Showing papers on "Boundary value problem published in 1970"

Numerical Methods That Work

Abstract: Part I. Fundamental Methods: 1. The calculation of functions 2. Roots of transcendental equations 3. Interpolation - and all that 4. Quadrature 5. Ordinary differential equations - initial conditions 6. Ordinary differential equations - boundary conditions 7. Strategy versus tactics - roots of polynomials 8. Eigenvalues I 9. Fourier series Part II. Double Trouble: 10. Evaluation of integrals 11. Power series, continued fractions, and rational approximations 12. Economization of approximations 13. Eigenvalues II - rotational methods 14. Roots of equations - again 15. The care and treatment of singularities 16. Instability in extrapolation 17. Minimum methods 18. Laplace's equation - an overview 19. Network problems.

The finite element method for elliptic equations with discontinuous coefficients

Abstract: Numerical solutions of boundary value problems for elliptic equations with discontinuous coefficients are of special interest In the case when the interface (ie the surface of the discontinuity of the coefficients) is smooth enough, then also the solution is usually very smooth (except on the interface) To obtain a high order of accuracy presents some difficulty, especially if the interface does not fit with the elements (in the finite element method) In this case the norm of the error in the spaceW1/2 is of the orderh 1/2 (see eg [1]) and on one dimensional case it is easy to see that the accuracy cannot be improved In this paper we shall show an approach which avoids this difficulty The idea is similar to [2] We shall show the proposed approach on a model problem — theDirichlet problem with an interface forLaplace equation; this will avoid pure technical difficulties The boundary of the domain and the interface will be assumed smooth enough The sufficient condition for the smoothnees can be determined

Thermomechanical analysis of viscoelastic solids

Abstract: SUMMARY This paper is concerned with the development of a computational algorithm for the solution of the uncoupled, quasi-static boundary value problem for a linear viscoelastic solid undergoing thermal and mechanical deformation. The method evolves from a finite element discretization of a stationary value problem, leading to the solution of a system of linear integral equations determining the motion of the solid. An illustrative example is included.

Calculation of Coefficients of Capacitance of Multiconductor Transmission Lines in the Presence of a Dielectric Interface

Abstract: A method is given for the numerical determination of the coefficients of capacitance for a class of multiconductor transmission-line systems. This class includes systems without ground planes, or with one or two ground planes, with the lines embedded in one or two layers of dielectrics. The conductors can be of any cross section that can be approximated adequately by polygons. The method is a refinement of the subareas method in which the assumption of a "staircase function" surface charge density, that is, constant charge density over each subarea, is replaced by the assumption of a piecewise linear charge density over the conductor surfaces, and the charge density parameters are determined by making a least-squares fit to the potential to the boundary conditions of the problem.

Scattering by a two-dimensional periodic array of conducting plates

Abstract: The boundary value problem of an infinite array of thin plates arranged in a doubly periodic grid along any two coordinates is formulated in a general form for an arbitrarily polarized plane wave incident from any oblique angle. The induced current on the plate, the near-field distribution, and the distant reflected waves can be obtained to a very close accuracy. Both magnitudes and phases of the reflection coefficients for some specific examples are determined explicitly. For the case of a wave incident normally on a rectangular lattice array of narrow rectangular plates, the calculated values are in excellent agreement with the measurements in a previously published paper.

A method of solution for certain problems of transient heat conduction

Abstract: This paper develops a numerical treatment of classical boundary value problems for ar- bitrarily shaped plane heat conducting solids obeying Fourier's law. An exact integral formula defined on the boundary of an arbitrary body is obtained from a fundamental singular solu- tion to the governing differential equation. This integral formula is shown to be a means of numerically determining boundary data, complementary to given data, such that the Laplace transformed temperature field may subsequently be generated by a Green's type integral identity. The final step, numerical transform inversion, completes the solution for a given problem. All operations are ideally suited for modern digital computation. Three illustra- tive problems are considered. Steady-state problems, for which the Laplace transform is un- necessary, form a relatively simple special case. A FORMULATION of the various transient boundary value problems associated with isotropic solids obeying Fourier's law of heat conduction is developed. An exact in- tegral formula is derived relating boundary heat flux and boundary temperature, in the Laplace transform space, that corresponds to the same admissible transformed temperature field throughout the body. Part of the boundary data in the formula is known from the description of a well posed bound- ary value problem. As is shown, the remaining part of the boundary data is obtainable numerically from the formula it- self regarded as a singular integral equation. Once both trans- formed temperature and heat flux are known everywhere on the boundary, the transformed temperature throughout the body is obtainable by means of a Green's type integral identity. This identity yields the field directly in terms of the mentioned boundary data. The final step, transform in- version, although done approximately also, is accomplished by a technique particularly well suited to the class of problems under investigation. The main feature of the solution procedure suggested is its generality. It is applicable to solids occupying domains of rather arbitrary shape and connectivity. Boundary data may be prescription of temperature, or heat flux, or parts of each corresponding to a mixed type problem. Also, a linear combination of temperature and flux may be given corre- sponding to the so-called convection boundary condition. The same boundary formula described previously is applicable in every case. Approximations in the transform space are made only on the boundary, in contrast to finite difference procedures, and the approximations made are conceptually simple, natural to make, and give rise, as is shown, to very ac- curate data for a relatively crude boundary approximation pattern. Problems posed for composite bodies, i.e., two or more heat conducting solids bonded together, are particularly amenable to the present treatment. One computer program is employed which utilizes only data describing the domain geometry, boundary temperature or flux, material properties, and a sequence of values of the transform parameter neces- sary for the inversion scheme. Output is the transformed temperature at any desired field point. A second program in-

Dielectric Friction on a Moving Ion. II. Revised Theory

Abstract: The Born–Fuoss–Boyd–Zwanzig calculation of the dielectric friction coefficient on a moving ion is revised in two ways. The relative motion of the ion and its surrounding fluid is taken into account by standard hydrodynamic methods, with both sticking and slipping boundary conditions. Also, the electrostatic problem of finding the electric field due to a change in a moving medium is formulated and solved exactly. The magnitude of the resulting friction coefficient is smaller than in the B–F–B–Z calculation. When the slipping boundary condition is used, the maximum conductance predicted for a singly charged ion in water is about 46 conductance units.

Differential rotation in stellar convection zones.

Abstract: The analysis of an earlier paper (Busse l970a) in which the differential rotation of the Sun was explained as the result of large scale convection in the solar convection zone is extended to include the detailed spatial dependence of the meridional circulation as well as of the differential rotation The thin shell approximation for a rotating spherical convection layer of a Boussinesq fluid is used The asymptotic representation of large order spherical harmonlcs by Hermite functions permits a simple integration of the equations for the meridional circulation and the differential rotation Explicit analytical solutions are given for the case of stressfree boundaries and numerical results are shown in the case of a rigid inner and a free outer boundary The case of stressfree boundaries is rather exceptional and the magnitude of the differential rotation is reduced for more general boundary conditions However, the differential rotation still exhibits the characteristic equatorial acceleration in the case of an inner rigid boundary (auth)

The Perturbations of Alternating Geomagnetic Fields by Conductivity Anomalies

Abstract: Summary The two-dimensional problems of interest in studying the perturbation of alternating electric current by a sharp discontinuity of conductivity in a conductor are considered, and their applicability to geophysical problems discussed. A numerical method has been developed for solving the appropriate differential equations and boundary conditions. The method has been applied to a vertical discontinuity in conductivity such as at a continental-oceanic interface. The two polarization cases are solved, and the fields and current distributions are determined in detail.

A New Technique for the Analysis of the Dispersion Characteristics of Microstrip Lines

Abstract: Dispersion characteristics of shielded microstrip lines are investigated using a new technique. The method utilizes the well-known singular integral equation approach for deriving an alternate form of eigenvalue equation with superior convergence properties. It is shown that accurate numerical results may be obtained from this eigenvalue equation using only a 2x2 matrix equation. In comparison, the conventional formulation of the problem requires the use of matrices that are much larger in size. Aside from the numerical efficiency, the simplicity of the method makes it possible to conveniently extract higher order modal solutions for the propagation constants that affect the high-frequency application of microstrip lines. Even though the derivation of the determinantal equation requires some intricate mathematical manipulations, the user may bypass these completely and use the final eigenvalue equation which is programmable on the computer.

Boundary Conditions for Mathemagenic Behaviors1

Abstract: Herbert Spencer once said that when a man's knowledge is not in order, the more of it he has the .greater will be his confusion. There is more knowledge about instructional processes available today than ever before; and there is more confusion. Educational psychologists write that educational psychology seems to be “ . . . in a superficial, ill-digested, and typically disjointed and watered-down miscellany of general psychology, learning theory, developmental psychology, social psychology, psychological measurement, psychology of adjustment, mental hygiene, client-centered counseling and child-centered education . . . \" (Ausubel, 1968, p. 1), and that “ . . . the study of instruction has produced a tremendous quantity of empirical research studies, many of them without thoughtful conceptualizations, without explicit responsibility for developing theory of instruction, and without contribution to knowledge about instruction\" (Wittrock, 1967, p. 1). As Spencer warned, intellectual disorder (in educational psychology) has resulted in confusion. As suggested in the quotes above, this disorder stems in part from two afflictions that have plagued the research community.

Capacitance and Field Distributions for Interdigital Surface-Wave Transducers

Abstract: : A boundary-shift technique used in conjunction with relaxation solutions of Laplace's equation allows the convenient numerical evaluation of the potential in the neighborhood of interdigital comb structures. With this method, the area of computation in the unbounded problem can be restricted to the region of interest near the electrode and interface surfaces. Because of the point-by-point nature of the calculation, a wide range of geometries can be studied with the inclusion of the effects of finger thickness and shape, and of any layers present. Capacitance values for many single interface and layered configurations of surface-wave transducers are presented along with a few representative examples of potential and field maps. (Author)

The Effect of Boundary Conditions on the Response of Laminated Composites

Abstract: The effect of boundary conditions on the bending, vibrations, and buckling of unsymmetrically laminated rectangular plates is in vestigated. Five sets of boundary conditions corresponding to various clamped and simply-supported edges are treated. The effect of in- plane boundary conditions is shown to be a function of fiber orienta tion within the laminate. Numerical results also show that the effects of bending-extensional coupling can be severe for all the boundary conditions considered. The applicability of the reduced-bending- stiffness approximation is also explored.

On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems

Abstract: Consider a linear ordinary differential equation of the 2nd order which has a singularity at the origin; according to the nature of this singularity we must consider either the two-point boundary-value problem or the one-point boundary value problem. Finite-difference schemes are studied; results are given concerning error analysis and monotone convergence.

General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions

Abstract: In this paper, a general technique for solving nonlinear, two-point boundary-value problems is presented; it is assumed that the differential system has ordern and is subject top initial conditions andq final conditions, wherep+q=n. First, the differential equations and the boundary conditions are linearized about a nominal functionx(t) satisfying thep initial conditions. Next, the linearized system is imbedded into a more general system by means of a scaling factor α, 0≤α≤1, applied to each forcing term. Then, themethod of particular solutions is employed in order to obtain the perturbation Δx(t)=αA(t) leading from the nominal functionx(t) to the varied function $$\tilde x$$ (t); this method differs from the adjoint method and the complementary function method in that it employs only one differential system, namely, the nonhomogeneous, linearized system. The scaling factor (or stepsize) α is determined by a one-dimensional search starting from α=1 so as to ensure the decrease of the performance indexP (the cumulative error in the differential equations and the boundary conditions). It is shown that the performance index has a descent property; therefore, if α is sufficiently small, it is guaranteed that $$\tilde P$$

One-dimensional consolidation of layered systems

Abstract: An analytical solution to the layered consolidation problem for a general set of boundary conditions and an arbitrary load history is presented herein. An example is provided in which the soil profile consists of four compressible layers.

A modified mapping-collocation technique for accurate calculation of stress intensity factors

Abstract: A technique combining the advantages of conformal mapping and boundary collocation arguments for calculating stress intensity factors for cracks in plane problems is described. The difficulty of finding the mapping function on a rigidly prescribed parameter region is avoided at the expense of using boundary collocation methods on part of the boundary. Conventional collocation arguments are modified by prescribing stress, force, and moment conditions in a least-square collocation sense. These pseudo-redundant conditions provide a reasonable basis for estimation of the effects of inaccuracy of the boundary conditions. The technique is applied to the problem of a circular disk with an internal crack under a loading of external hydrostatic tension.

Axisymmetric buckling of hollow spheres and hemispheres

Abstract: : A nonlinear thin shell theory is derived for the axisymmetric buckling of spherical shells subjected to either a pressure or a centrally directed surface load. The theory is reduced to a boundary value problem for a system of four first order ordinary differential equations. Numerical solutions of this boundary value problem are obtained by the shooting and parallel shooting methods. An extensive numerical study is made of the nonlinear deformations of the shells. We find for example, that all solution branches that bifurcate from the eigenvalues of the linearized buckling theory are connected to each other by means of intermediate branches. Some implications of the numerical results concerning the buckling of spherical shells are discussed. (Author)