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.

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.

The effect of boundary conditions in the numerical solution of 3-D thermoelastic problems

After the extensive research on the capabilities of the Boundary Integral Equation Method produced during the past years the versatility of its applications has been well founded. Maybe the years to come will see the in-depth analysis of several conflictive points, for example, adaptive integration, solution of the system of equations, etc. This line is clear in academic research. In this paper we comment on the incidence of the manner of imposing the boundary conditions in 3-D coupled problems. Here the effects are particularly magnified: in the first place by the simple model used (constant elements) and secondly by the process of solution, i.e. first a potential problem is solved and then the results are used as data for an elasticity problem. The errors add to both processes and small disturbances, unimportant in separated problems, can produce serious errors in the final results. The specific problem we have chosen is especially interesting. Although more general cases (i.e. transient)can be treated, here the domain integrals can be converted into boundary ones and the influence of the manner in which boundary conditions are applied will reflect the whole importance of the problem.

P-adaptive boundary elements for three-dimensional potential problems

This paper deals with the boundary element method (BEM) p-convergence approach applied to three-dimensional problems governed by Laplace's equation. The advantages derived from the boundary discretization and hierarchical interpolation functions are collated in order to minimize human effort in preparation of input data and improve numerical results.

Reduced order models for industrial thermo-fluid problems

A new method is presented to generate reduced order models (ROMs) in Fluid Dynamics problems of industrial interest. The method is based on the expansion of the flow variables in a Proper Orthogonal Decomposition (POD) basis, calculated from a limited number of snapshots, which are obtained via Computational Fluid Dynamics (CFD). Then, the POD-mode amplitudes are calculated as minimizers of a properly defined overall residual of the equations and boundary conditions. The method includes various ingredients that are new in this field. The residual can be calculated using only a limited number of points in the flow field, which can be scattered either all over the whole computational domain or over a smaller projection window. The resulting ROM is both computationally efficient(reconstructed flow fields require, in cases that do not present shock waves, less than 1 % of the time needed to compute a full CFD solution) and flexible(the projection window can avoid regions of large localized CFD errors).Also, for problems related with aerodynamics, POD modes are obtained from a set of snapshots calculated by a CFD method based on the compressible Navier Stokes equations and a turbulence model (which further more includes some unphysical stabilizing terms that are included for purely numerical reasons), but projection onto the POD manifold is made using the inviscid Euler equations, which makes the method independent of the CFD scheme. In addition, shock waves are treated specifically in the POD description, to avoid the need of using a too large number of snapshots. Various definitions of the residual are also discussed, along with the number and distribution of snapshots, the number of retained modes, and the effect of CFD errors. The method is checked and discussed on several test problems that describe (i) heat transfer in the recirculation region downstream of a backwards facing step, (ii) the flow past a two-dimensional airfoil in both the subsonic and transonic regimes, and (iii) the flow past a three-dimensional horizontal tail plane. The method is both efficient and numerically robust in the sense that the computational effort is quite small compared to CFD and results are both reasonably accurate and largely insensitive to the definition of the residual, to CFD errors, and to the CFD method itself, which may contain artificial stabilizing terms. Thus, the method is amenable for practical engineering applications. Resumen Se presenta un nuevo método para generar modelos de orden reducido (ROMs) aplicado a problemas fluidodinámicos de interés industrial. El nuevo método se basa en la expansión de las variables fluidas en una base POD, calculada a partir de un cierto número de snapshots, los cuales se han obtenido gracias a simulaciones numéricas (CFD). A continuación, las amplitudes de los modos POD se calculan minimizando un residual global adecuadamente definido que combina las ecuaciones y las condiciones de contorno. El método incluye varios ingredientes que son nuevos en este campo de estudio. El residual puede calcularse utilizando únicamente un número limitado de puntos del campo fluido. Estos puntos puede encontrarse dispersos a lo largo del dominio computacional completo o sobre una ventana de proyección. El modelo ROM obtenido es tanto computacionalmente eficiente (en aquellos casos que no presentan ondas de choque reconstruir los campos fluidos requiere menos del 1% del tiempo necesario para calcular una solución CFD) como flexible (la ventana de proyección puede escogerse de forma que evite contener regiones con errores en la solución CFD localizados y grandes). Además, en problemas aerodinámicos, los modos POD se obtienen de un conjunto de snapshots calculados utilizando un código CFD basado en la versión compresible de las ecuaciones de Navier Stokes y un modelo de turbulencia (el cual puede incluir algunos términos estabilizadores sin sentido físico que se añaden por razones puramente numéricas), aunque la proyección en la variedad POD se hace utilizando las ecuaciones de Euler, lo que hace al método independiente del esquema utilizado en el código CFD. Además, las ondas de choque se tratan específicamente en la descripción POD para evitar la necesidad de utilizar un número demasiado grande de snapshots. Varias definiciones del residual se discuten, así como el número y distribución de los snapshots,el número de modos retenidos y el efecto de los errores debidos al CFD. El método se comprueba y discute para varios problemas de evaluación que describen (i) la transferencia de calor en la región de recirculación aguas abajo de un escalón, (ii) el flujo alrededor de un perfil bidimensional en regímenes subsónico y transónico y (iii) el flujo alrededor de un estabilizador horizontal tridimensional. El método es tanto eficiente como numéricamente robusto en el sentido de que el esfuerzo computacional es muy pequeño comparado con el requerido por el CFD y los resultados son razonablemente precisos y muy insensibles a la definición del residual, los errores debidos al CFD y al método CFD en sí mismo, el cual puede contener términos estabilizadores artificiales. Por lo tanto, el método puede utilizarse en aplicaciones prácticas de ingeniería.

Theoretical Analysis of the No-Slip Boundary Condition Enforcement in SPH Methods

The aim of the present work is to provide an in-depth analysis of the most representative mirroring techniques used in SPH to enforce boundary conditions (BC) along solid profiles. We specifically refer to dummy particles, ghost particles, and Takeda et al. [Prog. Theor. Phys. 92 (1994), 939] boundary integrals. The analysis has been carried out by studying the convergence of the first- and second-order differential operators as the smoothing length (that is, the characteristic length on which relies the SPH interpolation) decreases. These differential operators are of fundamental importance for the computation of the viscous drag and the viscous/diffusive terms in the momentum and energy equations. It has been proved that close to the boundaries some of the mirroring techniques leads to intrinsic inaccuracies in the convergence of the differential operators. A consistent formulation has been derived starting from Takeda et al. boundary integrals (see the above reference). This original formulation allows implementing no-slip boundary conditions consistently in many practical applications as viscous flows and diffusion problems.

A set of canonical problems in sloshing. Part 0: Experimental setup and data processing

In a series of attempts to research and document relevant sloshing type phenomena, a series of experiments have been conducted. The aim of this paper is to describe the setup and data processing of such experiments. A sloshing tank is subjected to angular motion. As a result pressure registers are obtained at several locations, together with the motion data, torque and a collection of image and video information. The experimental rig and the data acquisition systems are described. Useful information for experimental sloshing research practitioners is provided. This information is related to the liquids used in the experiments, the dying techniques, tank building processes, synchronization of acquisition systems, etc. A new procedure for reconstructing experimental data, that takes into account experimental uncertainties, is presented. This procedure is based on a least squares spline approximation of the data. Based on a deterministic approach to the first sloshing wave impact event in a sloshing experiment, an uncertainty analysis procedure of the associated first pressure peak value is described.

Tranversal motion and flow structure of fully nonlinear streaks in a laminar boundary layer

Typical streak computations present in the literature correspond to linear streaks or to small amplitude nonlinear streaks computed using DNS or nonlinear PSE. We use the Reduced Navier-Stokes (RNS) equations to compute the streamwise evolution of fully non-linear streaks with high amplitude in a laminar flat plate boundary layer. The RNS formulation provides Reynolds number independent solutions that are asymptotically exact in the limit $Re \gg 1$, it requires much less computational effort than DNS, and it does not have the consistency and convergence problems of the PSE. We present various streak computations to show that the flow configuration changes substantially when the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-streamwise plane) becomes more important and strongly distorts the streamwise velocity profiles, that end up being quite different from those of the linear case. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks and compare them with available experimental results.

Linear instability of orthogonal compressible leading-edge boundary layer flow

Instability analysis of compressible orthogonal swept leading-edge boundary layer flow was performed in the context of BiGlobal linear theory. 1, 2 An algorithm was developed exploiting the sparsity characteristics of the matrix discretizing the PDE-based eigenvalue problem. This allowed use of the MUMPS sparse linear algebra package 3 to obtain a direct solution of the linear systems associated with the Arnoldi iteration. The developed algorithm was then applied to efficiently analyze the effect of compressibility on the stability of the swept leading-edge boundary layer and obtain neutral curves of this flow as a function of the Mach number in the range 0 ≤ Ma ≤ 1. The present numerical results fully confirmed the asymptotic theory results of Theofilis et al. 4 Up to the maximum Mach number value studied, it was found that an increase of this parameter reduces the critical Reynolds number and the range of the unstable spanwise wavenumbers.

Single-Stage Grid-Connected Forward Microinverter with Boundary Mode Control

This paper presents a microinverter to be integrated into a solar module. The proposed solution combines a forward converter and a constant off-time boundary mode control, providing MPPT capability and unity power factor in a single-stage converter. The transformer structure of the power stage remains as in the classical DC-DC forward converter. Transformer primary windings are utilized for power transfer or demagnetization depending on the grid semi-cycle. Furthermore, bidirectional switches are used on the secondary side allowing direct connection of the inverter to the grid. Design considerations for the proposed solution are provided, regarding the inductance value, transformer turns ratio and frequency variation during a line semi-cycle. The decoupling of the twice the line frequency power pulsation is also discussed, as well as the maximum power point tracking (MPPT) capability. Simulation and experimental results for a 100W prototype are enclosed

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.