Practical applications of an integral equation method for the solution of Laplace's equation

An integral equation method for the solution of Laplace's equation, originally proposed for boundary value problems in a single medium, is here extended to problems involving multiple media. The extended method has been used to compute the internal thermal resistance of electric cables and some numerical results are presented.     

Kupradze's functional equation for the torsion problem of prismatic bars — part 2

The integral equation of part 1 obtained from Kupradze's functional equation exhibits singularities at the corners. In this part we investigate the singularities arising from the procedure, as well as singularities appearing in re-entrant corners, by using the known asymptotic character of the displacement field near the corner. Some examples showing the convergence of the numerical solution of the integral equation are given.     

The use of approximate functions for the numerical solution of Laplace' equation by an integral equation

In this paper an integral equation method will be outlined to solve Laplace' equation numerically in a finite area S. The method uses either a function which is an approximation of the unknown potential of a particular solution which is only a good approximation in a part of S. The method is also valid if the approximate function is not a solution of Laplace' equation.     

Kupradze's functional equation for the torsion problem of prismatic bars — Part 1

St. Venant's torsion problem for a prismatic bar is reduced to an integral equation of the Fredholm type with the help of Kupradze's method: i.e. for a given load distribution on the lateral surface of the cylinder (“actual body $\text{K}{}^{\mathrm{+}}$”) imbedded in an elastic medium (“basic body $\text{K}{}^0$”), a distribution of screw dislocations has to be determined on the lateral surface in such a manner that the external region of the infinite medium (“complementary body $\text{K}{}^{\mathrm{?}}$”) will be free of tension. The unknown function is the tangential derivative of the warping function. Part 1 demonstrates the method for a smooth boundary. In part 2 the influence of the corners on the convergence of the numerical solution of the integral equation is investigated. Prismatic bars with multiconnected cross sections as well as cross sections with corners can be treated with this method.     

Singularity and positioning constants in Poisson's equation with a point source of unit intensity

The singularity constants and positioning constants are described in the potential field governed by Poisson's equation with a point source of unit intensity. It is then demonstrated that the singularity constants play an important role in the integral equation formulations utilizing rather complicated fundamental solutions.     

Computation methods for hydrodynamic problems (Reynold's equation)

This paper deals with the application of numerical methods to finding approximate solutions for a two-dimensional Reynolds' equation.Pressure height parameters ($\text{P}{}^1$) are evaluated at a series of discrete points on a rectangular mesh in the circumferential direction. Using these parameters the Reynold's equation is transformed into its equivalent finite difference form.In this form the equation gives the value of the pressure height parameter $\text{P}{}^1$ in terms of the values at neighbouring points and the corresponding influence coefficients. The finite difference equation is solved using an interactive method.Jacobi, Gauss-Seidel and successive over-relaxation methods are discussed. It is observed that by using an optimum over-relaxation factor, much computer time can be saved. Finally a convergence criterion is discussed which decides the iteration process.     

Discrete hill's equations

In the work, based on the formulas of discrete mechanics in a rotating frame, a discretization of classical Hill's equations of the moon motion is worked out. It is proved that the proposed discretization conserves Jacobi's constant of motion. For computational purposes an algorithm to the solution of obtained discrete Hill's equations is given.     

An imbedded initialization of Newton's algorithm for matrix Riccati equation

An effective numerical computation of the steady-state Riccati matrix is based on the successive solutions of a Lyapunov equation using Newton's method. The requirements of this algorithm are an initial stabilizing matrix and the numerical solution of the associated Lyapunov equation. Computationally, the first requirement is the more influencing factor in solving the Riccati equation with reasonable accuracy and speed. In this paper an initial matrix, based on the parameter imbedded solution of the Riccati equation, is introduced for the Newton's algorithm. The imbedding Newton algorithm has been applied to a variety of system, both stable and unstable as well as high-dimensional, A matrices, one of which is reported here. The proposed modification has improved the required CPU time of previous initialization schemes by as much as a factor of 6 times for the same order of accuracy.     

Numerical solution of Laplace's equation as an integral equation of the first kind

A Fourier approximation method is developed for the simple layer potential reformulation of Laplace's equation. The efficacy of the method is demonstrated in computational examples, and also analyzed theoretically.     

Application of the galerkin method to the solution of burgers' equation

The Galerkin method, with an expansion of B-splines is used to solve Burgers' equation from an initial state involving a discontinuity in the space domain.In an initial series of calculations a fixed spatial distribution of knots is used to define the B-splines, and the effect upon accuracy of changes in the order of the spline and the number of terms in the expansion is examined.In a second series of calculations the spatial distribution of knots is varied as the solution advances through time so that an area of fine knot resolution is maintained in the region where the solution of Burgers' equation changes most rapidly. Results of high accuracy are obtained using this method, which avoids the necessity of having a fine resolution knot distribution over the whole spatial domain in order to achieve an accurate solution.     

An integral equation method for inversion of electromagnetic soundings

The analogy of the problem of electromagnetic induction sounding of the earth's conductivity-depth profile using a two perpendicular-loop system to the general remote sensing problem of the atmosphere is pointed out. The induction sounding problem has been reduced to the solution of a new integral equation of Fredholm type. Kernels of the integral equation are evaluated, and their characteristics are investigated. A nonlinear relaxation inversion technique is suggested for the solution. Convergence of the solution, information content of measurements, and the range of possible retrieval of the conductivity profile are discussed.     

Hoare's logic and Peano's arithmetic

We develop the proof theory of Hoare's logic for the partial correctness of while- programs applied to arithmetic as it is defined by Peano's axioms. By representing the strongest postcondition calculus in Peano arithmetic PA, we are able to show that Hoare's logic over PA is equivalent to PA itself.     

Triangular finite elements for Poisson's equation with nodal derivatives

Explicit values are given for the element stiffness matrices of two triangular finite elements for Poisson's equation in the plane. One with four nodal points including derivatives as nodal values at the vertices and another without a central point. These provide O(h⁶) andO(h⁴) accuracy in the energy, respectively.     

Substitution of semi-AFL's

S.A. Greibach proved in “Chains of full AFL's” a very useful property concerning substitution of full semi-AFL's.S. Ginsburg raises the question whether the property remains true when the families under considerations are supposed only to be semi-AFL's.We give here a negative answer to this question.     

Saint venant's principle in sandwich strip

The Saint Venant's principle is generally accepted to be valid in most problems in structural mechanics. But some earlier studies have drawn attention to the fact that routine application of Saint Venant's principle in the solution of elasticity problems of sandwich type structures is not justified in general. The aim of the investigation reported herein is to study the applicability of Saint Venant's principle to the problem of a sandwich strip quantitatively in detail. A simple finite element model, with linear strain triangles representing the core and bar elements representing the thin faces, is used for the analysis. Four different load cases are studied, and representative numerical results are presented. In all cases it is observed that significant edge disturbances do exist at relatively large distances from the load point, compared to homogeneous type of construction. Strength of Materials solutions are valid only beyond a distance of “4h” from the loaded end in the case of very soft cores, the distance being less for more rigid cores. Also very high values of face to core modulus ratio do not influence the solution significantly.     

The prisoner's dilemma tournament

A tournament style computer competition eminently suitable for students is described here. It is based on the celebrated “Prisoner's Dilemma” paradox and is an interesting mix of logical analysis and knowledge of computer programming. Such a tournament has been run at the author's institution.     

An alternative to Liapunov's stability method and its application to higher-order systems

This paper is an extension of [1] and should be read in conjunction with that reference. It proves that an alternative to Liapunov's stability method, which removes the obstacle of having to establish a Liapunov function, can be generalized to cover also those cases in which higher-order differential equation systems are involved.     

Making computational sense of Montague's intensional logic

Montague's difficult notation and complex model theory have tended to obscure potential insights for the computer scientist studying Natural Language. Despite his strict insistence on an abstract model-theoretic interpretation for his formalism, we feel that Montague's work can be related to procedural semantics in a fairly direct way. A simplified version of Montague's formalism is presented, and its key concepts are explicated in terms of computational analogues. Several examples are presented within Montague's formalism but with a view toward developing a procedural interpretation. We provide a natural translation from intensional logic into lisp. This allows one to express the composition of meaning in much the way Montague does, using subtle patterns of functional application to distribute the meanings of individual words throughout a sentence. The paper discusses some of the insights this research has yielded on knowledge representation and suggests some new ways of looking at intensionality, context, and expectation.     

Two theorems about the completeness of Hoare's logic

We prove two theorems about the completeness of Hoare's logic for the partial correctness of while-programs over an axiomatic specification. The first result is a completion theorem: any specification (Σ,E) can be refined to a specification (Σ₀, E₀), conservative over (Σ, E), whose Hoare's logic is complete. The second result is a normal form theorem: any complete specification (Σ, E) possessing some complete logic for partial correctness can be refined to an effective specification (Σ₀, E₀) conservative over (Σ, E), which generates all true partial correctness formulae with Hoare's standard rules.     

Quadratic convergence of crisfield's method

Crisfield's method is a geometrically appealing variant of Newton's method used to track equilibrium curves in nonlinear structural analysis. A rigorous proof of quadratic convergence in one dimension is given, and numerical results are given for a complicated 21-dimensional lamella dome problem.