A numerical experimental study of inverse preconditioning for the parallel iterative solution to 3D finite element flow equations

Integration of the subsurface flow equation by finite elements (FE) in space and finite differences (FD) in time requires the repeated solution to sparse symmetric positive definite systems of linear equations. Iterative techniques based on preconditioned conjugate gradients (PCG) are one of the most attractive tool to solve the problem on sequential computers. A present challenge is to make PCG attractive in a parallel computing environment as well. To this aim a key factor is the development of an efficient parallel preconditioner. FSAI (factorized sparse approximate inverse) and enlarged FSAI relying on the approximate inverse of the coefficient matrix appears to be a most promising parallel preconditioner. In the present paper PCG using FSAI, diagonal and pARMS (parallel algebraic recursive multilevel solvers) preconditioners is implemented on the IBM SP4/512 and CLX/768 supercomputers with up to 32 processors to solve underground flow problems of a large size. The results show that FSAI may allow for a parallel relative efficiency larger than 50% on the largest problems with p=32 processors. Moreover, FSAI turns out to be significantly less expensive and more robust than pARMS. Finally, it is shown that for p in the upper range may be much improved if PCG–FSAI is implemented on CLX.     

Erratum to “A study on preconditioning iterative methods and relativity of pressure in the numerical calculation of fluid flow” by Hiroyuki Hirano,Hideto Yoshida,Hiroyuki Ozoe,and Naotaka Okamoto,Numerical Methods for Partial Differential Equations (2004) 20(5) 789–802

The original article to which this Erratum refers was published in Numerical Methods for Partial Differential Equations (2004) 20(5) 789–802     

A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions

Let , , be a dimensional slab. Denote points by , where and . Denoting the boundary of the slab by , let

where is an ordered sequence of intervals on the right half line (that is, b_{n}$">). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let . Let and denote respectively the cone of bounded, positive harmonic functions in and the cone of positive harmonic functions in which satisfy the Dirichlet boundary condition on and the Neumann boundary condition on .

Letting , the main result of this paper, under a modest assumption on the sequence , may be summarized as follows when :

1. If , then and are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if with 2$">.

2. If and , then and is one-dimensional. In particular, this occurs if .

3. If , then and the set of minimal elements generating is isomorphic to (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if with .

When , as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for is as above.

     

Spectral theory and iterative methods for the Maxwell system in nonsmooth domains

We study spectral properties of boundary integral operators which naturally arise in the study of the Maxwell system of equations in a Lipschitz domain Ω ? ?³. By employing Rellich‐type identities we show that the spectrum of the magnetic dipole boundary integral operator (composed with an appropriate projection) acting on L²(?Ω) lies in the exterior of a hyperbola whose shape depends only on the Lipschitz constant of Ω. These spectral theory results are then used to construct generalized Neumann series solutions for boundary value problems associated with the Maxwell system and to study their rates of convergence (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Dirichlet,Neumann and Mixed Boundary Value Problems of the Theory of Elasticity in n-Dimensional Domains with Boundaries Containing Closed Cuspidal Edges

The Dirichlet, Neumann and mixed problems of statics of the theory of elasticity for anisotropic homogeneous media are studied in n-dimensional (n ≧ 2) domains with boundaries containing closed cuspidal edges. Theorems of the existence and uniqueness of solutions of these problems in the Besov and Bessel potential spaces are obtained. Smoothness of solutions in a closed domain occupied by elastic medium is investigated.     

Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.     

Burghelea-Friedlander-Kappeler's gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion

The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under an assumption that the product structure is given near the boundary. As applications of this result, we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.

     

Comment on “On the Neumann function and the method of images in spherical and ellipsoidal geometry”

In this note, we point out two errors in the article “On the Neumann function and the method of images in spherical and ellipsoidal geometry” by Dassios and Sten. Two corrections are then proposed.     

On homogenization of solutions of the poisson equation in a perforated domain with different types of boundary conditions on different cavities

In this paper we consider boundary value problems in perforated domains with periodic structures and cavities of different scales, with the Neumann condition on some of them and mixed boundary conditions on others. We take a case when cavities with mixed boundary conditions have so called critical size (see [1]) and cavities with the Neumann conditions have the scale of the cell. In the same way other cases can be studied, when we have the Neumann and the Dirichlet boundary conditions or the Dirichlet condition and the mixed boundary condition on the boundary of cavities.There is a large literature where homogenization problems in perforated domains were studied [2];-[7];     

A numerical solution of the dirichlet problem on some special doubly connected regions

The aim of this paper is to give a convergence proof of a numerical method for the Dirichlet problem on doubly connected plane regions using the method of reflection across the exterior boundary curve (which is analytic) combined with integral equations extended over the interior boundary curve (which may be irregular with infinitely many angular points).     

Effects of uncertainties in the domain on the solution of Neumann boundary value problems in two spatial dimensions

An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.

     

A comparative study of the numerical scales and the prioritization methods in AHP

In the analytic hierarchy process (AHP), a decision maker first gives linguistic pairwise comparisons, then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them, and finally derives a priority vector from the numerical pairwise comparisons. In particular, the validity of this decision-making tool relies on the choice of numerical scale and the design of prioritization method. By introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices (LPCMs), and by defining the deviation measures of LPCMs, we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods. Using these performance measure algorithms, we compare the most common numerical scales (the Saaty scale, the geometrical scale, the Ma–Zheng scale and the Salo–Hämäläinen scale) and the prioritization methods (the eigenvalue method and the logarithmic least squares method). In addition, we also discuss the parameter of the geometrical scale, develop a new prioritization method, and construct an optimization model to select the appropriate numerical scales for the AHP decision makers. The findings in this paper can help the AHP decision makers select suitable numerical scales and prioritization methods.     

A numerical study of 2D integrated RBFNs incorporating Cartesian grids for solving 2D elliptic differential problems

This article reports a numerical discretization scheme, based on two‐dimensional integrated radial‐basis‐function networks (2D‐IRBFNs) and rectangular grids, for solving second‐order elliptic partial differential equations defined on 2D nonrectangular domains. Unlike finite‐difference and 1D‐IRBFN Cartesian‐grid techniques, the present discretization method is based on an approximation scheme that allows the field variable and its derivatives to be evaluated anywhere within the domain and on the boundaries, regardless of the shape of the problem domain. We discuss the following two particular strengths, which the proposed Cartesian‐grid‐based procedure possesses, namely (i) the implementation of Neumann boundary conditions on irregular boundaries and (ii) the use of high‐order integration schemes to evaluate flux integrals arising from a control‐volume discretization on irregular domains. A new preconditioning scheme is suggested to improve the 2D‐IRBFN matrix condition number. Good accuracy and high‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On the Neumann function and the method of images in spherical and ellipsoidal geometry

The invention of an image system for a boundary value problem adds to a significant understanding of the structure of the problem, both at the mathematical and at the physical level. In this paper, the interior and exterior Neumann functions for the Laplacian in the cases of spherical and ellipsoidal domains are represented in terms of images. Besides isolated images, the presence of the normal derivative in the Neumann condition demands an additional continuous distribution of images, which in the spherical cases, can be restricted to a one‐dimensional manifold, whereas for the ellipsoid, both a one‐dimensional and a two‐dimensional distribution of images is needed. Copyright © 2012 John Wiley & Sons, Ltd.     

A comparison between iterative methods by using the basins of attraction

There exists a real competition between authors to construct improved iterative methods for solving nonlinear equations. In this paper, by using computer experiment, we study the basins of attraction for some of the iterative methods for solving the equation P(z) = 0, where P:C→C is a complex coefficients polynomial, and this allows us to compare their performances (the area of convergence and theirs speed). The beauty fractal pictures generated by these methods are presented too.     

A numerical method for the viscous incompressible Oseen flow in shape reconstruction

This paper is concerned with the shape reconstruction of a bounded domain with a viscous incompressible fluid driven by the Oseen equations. For the approximate solution of the ill-posed and nonlinear problem we propose a regularized Gauss-Newton method. A theoretical foundation for the method is given by establishing the differentiability of the boundary value problem with respect to the boundary in the sense of the domain derivative. The results of several numerical experiments show that our theory is useful for practical purpose, and the proposed algorithm is feasible.     

A numerical study of the dynamics of elastic articulated systems in a fluid

We propose a method and an algorithm for computing the dynamics of elastic structures of articulated form in a fluid flow taking account of the weakening in certain structural elements. In describing the motion we use two sets of radius-vectors, which are approximated in the computations by parametric local splines of first degree. The possibilities of the proposed method are illustrated using the example of the study of the dynamics of transition processes in an articulated anchor-buoy structure, which arise when there is an abrupt change in the direction of the fluid flow velocity. We determine the kinematic and force characteristics of the structure under various changes in the direction of the flow velocity. We determine the structural elements in which the weakening occurs. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 128–134.     

Short shots and industrial case studies: Understanding fluid flow and solidification in high pressure die casting

The geometric complexity and high fluid speeds involved in high pressure die casting (HPDC) combine to give strongly three dimensional fluid flow with significant free surface fragmentation and splashing. A simulation method that has proved particularly suited to modelling HPDC is Smoothed Particle Hydrodynamics (SPH). Materials are approximated by particles that are free to move around rather than by fixed grids, enabling more accurate prediction of fluid flows involving complex free surface motion. Three practical industrial case studies of SPH simulated HPDC flows are presented; aluminium casting of a differential cover (automotive), an electronic housing and zinc casting of a door lock plate. These show significant detail in the fragmented fluid free surfaces and allow us to understand the predisposition to create defects such as porosity in the castings. The validation of flow predictions coupled with heat transfer and solidification is an important area for such modelling. One powerful approach is to use short shots, where insufficient metal is used in the casting or the casting shot is halted part way through, to leave the die cavity only partially filled. The frozen partial castings capture significant detail about the order of fill and the flow structures occurring during different stages of filling. Validation can occur by matching experimental and simulated short shots. Here we explore the effect of die temperature, metal super-heat and volume fill on the short shots for the casting of a simple coaster. The bulk features of the final solid castings are found to be in good agreement with the predictions, but the fine details appear to depend on surface behaviour of the solidifying metals. This potentially has significant implications for modelling HPDC.     

Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations

Summary We discuss block matrices of the formA=[A _ij ], whereA _ij is ak×k symmetric matrix,A _ij is positive definite andA _ij is negative semidefinite. These matrices are natural block-generalizations of Z-matrices and M-matrices. Matrices of this type arise in the numerical solution of Euler equations in fluid flow computations. We discuss properties of these matrices, in particular we prove convergence of block iterative methods for linear systems with such system matrices.