exponentially with distance from a boundary or interface at a rate proportional to e−1. Similarly ψ0 decays at a rate proportional to e−1 with time after the initial time. For a near critical reactor, the leading term in ψi is determined by a diffusion equation. The leading term in ψb is determined by a half‐space problem with a plane boundary. The initial and boundary conditions for the diffusion equation are obtained by requiring ψ0 and ψb to decay away from ...
402 citations
Proceedings Article•
01 Jan 1974
TL;DR: The Second Edition of this popular text provides an insightful introduction to the use of finite difference and finite element methods for the computational solution of ordinary and partial differential equations.
Abstract: Learn to write programs to solve ordinary and partial differential equations The Second Edition of this popular text provides an insightful introduction to the use of finite difference and finite element methods for the computational solution of ordinary and partial differential equations. Readers g...
395 citations
TL;DR: A general theory of subsonic potential aerodynamic flow around a lifting body having arbitrary shape and motion is presented in this article, where an integral representation for the velocity potential is obtained by using the Green function method.
Abstract: A general theory of subsonic potential aerodynamic flow around a lifting body having arbitrary shape and motion is presented. By using the Green function method, an integral representation for the velocity potential is obtained for both supersonic and subsonic flow. Under the small perturbation assumption, the potential at any point in the field depends only upon the values of the potential and its normal derivative on the surface of the body. On the surface of the body, this representation reduces to an integro-differential equation relating the potential and its normal derivative (which is known from the boundary conditions) on the surface. The theory is applied to finite-thickness wings in subsonic steady and oscillatory flows.
329 citations
TL;DR: In this article, it was shown that the error in an interior domain 2 can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error of a weaker norm over a slightly larger domain which measures the effects from outside of the domain Q.
Abstract: Interior a priori error estimates in Sobolev norms are derived from interior RitzGalerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain 2 can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain Q. Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given. 0. Introduction. There are presently many methods which are available for computing approximate solutions of elliptic boundary value problems which may be classified as Ritz-Galerkin type methods. Many of these methods differ from each other in some respects (for example, in how they treat the boundary conditions) but have much in common in that they have what may be called "interior Ritz-Galerkin equations" which are the same. Here we shall be concerned with finding interior estimates for the rate of convergence for such a class of methods which are consequences of these interior equations. Let us briefly describe, in a special case, the type of question we wish to consider. Let &2 be a bounded domain in RN with boundary M2 and consider, for simplicity, the problem of finding an approximate solution of a boundary value problem (0.1) \u =f in Q2, (0.2) Au= g on U2, where A is some boundary operator. Suppose now that we are given a one-parameter family of finite-dimensional subspaces Sh (0 < h < 1) of an appropriate Hilbert space in which u lies and that, for each h, we have computed an approximate solution Uh c Sh to u using some Ritz-Galerkin type method. Here we have in mind, for example, methods such as the "engineer's" finite element method [8], [22], the Aubin-Babuska penalty method [2], [4], the methods of Nitsche [12], [13] or the Received October 15, 1973. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. Copyright i 1974, American Mathematical Society
TL;DR: In this article, the authors consider the boundary value problem and investigate the existence and stability properties of stationary solutions under suitable hypotheses concerning f. The approach is to interpret (*) as a dynamical system in an appropriately chosen Banach space.
Abstract: We Consider the boundary value problem [d] Here λ is a non negative parameter; f is a given real valuede function defined and a class C2 [d] is an arbitrarily specified function of class C1 on [0, n] satisfying [d] = 0. Under suitable hypotheses concerning f, we investigate the existence and stability properties of stationary solutions for (*). Our approach is to interpret (*) as a dynamical system in an appropriately chosen Banach space, and then to apply to (*) certain known results in the theory of Liapunov stability for general dynamical systems
TL;DR: In this article, the Dirichlet and Neumann conditions are applied to alternate components at the boundary of a model to eliminate reflections from the boundary by adding together the solutions of the Dirichelet and NN problems.
TL;DR: An improved numerical implementation of the boundaryintegral equation method for three dimensional stress analysis is reported in this article, which models the boundary data as piecewise-linear variations over the boundary segments.
Journal Article•
TL;DR: In this paper, the Toda chain with an arbitrary number of particles is shown to be completely integrable, and a scheme to integrate these systems and study the interaction between solitons is developed.
Abstract: We use the inverse scattering method to study a system of particles with exponential interaction (the Toda chain) and a set of equations describing induced scattering of plasma oscillations by ions. We show that a Toda chain with an arbitrary number of particles is completely integrable. We develop a scheme to integrate these systems and study the interaction between solitons. We indicate a class of completely integrable discrete systems, that is, systems which can not be stochastized.
TL;DR: The relation between various boundary integral equation formulations of Dirichlet and Neumann problems for the three-dimensional Helmholtz equation is clarified in this paper, where the connection between interior and exterior problems is demonstrated; the spectra of the associated integral operators are investigated; and various properties of interior eigenfunctions are derived.
Abstract: The relation between various boundary integral equation formulations of Dirichlet and Neumann problems for the three-dimensional Helmholtz equation is clarified The integral equations derived using single or double layer distributions as well as those based on the Helmholtz representation using an unmodified free space Green’s function are presented in uniform notation; the connection between interior and exterior problems is demonstrated; the spectra of the associated integral operators are investigated; and various properties of interior eigenfunctions are derived Assuming only that exterior problems have at most one solution, existence as well as uniqueness of the corresponding boundary integral equations is established even at eigenvalues of the adjoins interior problems In this case, supplementary conditions are required, and these are presented in the form of boundary rather than volume integral equations
TL;DR: In this article, conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form (3/3» 2 Aja/dxj ) were established.
Abstract: This paper establishes conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form (3/3» — 2 Aja/dxj — B)u = F on [0, r] X ß, Mu g on [0,7\"] x 3Í2, u(0, x) f(x), x e Q. Assuming that X1 a priori inequalities are known for this equation, it is shown that if F e H'([0, T] x Q), g e H'*l'2([0, T] X 3ñ),/ e H'(Q) satisfy the natural compatibility conditions associated with this equation, then the solution is of class C from [0, T] to H'~f(Q), 0 < p < s. These results are applied to mixed problems with distribution initial data and to quasi-linear mixed problems.
TL;DR: In this paper, it is shown that if the nonlinearity and initial data in A satisfy certain restrictions then no classical (or weak) solution of A can exist for all time.
TL;DR: In this paper, a nonperturbative solution is obtained by expanding the scattered and transmitted fields in terms of spherical vector wave functions, so that Maxwell's equations are satisfied exactly in the regions exterior and interior to the raindrop, and by combining point matching with least-squares fitting to satisfy the boundary conditions on the surface of the raindropper with sufficient accuracy.
Abstract: This paper gives details of the analytical and num$eLrical procedures used to solve the basic problem of the scattering of a plane electromagnetic wave by an axisymmetric raindrop. A nonperturbative solution is obtained by expanding the scattered and transmitted fields in terms of spherical vector wave functions, so that Maxwell's equations are satisfied exactly in the regions exterior and interior to the raindrop, and by combining point matching with least-squares fitting to satisfy the boundary conditions on the surface of the raindrop with sufficient accuracy. Numerical results are presented for scattering by oblate spheroidal raindrops, with eccentricity depending on (and increasing with) drop size, for two orthogonal polarizations of the incident wave. The calculations were made at 4, 11, 18.1, and 30 GHz, in the case in which the direction of propagation of the incident wave is perpendicular to the axis of symmetry of the raindrop, which is of interest for terrestrial microwave relay systems. At 30 GHz, the calculations were also made for the case in which the angle between the direction of propagation and the axis of symmetry is 70° and 50°, since different elevation angles are of interest for satellite systems. These basic results were summed earlier over the drop-size distribution to calculate the differential attenuation and differential phase shift caused by rain, which are of importance in the investigation of cross polarization in radio communication systems.
TL;DR: In this article, a theory for surface polaritons associated with the planar surface of a semi-infinite anisotropic dielectric medium is presented for magnetic fields either perpendicular or parallel to the surface.
Abstract: A theory is presented for surface polaritons associated with the planar surface of a semi-infinite anisotropic dielectric medium. Retardation is included. In general, two attenuating components with different attenuation constants must be superposed within the medium in order to satisfy the boundary conditions, and the macroscopic electric field vector does not lie in the sagittal plane. For special cases, however, only one attenuating component is required, and the electric vector does lie in the sagittal plane. The theory is applied to the specific case of surface magnetoplasmons in a semiconductor for magnetic fields either perpendicular or parallel to the surface. In the latter case, propagation directions parallel and perpendicular to the magnetic field are considered. Possibilities for the experimental observation of the effects predicted are discussed.
DOI•
29 Jan 1974TL;DR: In this paper, the interaction of linear, plane water waves with stationary groups of rigid, vertical, circular cylinders under conditions in which the inertial forces on the cylinders dominate over the drag forces was studied.
Abstract: This study deals with the interaction of linear, plane water waves with stationary groups of rigid, vertical, circular cylinders under conditions in which the inertial forces on the cylinders dominate over the drag forces. A direct matrix solution as well as multiple scattering as suggested by Twersky (1952) are used to obtain the velocity potential in the vicinity of the cylinders. The groups may consist of a number of cylinders having any geometric arrangement, may have Dirichlet, Neumann, or mixed boundary conditions, and need not have identical diameters. The study represents an extension of the single cylinder case presented by MacCamy and Fuchs in 1954. Basic scattering coefficients for 192 different arrangements of two cylinders are obtained with the aid of a Bessel coordinate transformation and a matrix inversion procedure. The resulting potential function is then applied to calculate force components in the direction of wave advance and orthogonal to it. For the cases considered the former departs as much as 65% from the force on a single cylinder and the mass coefficient is found to range from 1.19 to 3.38 - a not insignificant departure from the often used value of 2.0. Furthermore the orthogonal force may be as large as 67% of the single-cylinder force.
TL;DR: In this paper, the problem of the harmonic forced vibrations of a massless rigid disc supported on an elastic layered medium is studied and the resulting mixed boundary value problems are reduced to sets of Fredholm integral equations that are solved numerically for a wide range of frequencies.
01 Jan 1974
TL;DR: In this article, the boundary value problem associated with the shielded inicrostrip-liie structure is formulated in terms of a rigorous hybrid-mode representation, and the resulting equations are subsequently transformed, via the application of Galerkink method in the spectral domain, to yield a characteristic equation for the dispersion properties of shielded microstrip lines.
Abstract: The boundary value problem associated with the shielded inicrostrip-liie structure is formulated in terms of a rigorous hybrid-mode representation. The resulting equations are subsequently transformed, via the application of Galerkink method in the spectral domain, to yield a characteristic equation for the dispersion properties of shielded microstrip lines. Among the ad- vantages of the method are its simplicity and rapid convergence. Numerical results are included for several d~erent structural parameters. These are compared" with other available data and with some experimental results.
01 Dec 1974
TL;DR: In this article, a general theory of potential aerodynamic flow around a lifting body having arbitrary shape and motion is presented, and an integral representation for the potential is obtained for both supersonic and subsonic flow.
Abstract: The general theory of potential aerodynamic flow around a lifting body having arbitrary shape and motion is presented. By using the Green function method, an integral representation for the potential is obtained for both supersonic and subsonic flow. Under small perturbation assumption, the potential at any point, P, in the field depends only upon the values of the potential and its normal derivative on the surface, sigma, of the body. Hence, if the point P approaches the surface of the body, the representation reduces to an integro-differential equation relating the potential and its normal derivative (which is known from the boundary conditions) on the surface sigma. For the important practical case of small harmonic oscillation around a rest position, the equation reduces to a two-dimensional Fredholm integral equation of second-type. It is shown that this equation reduces properly to the lifting surface theories as well as other classical mathematical formulas. The question of uniqueness is examined and it is shown that, for thin wings, the operator becomes singular as the thickness approaches zero. This fact may yield numerical problems for very thin wings.
TL;DR: In this article, the even-parity perturbations of the Reissner-Nordstrom family of black holes are considered and the wave equations are decoupled by a simple transformation.
Abstract: Extending some previous work, we here consider the even-parity perturbations of the Reissner-Nordstr\"om family of black holes. Gauge-invariant functions of the gravitational and electromagnetic perturbations are defined in terms of the standard Regge-Wheeler expansion functions, and the coupled wave equations obeyed by these quantities are given. The wave equations are decoupled by a simple transformation, and the decoupled equations are shown to admit no unstable normal-mode solutions obeying the appropriate boundary conditions.
TL;DR: In this paper, a system of three quasi-linear partial differential equations is considered, as a simplified model of the transport of mobile carriers in a semiconductor device, and it is shown that the initial value problem is well-posed, and that the steady state solution is unique and stable.
Abstract: A system of three quasi-linear partial differential equations is considered, as a simplified model of the transport of mobile carriers in a semiconductor device. Assuming a convenient form of the boundary conditions, it is shown that the initial value problem is well-posed, and that the steady state solution is unique and stable. A finite difference approximation preserving reasonable bounds on the numerical solutions is also described.
TL;DR: This paper extends techniques that have been used in piecewise polynomial approximation which permit the construction of equidistributing meshes to meshes on which the local truncation error of the method is approximately constant in some norm.
Abstract: In order to use finite difference approximations with non-uniform meshes in boundary value problems, it is necessary to develop procedures for mesh selection. In this paper we extend techniques that have been used in piecewise polynomial approximation which permit the construction of equidistributing meshes. By this term we mean meshes on which the local truncation error of the method is approximately constant in some norm. Improved error estimates for methods which use equidistributing meshes are obtained.