The internal structure of hydrogen-air difussion flames

The Internal Structure of Hydrogen-Air Diffusion Flames. Tho purpose of this paper is to study finite rate chemistry effects in diffusion controlled hydrogenair flames undor conditions appearing in some cases in a supersonic combustor. Since for large reaction rates the flame is close to chemical equilibrium, the reaction takes place in a very thin region, so thata "singular perturbation "treatment" of the problem seems appropriate. It has been shown previously that, within the inner or reaction zone, convection effects may be neglocted, the temperature is constant across the flame, and tho mass fraction distributions are given by ordinary differential equations, whore tho only independent variable involved is tho coordinate normal to the flame surface. Tho solution of the outer problom, which is a pure mixing problem with the additional condition that fuol and oxidizer do not coexist in any zone, provides t h e following information: tho flame position, rates of fuel consumption, temperature, concentrators of species, fluid velocity outside of tho flame, and the boundary conditions required to solve the "inner problem." The main contribution of this paper consists in the introduction of a fairly complicated chemical kinetic scheme representing hydrogen-oxygen reaction. The nonlinear equations expressing the conservation of chemical species are approximately integrated by means of an integral method. It has boen found that, in the case considered of a near-equilibrium diffusion flame, tho role played by the dissociation-recombination reactions is purely marginal, and that somo of the second order "shuffling" reactions are close to equilibrium. The method shown here may be applied to compute the distanco from the injector corresponding to a given separation from equilibrium, say ten to twenty percent. For the casos whore this length is a small fraction of the combustion zone length, the equilibrium treatment describes properly tho flame behavior.

An efficient ‘a priori’ model reduction for boundary element models

The Boundary Element Method (BEM) is a discretisation technique for solving partial differential equations, which offers, for certain problems, important advantages over domain techniques. Despite the high CPU time reduction that can be achieved, some 3D problems remain today untreatable because the extremely large number of degrees of freedom—dof—involved in the boundary description. Model reduction seems to be an appealing choice for both, accurate and efficient numerical simulations. However, in the BEM the reduction in the number of degrees of freedom does not imply a significant reduction in the CPU time, because in this technique the more important part of the computing time is spent in the construction of the discrete system of equations. In this way, a reduction also in the number of weighting functions, seems to be a key point to render efficient boundary element simulations.

The P-Adaptive BIEM Version in Elastostatics

This paper introduces the p-adaptive version of the boundary element method as a natural extension of the homonymous finite element approach. After a brief introduction to adaptive techniques through their finite element formulation in elastostatics, the concepts are cast into the boundary element environment. Thus, the p-adaptive version of boundary integral methods is shown to be a generalization of already well known ideas. In order to show the power of these numerical procedures, the results of two practical analysis using both methods are presented.

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.

Remarks on methods for the computation of boundary-element integrals by co-ordinate transformation

It is well known that the evaluation of the influence matrices in the boundary-element method requires the computation of singular integrals. Quadrature formulae exist which are especially tailored to the specific nature of the singularity, i.e. log(*- x0)9 Ijx- JC0), etc. Clearly the nodes and weights of these formulae vary with the location Xo of the singular point. A drawback of this approach is that a given problem usually includes different types of singularities, and therefore a general-purpose code would have to include many alternative formulae to cater for all possible cases. Recently, several authors1"3 have suggested a type independent alternative technique based on the combination of standard Gaussian rules with non-linear co-ordinate transformations. The transformation approach is particularly appealing in connection with the p.adaptive version, where the location of the collocation points varies at each step of the refinement process. The purpose of this paper is to analyse the technique in eference 3. We show that this technique is asymptotically correct as the number of Gauss points increases. However, the method possesses a 'hidden' source of error that is analysed and can easily be removed.

A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods

The numerical strategies employed in the evaluation of singular integrals existing in the Cauchy principal value (CPV) sense are, undoubtedly, one of the key aspects which remarkably affect the performance and accuracy of the boundary element method (BEM). Thus, a new procedure, based upon a bi-cubic co-ordinate transformation and oriented towards the numerical evaluation of both the CPV integrals and some others which contain different types of singularity is developed. Both the ideas and some details involved in the proposed formulae are presented, obtaining rather simple and-attractive expressions for the numerical quadrature which are also easily embodied into existing BEM codes. Some illustrative examples which assess the stability and accuracy of the new formulae are included.

P-Adaptive Boundary Elements

This paper presents the implementation of an adaptive philosophy to plane potential problems, using the direct boundary element method. After some considerations about the state of the art and a discussion of the standard approach features, the possibility of separately treating the modelling of variables and their interpolation through hierarchical shape functions is analysed. Then the proposed indicators and estimators are given, followed by a description of a small computer program written for an IBM PC. Finally, some examples show the kind of results to be expected.

Effective and neutral stresses in soils using boundary element methods

The evaluation of neutral pressures in soil mechanics problems is a fundamental step to evaluate deformations in soils. In this paper, we present some results obtained by using the boundary element method for plane problems, describing the undrained situation as well as the consolidation problem.

Improved Integration Methods for P-Adaptive Boundary Element Techniques

Non linear transformations are a good alternative for the numerical evaluation of singular and quasisingular integrals appearing in Boundary Element Method specially in the p-adaptive version. Some aspects of its numerical implementation in 2-D Potential codes is discussed and some examples are shown.

Implementation of Boundary Elements in Microcomputers

In this chapter we are going to develop some aspects of the implementation of the boundary element method (BEM)in microcomputers. At the moment the BEM is established as a powerful tool for problem-solving and several codes have been developed and maintained on an industrial basis for large computers. It is also well known that one of the more attractive features of the BEM is the reduction of the discretization to the boundary of the domain under study. As drawbacks, we found the non-bandedness of the final matrix, wich is a full asymmetric one, and the computational difficulties related to obtaining the integrals which appear in the influence coefficients. Te reduction in dimensionality is crucial from the point of view of microcomputers, and we believe that it can be used to obtain competitive results against other domain methods. We shall discuss two applications in this chapter. The first one is related to plane linear elastostatic situations, and the second refers to plane potential problems. In the first case we shall present the classical isoparametric BEM approach, using linear elements to represent both the geometry and the variables. The second case shows how to implement a p-adaptive procedure using the BEM. This latter case has not been studied until recently, and we think that the future of the BEM will be related to its development and to the judicious exploitation of the graphics capabilities of modern micros. Some examples will be included to demonstrate the kind of results that can be expected and sections of printouts will show useful details of implementation. In order to broaden their applicability, these printouts have been prepared in Basic, although no doubt other languages may be more appropiate for effective implementation.

Improvements for B.E.M. implementation in micros

This paper presents a computer program developed to run in a micro I.B.M.-P.C. wich incorporates some features in order to optimize the number of operations needed to compute the solution of plane potential problems governed by Laplace's equation by using the Boundary Integral Equation Method (B.I.E.M.). Also incorporated is a routine to plot isolines inside the domain under study.

Numerical treatment of thick shells with holes

A Boundary Integral Equation Method (B.I.E.M.)formulation is presented. After a general situation of the method among other usual numerical ones, the possibilities of discretization are developed. As this is done only in the boundary the treatment of tridimensional problems is greatly simplified in comparison with other methods. Some results on a simple shell with holes are finally presented.

Improved boundary elements in torsion problems

Since the epoch-making "memoir" of Saint-Venant in 1855 the torsion of prismatic and cilindrical bars has reduced to a mathematical problem: the calculation of an analytical function satisfying prescribed boundary values. For over one century, till the first applications of the F.E.M. to the problem, the only possibility of study in irregularly shaped domains was the beatiful, but limitated, theory of complex function analysis, several functional approaches and the finite difference method. Nevertheless in 1963 Jaswon published an interestingpaper which was nearly lost between the splendid F. E.M. boom. The method was extended by Rizzo to more complicated problems and definitively incorporated to the scientific community background through several lecture-notes of Cruse recently published, but widely circulated during past years. The work of several researches has shown the tremendous possibilities of the method which is today a recognized alternative to the well established F .E. procedure. In fact, the first comprehensive attempt to cover the method, has been recently published in textbook form. This paper is a contribution to the implementation of a difficulty which arises if the isoparametric elements concept is applicated to plane potential problems with sharp corners in the boundary domain. In previous works, these problems was avoided using two principal approximations: equating the fluxes round the corner or establishing a binode element (in fact, truncating the corner). The first approximation distortes heavily the solution in thecorner neighbourhood, and a great amount of element is neccesary to reduce its influence. The second is better suited but the price payed is increasing the size of the system of equations to be solved. In this paper an alternative formulation, consistent with the shape function chosen in the isoparametric representation, is presented. For ease of comprehension the formulation has been limited to the linear element. Nevertheless its extension to more refined elements is straight forward. Also a direct procedure for the assembling of the equations is presented in an attempt to reduce the in-core computer requirements.

Los métodos proyectivos en la mecánica de los medios contínuos

El Método de las Ecuaciones Integrales es una potente alternativa a los Métodos de Dominio tales como el Método de los Elementos Finitos. La idea ensencial es la combinación de la clásica relación de la reciprocidad con la filosofía de la discretización del F.E.M. La aplicación a algunos problemas reales ha demostrado que en ciertos casos el B.I.E.M. es preferiole al F.E.M. y ello es especialmente así cuando los problemas a tratar son tridimensionales y con geometría complicada. En esta ocasión se analizan comparativamente algunos aspectos matemáticos del procedimiento = Boundary integral equation method (B.I.E.M.)is a powerful alternative to the domain methods, as the well know Finite Element Method (F .E.M.) The esential idea, are the combination of the classical reciprocity re!ations with the discretization phylosophy of F.E.M. The reduction in dimension of the domain to be discretized, the easy treatment of infinite domains and the high accuracy of the results are the main adventages of B.I.E.M. Between the drawacks the nonsymetry and non sparseness of the matrices to be treated are worth remembering. Application to several real problems has shown that in certain cases B.I.E.M. is better than F.E.M. and this is specially true when tridimensional problems of complicated geometries have to be treated. Active research is in progress of its extensión to non linear and time dependent problems.

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.