A new boundary condition solved with B.I.E.M.

Among the classical operators of mathematical physics the Laplacian plays an important role due to the number of different situations that can be modelled by it. Because of this a great effort has been made by mathematicians as well as by engineers to master its properties till the point that nearly everything has been said about them from a qualitative viewpoint. Quantitative results have also been obtained through the use of the new numerical techniques sustained by the computer. Finite element methods and boundary techniques have been successfully applied to engineering problems as can be seen in the technical literature (for instance [ l ] , [2], [3] . Boundary techniques are especially advantageous in those cases in which the main interest is concentrated on what is happening at the boundary. This situation is very usual in potential problems due to the properties of harmonic functions. In this paper we intend to show how a boundary condition different from the classical, but physically sound, is introduced without any violence in the discretization frame of the Boundary Integral Equation Method. The idea will be developed in the context of heat conduction in axisymmetric problems but it is hoped that its extension to other situations is straightforward. After the presentation of the method several examples will show the capabilities of modelling a physical problem.

SERBA: a B.I.E. program with linear elements for 2-D elastostatics analysis

The great developments that have occurred during the last few years in the finite element method and its applications has kept hidden other options for computation. The boundary integral element method now appears as a valid alternative and, in certain cases, has significant advantages. This method deals only with the boundary of the domain, while the F.E.M. analyses the whole domain. This has the following advantages: the dimensions of the problem to be studied are reduced by one, consequently simplifying the system of equations and preparation of input data. It is also possible to analyse infinite domains without discretization errors. These simplifications have the drawbacks of having to solve a full and non-symmetric matrix and some difficulties are incurred in the imposition of boundary conditions when complicated variations of the function over the boundary are assumed. In this paper a practical treatment of these problems, in particular boundary conditions imposition, has been carried out using the computer program shown below. Program SERBA solves general elastostatics problems in 2-dimensional continua using the boundary integral equation method. The boundary of the domain is discretized by line or elements over which the functions are assumed to vary linearly. Data (stresses and/or displacements) are introduced in the local co-ordinate system (element co-ordinates). Resulting stresses are obtained in local co-ordinates and displacements in a general system. The program has been written in Fortran ASCII and implemented on a 1108 Univac Computer. For 100 elements the core requirements are about 40 Kwords. Also available is a Fortran IV version (3 segments)implemented on a 21 MX Hewlett-Packard computer,using 15 Kwords.

B.I.E.M. in solid mechanics

In solid mechanics the weak formulation produces an integral equation ready for a discretization and with less restrictive requiremets than the standard field equations. Fundamentally the weak formulation is a expresion of a green formula. An alternative is to choose another green formula materializing a reciprocity relationship between the basis unknowns and an auxiliary family of functions. The degree of smoothness requiered to practice the discretization is then translated to the auxiliar functions. The subsequent discretization (constant, linear etc.)produces a set of equations on the boundary of the domain. For linear 3-D problems the BIEM appears then as a powerful alternative to FEM, because of the reduction to 2-D thanks to the features previously described.

B.I.E.M. y estudio de singularidades en problemas de potencial

El objeto del presente artículo es el estudio de singularidades en problemas de Potencial mediante el uso del Método de las Ecuaciones Integrales sobre el contorno del dominio en estudio. Frente a soluciones basadas en la mejora de la discretización, análisis asintótico o introducción de funciones de forma que representen mejor la evolución de la función, una nueva hipótesis es presentada: el término responsable de la singularidad es incluido en la integral sobre el contorno de la función auxiliar. Los resultados obtenidos mejoran los de soluciones anteriores simplificando también el tiempo de cálculo = The subject of this paper is the modelling of singularities in potential problems, using the Boundary Integral Equation Method. As a logical alternative to classical methods (discretization refinement, asymptotic analysis, high order interpolatory functions) a new hypothesis is presented: the singularity responsible term is included in the interpolatory shape function. As shown by several exemples results are splendid and computer time radically shortened.