Boundary Value Problems Of Potential Theory Research Articles

Uncertainties associated with integral-based solutions to geodetic boundary-value problems

Physical geodesy applies potential theory to study the Earth’s gravitational field in space outside and up to a few km inside the Earth’s mass. Among various tools offered by this theory, boundary-value problems are particularly popular for the transformation or continuation of gravitational field parameters across space. Traditional problems, formulated and solved as early as in the nineteenth century, have been gradually supplemented with new problems, as new observational methods and data are available. In most cases, the emphasis is on formulating a functional relationship involving two functions in 3-D space; the values of one function are searched but unobservable; the values of the other function are observable but with errors. Such mathematical models (observation equations) are referred to as deterministic. Since observed data burdened with observational errors are used for their solutions, the relevant stochastic models must be formulated to provide uncertainties of the estimated parameters against which their quality can be evaluated. This article discusses the boundary-value problems of potential theory formulated for gravitational data currently or in the foreseeable future used by physical geodesy. Their solutions in the form of integral formulas and integral equations are reviewed, practical estimators applicable to numerical solutions of the deterministic models are formulated, and their related stochastic models are introduced. Deterministic and stochastic models represent a complete solution to problems in physical geodesy providing estimates of unknown parameters and their error variances (mean squared errors). On the other hand, analyses of error covariances can reveal problems related to the observed data and/or the design of the mathematical models. Numerical experiments demonstrate the applicability of stochastic models in practice.

Downward continuation of gravitational field quantities to an irregular surface by spectral weighting

In geophysical and geodetic studies, gravity inversion is typically performed such that observed gravity values are first continued downward onto a regular (planar, spherical or spheroidal) surface by solving an inverse integral transform, which originates from a classical solution to the first boundary-value problem in potential theory. A typical example is continuing gravity observed at the topographic surface down to the mean sea level (geoid). Nowadays, gravity-dedicated satellite missions and aerial gravimetry provide gravity data above the topographic surface in addition to classical terrestrial gravity observations. For specific purposes (such as gravity data combination and validation, or quasigeoid determination), satellite and aerial gravity observations have to be continued to the irregular topographic surface. In this study, we address this issue by formulating a functional model for a spectral downward continuation of selected gravitational field quantities to an irregular topographic surface. Moreover, we generalize this functional model to allow for transformation between different types of gravitational field quantities. In particular, we derive spectral weights for estimation of the disturbing potential or disturbing/anomalous gravity at the Earth’s surface by combining the first-, second- and third-order radial gradients of the disturbing potential (disturbing gradients). The correctness of the developed combined spectral estimator is verified in a closed-loop test based on synthetic satellite disturbing gradients. The combined spectral estimator is applied to simulated satellite disturbing gradients polluted by a realistic Gaussian noise. Results of the numerical experiments show that the combined spectral estimator puts the highest importance on the least polluted disturbing gradient, while the contribution of the least accurate disturbing gradient is negligible. An important advantage of this spectral combination method is that no matrix inversion with numerical instabilities requiring regularization is needed.

Dynamics of Separation Points Upon Impact of a Floating Circular Cylinder

The two-dimensional problem of vertical separation impact of a circular cylinder under the free surface of a heavy liquid is considered. The problem is studied in a linearized formulation corresponding to low velocities of the body and liquid and taking into account the dynamics of the points of separation of the cavitation zone. A coupled nonlinear problem is formulated, which includes a mixed boundary-value problem of potential theory with one-sided constraints on the surface of the body and an equation defining the law of motion of the cylinder. Examples demonstrating the dynamics of separation points during forced or free cylinder motions are considered. Numerical results obtained using the proposed mathematical model are compared with the results of asymptotic analysis of the initial nonlinear problem for small times.

Geoid determination using one-step integration

A residual (high-frequency) gravimetric geoid is usually computed from geographically limited ground, sea and/or airborne gravimetric data. The mathematical model for its determination from ground gravity is based on the transformation of observed discrete values of gravity into gravity potential related to either the international ellipsoid or the geoid. The two reference surfaces are used depending on height information that accompanies ground gravity data: traditionally orthometric heights determined by geodetic levelling were used while GPS positioning nowadays allows for estimation of geodetic (ellipsoidal) heights. This transformation is usually performed in two steps: (1) observed values of gravity are downward continued to the ellipsoid or the geoid, and (2) gravity at the ellipsoid or the geoid is transformed into the corresponding potential. Each of these two steps represents the solution of one geodetic boundary-value problem of potential theory, namely the first and second or third problem. Thus two different geodetic boundary-value problems must be formulated and solved, which requires numerical evaluation of two surface integrals. In this contribution, a mathematical model in the form of a single Fredholm integral equation of the first kind is presented and numerically investigated. This model combines the solution of the first and second/third boundary-value problems and transforms ground gravity disturbances or anomalies into the harmonically downward continued disturbing potential at the ellipsoid or the geoid directly. Numerical tests show that the new approach offers an efficient and stable solution for the determination of the residual geoid from ground gravity data.

Homogeneous solutions of the dirichlet problem for an anisotropic layer

A complete system of homogeneous solutions of the Dirichlet problem for an anisotropic layer is constructed. These solutions represent series containing metaharmonic functions of a complex argument which depends on all three coordinates. The solution obtained can be used when considering boundary-value problems of potential theory for a piecewise-homogeneous layer.

Torsional rigidity of reinforced concrete bars with arbitrary sectional shape

In this paper, the torsional rigidity of arbitrarily shape bar made of different materials is studied on the basis of theory of elasticity and finite element approach. With additional boundary conditions for the common boundaries of different materials from the continuous conditions of deformation and traction across the interior boundary, the torsion function can be solved numerically from the second boundary-value problem of potential theory. The traction jump boundary conditions across the interior surfaces are enforced in the alternate finite element approach. Several examples are shown to check the computational approach proposed, and the approach, at last, is applied to calculate the torsional rigidity of reinforced concrete bar and some multiply connected cross sections such as tower leg section of the Tsing Ma Bridge and other engineering structures.

Wavelet approximations on closed surfaces and their application to boundary-value problems of potential theory

Wavelets on closed surfaces in Euclidean space R3 are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of functional values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at infinity. Furthermore, an exact fully discrete wavelet approximation is developed in case of band-limited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem.

The method of non-linear time transformation in boundary-value problems of potential theory with moving boundaries for the non-linear wave equation

A new method for solving boundary-value problems for the wave equation [1–3] with moving boundaries is used to obtain a solution of a boundary-value problem with boundary conditions of three types [4].

A potential problem arising from the strip-punch problem in elasticity

Three-dimensional contact problems in the classical theory of linear elasticity can often be regarded as mixed boundary-value problems of potential theory. In this paper we examinethe problem where contact between the indenting object (called a punch) and the elastic medium is maintained over an infinite strip. It is assumed that a rigid frictionless punch with a known profile has indented a homogeneous,isotropic and linearly elastic half-space. Applying the theory of Mathieu functions, an analytic solution of Laplace's equation is obtained through separation of variables in the elliptic cylinder coordinate system. Finally three examples are discussed where in each case the normal component of stress under the punch is numerically evaluated.

Complete model of the process of propagation of long waves and their interaction with a vertical wall

Asymptotic analysis of the nonlinear three-dimensional boundary-value problem of potential theory is carried out and a complete system of equations describing the process of propagation of long surface waves is obtained. Approximate solutions of the problems for both traveling and standing waves are constructed to the third approximation.

INDENTATION OF AN ELASTIC HALF-SPACE BY A COOLED FLAT PUNCH

A solution is given for the frictionless thermoelastic contact of a cooled flat cylindrical rigid punch and an elastic half-space. If the thermal distortion is sufficiently small, contact is retained over the entire punch face and the solution can be written down from known results. For larger temperature differences the conventional boundary conditions do not permit a solution, but a solution is here developed in which a central circle is in imperfect contact, i.e. a non-zero contact resistance exists between the surfaces. The problem is reduced to a boundary-value problem in potential theory using a solution of the thermoelastic equations in two harmonic functions. These functions are written as the sum of a number of components each of which satisfies two-part boundary conditions. The resulting boundary conditions lead to coupled Abel integral equations which are reduced to a single Fredholm equation by substitution. Results are given for typical contact pressure distributions and for the relationship between load, temperature difference, heat flux and the radius of the imperfect contact region.

Love's integral and other relations between solutions to mixed boundary-value problems in potential theory

AbstractIn a paper published in 1949, E. R. Love [10] found an integral relation between a separated solution (in oblate spheroidal coordinates) to a particular mixed boundary-value problem and a solution to the same problem using an integral representation. This note examines further representations of the same type which occur in some simple two and three-dimensional potential problems.

Methods for numerical conformal mapping

Nonlinear integral equations for the boundary functions which determine conformal transformations in two dimensions are developed and analyzed. One of these equations has a nonsingular logarithmic kernel and is especially well suited for numerical computations of conformal maps including those which deal with regions having highly distorted boundaries. Numerical procedures based on interspersed Gaussian quadrature for approximating the integrals and a Newton-Raphson technique to solve the resulting nonlinear algebraic equations are described. The Newton-Raphson iteration converges reliably with very crude initial approximations. Numerical examples are given for the mapping of a half-infinite region with periodic boundary onto a half plane, with up to nine-figure accuracy for values of the map function on the boundary and for its first derivatives. The examples include regions bounded by “spike” curves characteristic of Rayleigh-Taylor instability phenomena. A differential equation is derived which relates changes in the map function to changes of the boundary. This is relevant to potential problems for regions with time-dependent boundaries. Further nonsingular integral formulas are derived for conformal mapping in a variety of geometries and for application to the boundary-value problems of potential theory.

Illustrative solution of a mixed problem of steady-state heat-conduction theory for a half-plane with a boundary condition of the third kind

A mixed boundary-value problem of potential theory is treated for a half-plane in the case in which two equal parts of the boundary are at constant but different temperatures, while there is heat transfer in accordance with Newton's law over the rest of the surface.

Existence of solutions of plane traction problems for inextensible transversely isotropic elastic solids

AbstractA plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic solid with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and mixed displacement-traction boundary conditions yield standard boundary-value problems of potential theory for which uniqueness and existence of solutions are well established. However, the important case of prescribed tractions at each boundary point gives a non-standard potential problem involving linking of boundary values at opposite ends of chords parallel to the axis of material symmetry. Uniqueness and existence of solutions, within arbitrary rigid motions, are now established for the traction problem for general domains.

Numerical results for a mixed boundary-value problem of potential theory using invariant imbedding

A mixed boundary-value problem of potential theory is solved numerically. The problem is reduced to the solution of an integral equation using the theory of dual integral equations. The invariant imbedding treatment of that integral equation leads to the solution of an initial-value problem in ordinary differential equations. Numerical results are presented and discussed.

Note on the reduction of dual and triple series equations to dual and triple integral equations

Dual integral equations involving Bessel functions occur in the solution of some boundary-value problems in potential theory with conditions prescribed on a circular disc and a considerable amount of attention has been given to the solution of such equations (cf. (1)). The method of solving these dual integral equations is very similar to that employed in the solution of certain dual series equations involving Legendre functions. Equations of this type occur in problems in potential theory with conditions prescribed on a spherical cap and their solution has been obtained by Collins (2). No definite mathematical connexion has, however, been established between these dual series and dual integral equations and the object of this note is to establish such a connexion.