The superconvergence of the composite midpoint rule for the finite-part integral

AbstractThe composite midpoint rule is probably the simplest one among the Newton–Cotes rules for Riemann integral. However, this rule is divergent in general for Hadamard finite-part integral. In this paper, we turn this rule to a useful one and, apply it to evaluate Hadamard finite-part integral as well as to solve the relevant integral equation. The key point is based on the investigation of its pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate. By applying the midpoint rule to approximate the finite-part integral and by choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving the finite-part integral equation. More interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression, by which an optimal error estimate is established. Some numerical examples are provided to validate the theoretical analysis

On a novel numerical quadrature based on cycle index of symmetric group for the Hadamard finite-part integrals

To evaluate the Hadamard finite-part integrals accurately, a novel interpolatory-type quadrature is proposed in this article. In our approach, numerical divided difference is utilized to represent the high order derivatives of the integrated function, which make it possible to reduced the numerical quadrature into a concise formula based on the cycle index for symmetric group. In addition, convergence analysis is presented and the error estimation is given. Numerical results are presented on cases with different weight functions, which substantiate the performance of the proposed method

The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel cot((x-s)/2) is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis

The Modified Trapezoidal Rule for Computing Hypersingular Integral on Interval

The modified trapezoidal rule for the computation of hypersingular integrals in boundary element methods is discussed. When the special function of the error functional equals zero, the convergence rate is one order higher than the general case. A new quadrature rule is presented and the asymptotic expansion of error function is obtained. Based on the error expansion, not only do we obtain a high order of accuracy, but also a posteriori error estimate is conveniently derived. Some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms

Numerical methods to compute hypersingular integral in boundary element methods

超奇异积分的近似计算是边界元方法,特别是自然边界元理论中必须面对的难题之一.经典的数值方法,如gAuSS求积公式和nEWTOn-COTES积分公式等数值方法,都不能直接用于超奇异积分的近似计算.本文将介绍超奇异积分基于不同定义的gAuSS积分公式、S型变换公式、nEWTOn-COTES积分公式和外推法近似计算超奇异积分的思路,重点阐述nEWTOn-COTES积分公式和基于有限部分积分定义的外推法近似计算超奇异积分的主要结论.The computation of hypersingular integral is one of the important subjects in boundary element methods especially in natural boundary element methods.Classical numerical methods such as Gauss methods,Newton-Cotes methods cannot be used to approximate the hypersingular integral directly.In this paper, we introduce the numerical methods such as Gauss methods, Newton-Cotes methods, S transformation methods and extrapolation methods which are based on different definitions; then we mainly present the results of Newton-Cotes methods and extrapolation methods which are used to compute the hypersingular integral.国家自然科学基金(批准号:11471195;11101247;11201209和91330106); 中国博士后科学基金(批准号:2013M540541)资助项

Numerical method for hypersingular integrals of highly oscillatory functions on the positive semiaxis

This paper deals with a quadrature rule for the numerical evaluation of hypersingular integrals of highly oscillatory functions on the positive semiaxis. The rule is of product type and consists in approximating the density function f by a truncated interpolation process based on the zeros of generalized Laguerre polynomials and an additional point. We prove the stability and the convergence of the rule, giving error estimates for functions belonging to weighted Sobolev spaces equipped with uniform norm. We also show how the proposed rule can be used for the numerical solution of hypersingular integral equations. Numerical tests which confirm the theoretical estimates and comparisons with other existing quadrature rules are presented

Discrétisation Espace-Temps d'Équations d'Ondes Élasto-Acoustiques dans des Bases Trefftz-DG Polynomiales

Discontinuous Finite Element Methods (DG FEM) have proven flexibility and accuracy for solving wave problems in complex media. However, they require a large number of degrees of freedom, which increases the corresponding computational cost compared with that of continuous finite element methods. Among the different variational approaches to solve boundary value problems, there exists a particular family of methods, based on the use of trial functions in the form of exact local solutions of the governing equations. The idea was first proposed by Trefftz in 1926, and since then it has been further developed and generalized. A Trefftz-DG variational formulation applied to wave problems reduces to surface integrals that should contribute to decreasing the computational costs.Trefftz-type approaches have been widely used for time-harmonic problems, while their implementation for time-dependent simulations is still limited. The feature of Trefftz-DG methods applied to time-dependent problems is in the use of space-time meshes. Indeed, standard DG methods lead to the construction of a semi-discrete system of ordinary differential equations in time which are integrated by using an appropriate scheme. But Trefftz-DG methods applied to wave problems lead to a global matrix including time and space discretizations which is huge and sparse. This significantly hampers the deployment of this technology for solving industrial problems.In this work, we develop a Trefftz-DG framework for solving mechanical wave problems including elasto-acoustic equations. We prove that the corresponding formulations are well-posed and we address the issue of solving the global matrix by constructing an approximate inverse obtained from the decomposition of the global matrix into a block-diagonal one. The inversion is then justified under a CFL-type condition. This idea allows for reducing the computational costs but its accuracy is limited to small computational domains. According to the limitations of the method, we have investigated the potential of Tent Pitcher algorithms following the recent works of Gopalakrishnan et al. It consists in constructing a space-time mesh made of patches that can be solved independently under a causality constraint. We have obtained very promising numerical results illustrating the potential of Tent Pitcher in particular when coupled with a Trefftz-DG method involving only surface terms. In this way, the space-time mesh is composed of elements which are 3D objects at most. It is also worth noting that this framework naturally allows for local time-stepping which is a plus to increase the accuracy while decreasing the computational burden.Les méthodes d'éléments finis de type Galerkine discontinu (DG FEM) ont démontré précision et efficacité pour résoudre des problèmes d'ondes dans des milieux complexes. Cependant, elles nécessitent un très grand nombre de degrés de liberté, ce qui augmente leur coût de calcul en comparaison du coût des méthodes d'éléments finis continus. Parmi les différentes approches variationnelles pour résoudre les problèmes aux limites, se distingue une famille particulière, basée sur l'utilisation de fonctions tests qui sont des solutions locales exactes des équations à résoudre. L'idée vient de E.Trefftz en 1926 et a depuis été largement développée et généralisée. Les méthodes variationnelles de type Trefftz-DG appliquées aux problèmes d'ondes se réduisent à des intégrales de surface, ce qui devrait contribuer à réduire les coûts de calcul.Les approches de type Trefftz ont été largement développées pour les problèmes harmoniques, mais leur utilisation pour des simulations en domaine transitoire est encore limitée. Quand elles sont appliquées dans le domaine temporel, les méthodes de Trefftz utilisent des maillages qui recouvrent le domaine espace-temps. C'est une des paraticularités de ces méthodes. En effet, les méthodes DG standards conduisent à la construction d'un système semi-discret d'équations différentielles ordinaires en temps qu'on intègre avec un schéma en temps explicite. Mais les méthodes de Trefftz-DG appliquées aux problèmes d'ondes conduisent à résoudre une matrice globale, contenant la discrétisation en espace et en temps, qui est de grande taille et creuse. Cette particularité gêne considérablement le déploiement de cette technologie pour résoudre des problèmes industriels.Dans ce travail, nous développons un environnement Trefftz-DG pour résoudre des problèmes d'ondes mécaniques, y compris les équations couplées de l'élasto-acoustique. Nous prouvons que les formulations obtenues sont bien posées et nous considérons la difficulté d'inverser la matrice globale en construisant un inverse approché obtenu à partir de la décomposition de la matrice globale en une matrice diagonale par blocs. Cette idée permet de réduire les coûts de calcul mais sa précision est limitée à de petits domaines de calcul. Etant données les limitations de la méthode, nous nous sommes intéressés au potentiel du "Tent Pitcher", en suivant les travaux récents de Gopalakrishnan et al. Il s'agit de construire un maillage espace-temps composé de macro-éléments qui peuvent être traités indépendamment en faisant une hypothèse de causalité. Nous avons obtenu des résultats préliminaires très encourageants qui illustrent bien l'intérêt du Tent Pitcher, en particulier quand il est couplé à une méthode de Trefftz-DG formulée à partir d'intégrales de surface seulement. Dans ce cas, le maillage espace-temps est composé d'éléments qui sont au plus de dimension 3. Il est aussi important de noter que ce cadre se prête à l'utilisation de pas de temps locaux ce qui est un plus pour gagner en précision avec des coûts de calcul réduits

Numerical analysis of particle-laden flows with the finite element method

In this work we study the numerical simulation of particle-laden fluids, with a focus on Newtonian fluids and spherical, rigid particles. We are thus dealing with a multi-phase (more precisely, a multi-component) problem, with two phases: the fluid (continuous phase) and the the particles (disperse phase). Our general strategy consists in using the discrete element method (DEM) to model the particles and the finite element method (FEM) to discretize the Navier-Stokes equations, which model the continuous phase. The interaction model between both phases is (must be) based on a multiscale concept, since the smallest scales resolved of the continuous phase are considered much bigger than the particles. In other words, the resolution of the numerical model for the particles is finer than that used for the fluid. Consequently, whether implicit or explicit, there must be a filtering or averaging operation involved in the interaction between both phases, where the details of their motions smaller than the smallest resolution scale of the fluid are soothed out, since the latter is the coarsest of the two different resolutions considered. The spatial discretization of the continuous phase is performed with the FEM, using equal-order spaces of shape functions for the velocity and for the pressure. It is a well-known fact that this type of combination involves the violation of the Ladyzenskaja-Babuška-Brezzi (LBB) condition, resulting in an unstable numerical method. Moreover, the presence of the convective term in Eulerian description of the flow also leads to numerical instabilities. Both effects are treated with the sub-grid scale stabilization methods here. About the disperse phase, the trajectory of each particle is calculated based both on the fluid-interaction forces and on the contact forces between them and the surrounding rigid boundaries. The differential equation that describes the motion of particles in between successive collisions, given the mean (averaged) far field and for particles much smaller than the smallest scales of the flow (the Kolmogorov scale in turbulence) is the Maxey-Riley equation (MRE). This equation is the subject of chapter 2. The objective of this theoretical study is to establish quantitative (up to order-of-magnitude accuracy) limits to its range of validity and to the relative importance of its various terms. The method employed is dimensional analysis, which is systematically applied to derive the 'first effects' of a series of phenomena that are neglected in the derivation of the MRE. Chapter 3 is dedicated to the numerical resolution of the MRE. Here we present improvements to the method of van Hinsberg et al. (2011) for the calculation of the history term and analyse the method thoroughly. We include several tests to show the efficiency and utility of the proposed approach. The MRE is directly applicable to flows where the particle-based Reynolds number is Re << 1. But its relevance reaches further, as its structure is the basis for the majority of extensions that model the movement of suspended particles outside the range of validity of the MRE. Chapter 4 is markedly more applied than the two preceding ones. It treats various industrial flux types with particles where we employ several extensions of the MRE of the type mentioned above. In the first part of this chapter we review the most important of these extensions and study the process of derivative recovery, necessary to calculate several terms in the equation of motion. The tests examples considered include bubble trapping in 'T'-junction tubes, the simulation of drilling systems of the oil industry based on the bombardment of steel particles and fluidized beds. For the latter we use a discrete filtering-based coupling approach, that mirrors the continuous theory sketched above. This set of three chapters (2, 3, 4) is the core of the Thesis, which is completed with an introduction (chapter 1) and the conclusions (chapter 5).En este trabajo se estudia la simulación numérica de fluidos con partículas en suspensión, con énfasis en fluidos newtonianos y partículas esféricas y rígidas. El problema es, pues, multi-fásico (o, más precisamente, multi-componente) en donde dos son las fases: el fluido (fase continua) y las partículas (fase dispersa). La estrategia general consiste en la modelización de las partículas mediante el método de los elementos discretos (DEM) y el método de los elementos finitos (FEM) para la discretización de las ecuaciones de Navier-Stokes, que modelan la fase continua. El modelo de interacción entre fases se basa (debe basarse) en una concepción multiescala del sistema, puesto que las escalas más pequeñas resueltas para el fluido se consideran mucho mayores a las partículas. Dicho de otro modo, ya sea implícita o explícitamente, en la interacción interviene un proceso de filtrado o promediado en que se suavizan los detalles del movimiento más pequeños que la escala de resolución del fluido. Par la fase continua la discretización del dominio se realiza con el FEM, con espacios de funciones de forma de igual orden para la velocidad y para la presión. Como es bien sabido, ello conlleva la violación de la condición de Ladyzenskaja-Babuška-Brezzi (LBB), dando un método numérico inestable. Además, la presencia del término convectivo en la descripción euleriana del flujo también resulta en inestabilidad. Ambos son tratados con métodos de estabilización basada en la modelización de 'escalas sub-malla'. En cuanto a la fase dispersa, se calcula la trayectoria de cada una de las partículas en función de fuerzas de contacto con las demás partículas y las superficies sólidas que limitan el dominio de cálculo por un lado, y de las fuerzas de interacción con el fluido por otro. La ecuación que describe el movimiento entre colisiones para partículas menores que las escalas más pequeñas del flujo (escala de Kolmogorov en flujos turbulentos), dado el campo lejano (promediado) de velocidades es la de Maxey-Riley (MRE). Esta ecuación es el objeto de estudio del capítulo 2. El objetivo de este estudio teórico es establecer de forma cuantitativa (en orden de magnitud) su rango de validez y la importancia relativa de sus distintos términos. El método empleado es el análisis dimensional aplicado sistemáticamente al estudio de los 'primeros efectos' de distintos fenómenos físicos que se desprecian en el planteamiento de la ecuación. El capítulo 3 se centra en la resolución numérica de la MRE. En él se presenta una mejora y estudio sistemático del método de van Hinsberg et al. (2011) para el cálculo del término histórico de la ecuación. Se incluyen distintos tests para demostrar la eficiencia del método y su aplicabilidad práctica. La MRE es de directa aplicación en flujos en los que el número de Reynolds relativo a la partícula es Re << 1. Sin embargo, su relevancia va más allá, pues en su estructura se basan la mayoría de modelos para el movimiento de partículas en suspensión, fuera del rango de aplicación de la MRE. El capítulo 4 es de índole más aplicada que los dos anteriores, y trata diversos ejemplos industriales de flujos con partículas en los que se emplean extensiones de la MRE de este tipo. En la primera parte se revisan las extensiones más importantes y la recuperación de derivadas, proceso necesario para el cálculo de varios términos de la ecuación de movimiento de las partículas. Las aplicaciones prácticas tratadas incluyen el aprisionamiento de burbujas en juntas en 'T', la simulación de sistemas de perforación petrolífera basados en el bombardeo con partículas de acero y los lechos fluidificados. Para esta última, se usa una técnica de filtrado discreto inspirada en la teoría esbozada más arriba. Estos tres capítulos (2, 3, 4) se completan con la introducción (capítulo 1) y las conclusiones (capítulo 5)