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.

Computation of nonlinear streaky structures in boundary layer flow

En esta tesis se integran numéricamente las ecuaciones reducidas de Navier Stokes (RNS), que describen el flujo en una capa límite tridimensional que presenta también una escala característica espacial corta en el sentido transversal. La formulación RNS se usa para el cálculo de “streaks” no lineales de amplitud finita, y los resultados conseguidos coinciden con los existentes en la literatura, obtenidos típicamente utilizando simulación numérica directa (DNS) o nonlinear parabolized stability equations (PSE). El cálculo de los “streaks” integrando las RNS es mucho menos costoso que usando DNS, y no presenta los problemas de estabilidad que aparecen en la formulación PSE cuando la amplitud del “streak” deja de ser pequeña. El código de integración RNS se utiliza también para el cálculo de los “streaks” que aparecen de manera natural en el borde de ataque de una placa plana en ausencia de perturbaciones en la corriente uniforme exterior. Los resultados existentes hasta ahora calculaban estos “streaks” únicamente en el límite lineal (amplitud pequeña), y en esta tesis se lleva a cabo el cálculo de los mismos en el régimen completamente no lineal (amplitud finita). En la segunda parte de la tesis se generaliza el código RNS para incluir la posibilidad de tener una placa no plana, con curvatura en el sentido transversal que varía lentamente en el sentido de la corriente. Esto se consigue aplicando un cambio de coordenadas, que transforma el dominio físico en uno rectangular. La formulación RNS se integra también expresada en las correspondientes coordenadas curvilíneas. Este código generalizado RNS se utiliza finalmente para estudiar el flujo de capa límite sobre una placa con surcos que varían lentamente en el sentido de la corriente, y es usado para simular el flujo sobre surcos que crecen en tal sentido. Abstract In this thesis, the reduced Navier Stokes (RNS) equations are numerically integrated. This formulation describes the flow in a three-dimensional boundary layer that also presents a short characteristic space scale in the spanwise direction. RNS equations are used to calculate nonlinear finite amplitude “streaks”, and the results agree with those reported in the literature, typically obtained using direct numerical simulation (DNS) or nonlinear parabolized stability equations (PSE). “Streaks” simulations through the RNS integration are much cheaper than using DNS, and avoid stability problems that appear in the PSE when the amplitude of the “streak” is not small. The RNS integration code is also used to calculate the “streaks” that naturally emerge at the leading edge of a flat plate boundary layer in the absence of any free stream perturbations. Up to now, the existing results for these “streaks” have been only calculated in the linear limit (small amplitude), and in this thesis their calculation is carried out in the fully nonlinear regime (finite amplitude). In the second part of the thesis, the RNS code is generalized to include the possibility of having a non-flat plate, curved in the spanwise direction and slowly varying in the streamwise direction. This is achieved by applying a change of coordinates, which transforms the physical domain into a rectangular one. The RNS formulation expressed in the corresponding curvilinear coordinates is also numerically integrated. This generalized RNS code is finally used to study the boundary layer flow over a plate with grooves which vary slowly in the streamwise direction; and this code is used to simulate the flow over grooves that grow in the streamwise direction.

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.

Transient heat conduction problems using B.I.E.M

A method, using boundary elements, is presented as a solution to plane transient heat conduction. The proposed method considers the governing equation to be a Helmholtz's equation and solves the problem of time variation using step by step integration. A numerical procedure is developed and its effectiveness verified. Several examples are provided and their results compared with the theoretical ones.

Boundary element approach to the dynamic stiffness functions of circular foundations

A boundary element approach for time harmonic axisymmetric problems using the complete space point load fundamental solution is presented. The fundamental solution is integrated numerically along the azimuthal co-ordinate of each axisymmetric element. To increase the accuracy of the numerical integration a simple co-ordinate transformation is proposed. The approach is applied to the computation of the dynamic stiffness functions of rigid circular foundations on layered viscoelastic soils. Three different sites are considered: a uniform half-space, a soil layer on a half-space, and a soil consisting of four horizontal layers and a compliant half-space. The numerical results obtained by the proposed approach for surface circular foundations are very close to corresponding published results obtained by different procedures.

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.

Resonant behaviour of the barrier of YBa2Cu3O7 grain boundary Josephson junctions fabricated on bicrystalline substrates with different geometries

We have analyzed a resonant behavior in the dielectric constant associated to the barrier of YBa2Cu3O7 (YBCO) grain boundary Josephson junctions (GBJJs) fabricated on a wide variety of bicrystalline substrates: 12° [0 0 1] tilt asymmetric, 24° [0 0 1] tilt asymmetric, 24° [0 0 1] tilt symmetric, 24° [1 0 0] tilt asymmetric, 45° [1 0 0] tilt asymmetric and 24° [0 0 1] tilt symmetric +45° [1 0 0] tilt asymmetric bicrystals. The resonance analysis allows us to estimate a more appropriate value of the relative dielectric constant, and so a more adequate value for the length L of the normal N region assuming a SNINS model for the barrier. In this work, the L dependence on the critical current density Jc has been investigated. This analysis makes possible a single representation for all the substrate geometries independently on around which axes the rotation is produced to generate the grain boundary. On the other hand, no clear evidences exist on the origin of the resonance. The resonance frequency is in the order of 1011 Hz, pointing to a phonon dynamic influence on the resonance mechanism. Besides, its position is affected by the oxygen content of the barrier: a shift at low frequencies is observed when the misorientation angle increases.

Combining a Similar Coefficient Identification Algorithm with the Boundary Element Method

The boundary element method (BEM) has been applied successfully to many engineering problems during the last decades. Compared with domain type methods like the finite element method (FEM) or the finite difference method (FDM) the BEM can handle problems where the medium extends to infinity much easier than domain type methods as there is no need to develop special boundary conditions (quiet or absorbing boundaries) or infinite elements at the boundaries introduced to limit the domain studied. The determination of the dynamic stiffness of arbitrarily shaped footings is just one of these fields where the BEM has been the method of choice, especially in the 1980s. With the continuous development of computer technology and the available hardware equipment the size of the problems under study grew and, as the flop count for solving the resulting linear system of equations grows with the third power of the number of equations, there was a need for the development of iterative methods with better performance. In [1] the GMRES algorithm was presented which is now widely used for implementations of the collocation BEM. While the FEM results in sparsely populated coefficient matrices, the BEM leads, in general, to fully or densely populated ones, depending on the number of subregions, posing a serious memory problem even for todays computers. If the geometry of the problem permits the surface of the domain to be meshed with equally shaped elements a lot of the resulting coefficients will be calculated and stored repeatedly. The present paper shows how these unnecessary operations can be avoided reducing the calculation time as well as the storage requirement. To this end a similar coefficient identification algorithm (SCIA), has been developed and implemented in a program written in Fortran 90. The vertical dynamic stiffness of a single pile in layered soil has been chosen to test the performance of the implementation. The results obtained with the 3-d model may be compared with those obtained with an axisymmetric formulation which are considered to be the reference values as the mesh quality is much better. The entire 3D model comprises more than 35000 dofs being a soil region with 21168 dofs the biggest single region. Note that the memory necessary to store all coefficients of this single region is about 6.8 GB, an amount which is usually not available with personal computers. In the problem under study the interface zone between the two adjacent soil regions as well as the surface of the top layer may be meshed with equally sized elements. In this case the application of the SCIA leads to an important reduction in memory requirements. The maximum memory used during the calculation has been reduced to 1.2 GB. The application of the SCIA thus permits problems to be solved on personal computers which otherwise would require much more powerful hardware.

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.

Indicators and Estimators in P-Adaptive Boundary Elements

The paper presents the possibility of implementing a p-adaptive process with the B.E.M. Although the exemples show that good results can be obtained with a limited amount of storage and with the simple ideas explained above, more research is needed in order to improve the two main problems of the method, i.e.: the criteria of where to refine and until what degree. Mathematically based reasoning is still lacking and will be useful to simplify the decission making. Nevertheless the method seems promising and, we hope, opens a path for a series of research lines of maximum interest. Although the paper has dealt only with plane potential problem the extension to plane elasticity as well as to 3-D potential problem is straight-forward.

Consolidation problems

The analysis of deformation in soils is of paramount importance in geotechnical engineering. For a long time the complex behaviour of natural deposits defied the ingenuity of engineers. The time has come that, with the aid of computers, numerical methods will allow the solution of every problem if the material law can be specified with a certain accuracy. Boundary Techniques (B.E.) have recently exploded in a splendid flowering of methods and applications that compare advantegeously with other well-established procedures like the finite element method (F.E.). Its application to soil mechanics problems (Brebbia 1981) has started and will grow in the future. This paper tries to present a simple formulation to a classical problem. In fact, there is already a large amount of application of B.E. to diffusion problems (Rizzo et al, Shaw, Chang et al, Combescure et al, Wrobel et al, Roures et al, Onishi et al) and very recently the first specific application to consolidation problems has been published by Bnishi et al. Here we develop an alternative formulation to that presented in the last reference. Fundamentally the idea is to introduce a finite difference discretization in the time domain in order to use the fundamental solution of a Helmholtz type equation governing the neutral pressure distribution. Although this procedure seems to have been unappreciated in the previous technical literature it is nevertheless effective and straightforward to implement. Indeed for the special problem in study it is perfectly suited, because a step by step interaction between the elastic and flow problems is needed. It allows also the introduction of non-linear elastic properties and time dependent conditions very easily as will be shown and compares well with performances of other approaches.

On the Use of a 3-D Fundamental Solution for Axisymmetric Steady-State Dynamic Problems

The use of 3-D fundamental solution is some axisymmetric problems is straightforward. The resulting algorithms seem to work better than the usual ones (at least for static solutions) and for dynamic cases, than those presented in the previous paragraph. The robustness of the method allows the computations for very high and very low frequencies without any noticeable difficulty.

Hierarchical boundary elements

In this paper we show the possibility of applying adaptive procedures as an alternative to the well-known philosophy of standard Boundary Elements. The three characteristic steps of adaptive procedures, i.e. hierarchical shape functions families, indicator criteria, and a posteriori estimation, can be defined in order to govern an automatic refinement and stopping of the solution process. A computer program to treat potential problems, called QUEIMADA, has been developed to show the capabilities of the new idea.

Some minor problems with B.E.M.

As it is well known B.E.M. works efficiently in the treatment of a bread class of potential and elasticity problems. In this paper we present the results of several runs established with linear elements in plane potential theory and treating the importance of singularities and the pattern and size of elements used in the boundary discretization.