Data continuation for the explicit solution of an inverse biomagnetic problem

This paper considers the problem of data continuation for an explicit identification method for electric dipoles situated in a conductor, and shows that the magnetic field measured over a part of a closed surface (which encloses the conductor) can be analytically continued to the rest of the surface. Such a continuation problem has a unique solution for an arbitrarily shaped conductor and observation surface. The paper proposes two straightforward methods for the continuation - the spherical harmonic expansion and the integral equation approach - and presents continuation examples for the case of a single dipole in a spherical conductor with the data measured on a part of a spherical observation surface. The spherical harmonic expansion method works well when the field is continued relatively far from the conductor and when the measured data are clean; however, the spherical harmonic expansion fails when the data are noisy. On the other hand, the integral equation approach, although it suffers from numerical integration errors, shows a much better continuation with quite noisy data. (A method to reduce the numerical integration errors is proposed.) The latter approach is also much more accurate than the spherical harmonic expansion for continuing the field in the proximity of the conductor. Finally, the paper discusses the applicability of both methods to an explicit identification method.     

An approximate solution of the annular crack problem

A method is presented for the approximate solution of the axisymmetric problem of an annular crack Embedded in an infinite elastic solid and subjected to a normal internal pressure. The numerical results Obtained for the stress intensity factors are in good agreement with those obtained by Smetanin².     

On the coupled differential-integral equations for the solution of the general magnetostatic problem

The paper discusses some coupled differential-integral formulations of the magnetostatic problem. These formulations are derived from the original integral equation. Their main features are differential equations inside the iron; and an integral equation on the iron boundary, that is coupled with the differential equations. An application of perturbation estimates is illustrated. Preliminary computational results support our belief that the methodology described is functional.     

A general solution to the necklace model problem in the rheology of macromolecules

Summary A very general solution is given of a problem which plays a vital role in certain molecular theories on the deformation of macromolecules. In these theories use is made of the necklace model, in which the macromolecule in solution is supposed to be built up of submolecules or segments. The total hydrodynamic resistance of the monomers in a segment is assumed to be concentrated in the end points of the segments: the beads. By applying a macroscopic field of flow to the solution, forces are exerted on the beads which are compensated by entropy-elastic forces in the segment. The configuration of the beads is described by a distribution function which must satisfy an equation of continuity. This equation is now solved by means of Fourier transformation for any time-dependent field of flow for which the velocities are linear functions of the coordinates. The solution appears to be a Gaussian distribution whose second moments have to satisfy a system of linear first-order time-dependent differential equations. Once the distribution function is known, all kinds of macroscopically measurable quantities of the solution can in principle be calculated.     

An explicit elastic solution for a brittle film with periodic cracks

A two-dimensional explicit elastic solution is derived for a brittle film bonded to a ductile substrate through either a frictional interface or a fully bonded interface, in which periodically distributed discontinuities are formed within the film due to the applied tensile stress in the substrate and consideration of a “weak form stress boundary condition” at the crack surface. This solution is applied to calculate the energy release rate of three-dimensional channeling cracks. Fracture toughness and nominal tensile strength of the film are obtained through the relation between crack spacing and tensile strain in the substrate. Comparisons of this solution with finite element simulations show that the proposed model provides an accurate solution for the film/substrate system with a frictional interface; whereas for a fully bonded interface it produces a good prediction only when the substrate is not overly compliant or when the crack spacing is large compared with the thickness of the film. If the section is idealized as infinitely long, this solution in terms of the energy release rate recovers Beuth’s exact solution for a fully cracked film bonded to a semi-infinite substrate. Interfacial shear stress and the edge effect on the energy release rate of an asymmetric crack are analyzed. Fracture toughness and crack spacing are calculated and are in good agreement with available experiments.     

Direct method of solution for general boundary value problem of the Laplace equation

A general boundary value problem for two-dimensional Laplace equation in the domain enclosed by a piecewise smooth curve is considered. The Dirichlet and the Neumann data are prescribed on respective parts of the boundary, while there is the second part of the boundary on which no boundary data are given. There is the third part of the boundary on which the Robin condition is prescribed. This problem of finding unknown values along the whole boundary is ill posed. In this sense we call our problem an inverse boundary value problem. In order for a solution to be identified the inverse problem is reformulated in terms of a variational problem, which is then recast into primary and adjoint boundary value problems of the Laplace equation in its conventional form. A direct method for numerical solution of the inverse boundary value problem using the boundary element method is presented. This method proposes a non-iterative and unified treatment of conventional boundary value problem, the Cauchy problem, and under- or over-determined problems.     

An iterative filter for solution of the inverse heat-conduction problem

An iterative modification of the nonlinear Kalman filter is proposed for the determination of time-variable heat-transfer coefficients.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 35, No. 5, pp. 916–923, November, 1978.     

An explicit closed-form solution for linear systems subjected to nonstationary random excitation

Explicit, closed-form solutions are presented for the correlation matrix and evolutionary power spectral density matrix of the response of a linear, classically damped MDOF system subjected to a uniformly modulated random process with the gamma envelope function. The effects of the statistical cross-modal correlations on the evolutionary mean square responses are investigated. A simple MDOF system subjected to ground motion is used as an illustrative example. Through this study, additional insight is gained into the nonstationary behavior of linear dynamic systems.     

An algorithm for the solution of a third-order eigenvalue problem

A simple, rapidly convergent procedure is described for solving a third-order symmetric eigenvalue problem Au = λ Bu typically arising in vibration analysis. The eigenvalue problem is represented in terms of its variational dual, the Rayleigh quotient, and the eigenosolution is obtained through a topographical search for points of quotient stationarity. The associated computer routine is compact and can easily be incorporated within the calling program. Degenerate eigensolutions cause no difficulty. An example FORTRAN routine is given.     

An iterative solution of the two-dimensional electromagnetic inverse scattering problem

A new method, based on an iterative procedure, for solving the two-dimensional inverse scattering problem is presented. This method employs an equivalent Neumann series solution in each iteration step. The purpose of the algorithm is to provide a general method to solve the two-dimensional imaging problem when the Born and the Rytov approximations break down. Numerical simulations were calculated for several cases where the conditions for the first order Born approximation were not satisfied. The results show that in both high and low frequency cases, good reconstructed profiles and smoothed versions of the original profiles can be obtained for smoothly varying permittivity profiles (lossless) and discontinuous profiles (lossless), respectively. A limited number of measurements around the object at a single frequency with four to eight plane incident waves from different directions are used. The method proposed in this article could easily be applied to the three-dimensional inverse scattering problem, if computational resources are available.     

The collinear one-bumper, two-body problem with unequal masses

We present a special model which is a caricature of the collinear three-body problem. Near triple-collision behavior for the model is governed by the collision manifold. Using the dynamics on the collision manifold, we study the allowed sequences of bounces by generating a partition on a Poincare slice in the flow. When the ratio of masses is small, we use two subshifts of finite type to describe which sequences of bounces are guaranteed and which are excluded. There is a third class of sequences which are neither guaranteed nor excluded. We also describe the bifurcations of this system as the ratio of masses approaches zero.     

An explicit eighth order method with minimal phase-lag for the numerical solution of the Schrödinger equation

A new explicit hybrid eighth algebraic order two-step method with phase-lag of order twelve is developed for computing eigenvalues and resonances of the one-dimensional Schrödinger equation. Based on this new method and on the method developed recently by Simos we obtain a new variable-step procedure for the numerical integration of the Schrödinger equation. Numerical results obtained for the integration of the resonance problem for the well known case of the Woods-Saxon potential and for the integration of the eigenvalue problem for the well known case of the Morse potential show that this new method is better than other variable-step methods.     

An asymptotic solution of the problem of nonlinear waves in a viscous liquid

A strict solution of the problem of temporal evolution of the shape of a periodic wave on the surface of a viscous infinitely deep liquid is obtained for the first time in an approximation quadratic in the wave amplitude.     

An iteration algorithm for the solution of the inverse boundary-value problem of heat conduction

An iterative procedure is constructed for solving the inverse boundary-value problem of heat conductivity in an extremal formulation on the basis of solving a Cauchy problem.