Statistical Methods of Quantitative Assessment of Material Damageability, Their Numerical Implementation and Convergence Analysis

A GPU-based line-by-line method for thermal radiation transfer of H2O, CO2, and H2O/CO2 mixture

A new method is presented in this paper for modeling the 3-D terrain effect that, in turn, can be approximately removed by applying a terrain correction to resistivity surveys. First, the 3-D electrical boundary value problem is transformed into an integral equation problem by use of Green’s theorem. Then the boundary element method is used to solve the integral equation. The ground surface and the boundary of the anomalous body are divided into triangular elements. Linear variation of quantities is assumed within each element and a Gaussian quadrature formula is used to calculate the integral. In this way, the integral equation is converted into a set of linear equations. The potential field value on the ground surface is obtained by solving the linear equation system with Gaussian elimination. This method uses a special triangular element mesh for division of the ground surface which is moved simultaneously with the movement of the source. Therefore, the number of computational nodes is greatly reduced. To account for the influence of 3-D terrain on the apparent resistivity, only 68 nodes are required in typical situations. Since this method does not require a large computational effort, it can be conveniently run on a microcomputer. Calculation examples for two models show that the results obtained by this method compare well with an analytic solution and results from other numerical methods. An example of the 3-D terrain correction in a resistivity survey is also given; the example shows that applying the terrain correction can greatly attenuate the terrain effect.

Introduction. Mathematical modeling of hydrodynamic processes in shallow reservoirs of complex geometry in the presence of coastal engineering systems requires an integrated approach in the development of algorithms for constructing computational grids and methods for solving grid equations. The work is devoted to the description of algorithms that allow to reduce the time for solving SLAE by using an algorithm for processing overlapping geometry segments and organizing parallel pipeline calculations. The aim of the work is to compare the acceleration of parallel algorithms for the methods of Seidel, Jacobi, modified alternately triangular method and the method of solving grid equations with tridiagonal preconditioner depending on the number of computational nodes.Materials and Methods. The numerical implementation of the modified alternating-triangular iterative method for solving grid equations (MATM) of high dimension is based on parallel algorithms based on a conveyor computing process. The decomposition of the computational domain for the organization of the pipeline calculation process has been performed. A graph model is introduced that allows to fix the connections between neighboring fragments of the computational grid. To describe the complex geometry of a reservoir, including coastal structures, an algorithm for overlapping geometry segments is proposed.Results. It was found that the efficiency of implementing one step of the MATM on the GPU depends only on the number of threads along the Oz axis, and the step execution time is inversely proportional to the number of nodes of the computational grid along the Oz axis. Therefore, it is recommended to decompose the computational domain into parallelepipeds in such a way that the size along the Oz axis is maximum, and the size along the Ox axis is minimal. Thanks to the algorithm for combining geometry segments, it was possible to speed up the calculation by 14–27 %.Discussion and Conclusions. An algorithm has been developed and numerically implemented for solving a system of large-dimensional grid equations arising during the discretization of the shallow water bodies’ hydrodynamics problem by MATM, adapted for heterogeneous computing systems. The graph model of a parallel-pipeline computing process is proposed. The connection of water body’s geometry segments allowed to reduce the number of computational operations and increase the speed of calculations. The efficiency of parallel algorithms for the methods of Seidel, Jacobi, modified alternately triangular method and the method of solving grid equations for problems of hydrodynamics in flat areas, depending on the number of computational nodes, is compared.

The occurring mechanism of the ill-conditioned system due to degenerate scale in the direct boundary element method (BEM) and the indirect BEM is analytically examined by using degenerate kernels. Five regularization techniques to ensure the unique solution, namely hypersingular formulation, method of adding a rigid body mode, rank promotion by adding the boundary flux equilibrium (direct BEM), CHEEF method and the Fichera’s method (indirect BEM), are analytically studied and numerically implemented. In this paper, we examine the sufficient and necessary condition of boundary integral formulation for the uniqueness solution of 2D Laplace problem subject to the Dirichlet boundary condition. Both analytical study and BEM implementation are addressed. For the analytical study, we employ the degenerate kernel in the polar and elliptic coordinates to derive the unique solution by using five regularization techniques for any size of circle and ellipse, respectively. Full rank of the influence matrix in the BEM using Fichera’s method for both ordinary scale and degenerate scale is also analytically proved. In numerical implementation, the BEM programme developed by NTOU/MSV group is employed to see the validity of the above formulation. Finally, the circular and elliptic cases are numerically demonstrated by using five regularization techniques. Besides, a general shape of a regular triangle is numerically implemented to check the uniqueness solution of BEM.

1R1. Boundary Element Programming in Mechanics. - Xiao-Wei Gao (Dept of Mech and Aerospace Eng, Arizona State Univ, Tempe AZ) and TG Davies (Glasgow Univ, UK). Cambridge UP, Cambridge, UK. 2002. 254 pp. (CD-Rom included). ISBN 0-521-77359-8. $69.95. Reviewed by DE Beskos (Dept of Civil Eng, Univ of Patras, Patras, GR-26500, Greece).This is a really excellent textbook as well as a reference book on the numerical implementation and computer programing of the direct boundary element method as applied to two and three-dimensional problems of linear elasticity and nonlinear elastoplasticity. The book is aimed at both graduate students and researchers as well as practicing engineers in mechanical, aeronautical, and civil engineering fields. The book consists of 254 pages plus a CD-Rom with the computer programs described. The whole book can be divided in two main components: The first one dealing with linear problems (elasticity) and the second one with nonlinear problems (time independent elastostoplasticity). In both parts, the chapter breakdown is the same and consists of the theory (of elasticity or plasticity), the corresponding boundary integral formulation of the problem, the numerical implementation, the detailed (subroutine by subroutine) description of the computer program and a number of applications—numerical examples to illustrate the code and demonstrate its accuracy. The book is completed by an introduction and an epilogue, eight appendices, a list of references, and a subject index. The figures and tables are of very good quality. The list of references is comprehensive, but selective, and the subject index is informative, but somewhat short. The unusual features of this book, in order of importance, are the following: the book is very clearly written and the English language is not just correct and easy to understand, but lively and enjoyable. The authors have proved they are not just very good on technical matters, but they also know to handle the English language superbly. The various computational aspects of the boundary element method, such as singular integration, treatment of edges and corners, computation of boundary stresses, solution of systems of linear equations, return mapping algorithms in plasticity, etc, are all treated in detail. In particular, the authors in every case, first discuss the problem, mention the work of others, and then offer their solution which they consider to be the most effective. The computer programs are explained in detail on a subroutine-by-subroutine basis and serve to illustrate the implementation of the method in the best possible way. In addition, they can be modified by the user, if he wishes to add or replace things. There is an emphasis on three-dimensional problems both in elasticity and plasticity, which cannot be found in other books on the subject. The book succeeds completely relative to the author’s stated aims and the subject matter. As a matter of fact, the book can be the ideal vehicle to teach the boundary element method to engineers who want the theory to go hand-in-hand with the numerical implementation and are not so much interested in mathematical details. Thus, the book can be used either as an ideal introductory text on boundary elements, in general, or as a specialized book on boundary element methods in plasticity. There are many introductory books available on boundary elements, but most of them deal with potential theory and elasticity, and they do not emphasize either the numerical implementation of the method nor 3D problems. This reviewer predicts that this book will prove to be, for boundary element programing, what has been the case with the two texts by Hinton and Owen on finite elements. The only slightly negative about this book has to do with the title, which employs the general and misleading term mechanics instead of the more appropriate term solid mechanics, which would more exactly reflect the contents of the book. Boundary Element Programming in Mechanics is highly recommended for purchase by both individuals and libraries.

A coupled finite element and boundary element method is developed to predict the magnetic vector and scalar potential distributions in the droplets levitated in an alternating magnetic or electrostatic field. The computational algorithm entails the application of boundary elements in the region of free space and finite elements in the droplet region, the two being coupled along the droplet–air interface. The coupled boundary and finite element scheme is further integrated with a WRM-based algorithm to predict the free surface deformation of magnetically and electrostatically levitated droplets. Several corner treatments for the boundary and finite element coupling and their implications to free surface calculations are discussed. Detailed formulation and numerical implementation are given. Numerical results are compared with available analytical solutions whenever available. A selection of computed results is presented for mag- netically or electrostatically levitated droplets under both terrestrial and microgravity conditions. Copyright © 1999 John Wiley & Sons, Ltd.

A coupled finite element and boundary element method is developed to predict the magnetic vector and scalar potential distributions in the droplets levitated in an alternating magnetic or electrostatic field. The computational algorithm entails the application of boundary elements in the region of free space and finite elements in the droplet region, the two being coupled along the droplet–air interface. The coupled boundary and finite element scheme is further integrated with a WRM-based algorithm to predict the free surface deformation of magnetically and electrostatically levitated droplets. Several corner treatments for the boundary and finite element coupling and their implications to free surface calculations are discussed. Detailed formulation and numerical implementation are given. Numerical results are compared with available analytical solutions whenever available. A selection of computed results is presented for mag- netically or electrostatically levitated droplets under both terrestrial and microgravity conditions. Copyright © 1999 John Wiley & Sons, Ltd.

Recently, high-performance mobile computer devices such as smart phones and tablet devices spread rapidly. They have attracted attention as a new promising platform for parallel and distributed applications. Based on the background, we are developing a cluster computer system using mobile devices or single board computers running Android OS. However, since mobile devices can move anywhere, node computers might leave from the cluster and new nodes might join the cluster. In this paper, we present an Android Cluster system that can reconfigure the system's scale dynamically. Our system can automatically detect the change in the number of computation nodes and reconfigure the cluster's nodes, even while parallel and distributed application is running. Furthermore, we show preliminary performance results of our system. The results show that our cluster provides the scalable performance to the number of nodes in parallel computation. Finally, it is confirmed that the mechanism of load balancing per process basis and the mechanism of switching to efficient data communication method can reduce the execution time of parallel applications. Our evaluation result shows that the execution time can be reduced up to 11.8% by load balancing per process basis, as compared to the load balancing per node basis, and shows that the execution time can be reduced 68% at maximum, by switching the communication method between processes to efficient one.

In recent years, high-performance mobile devices such as smart phones and tablet devices spread rapidly. They have attracted attention as a new platform for parallel and distributed applications. Based on this background, we are developing a cluster computer system using wireless-connected mobile devices running Android OS. However, since mobile devices can move anywhere, node computers might leave from the cluster, new devices might join the cluster. In this paper, we present an Android cluster system that can reconfigure its system's scale statically and dynamically. The system can automatically detect the change in the number of computation nodes and reconfigure the cluster's nodes, even while parallel and distributed application is running. Furthermore, we show preliminary performance results of our system. It is shown that our cluster provides the scalable performance to the number of nodes in parallel computation. Also, we have confirmed that the runtime overhead caused by checkpointing varies highly depending upon the interval of checkpointing.

Study on vertical line source Green's function for hydrodynamic calculations of ocean structures in water with ice cover

3R4. Underlying Principles of the Boundary Element Method. - D Cartwright (Col of Eng, Bucknell Univ PA). WIT Press, Southampton, UK. 2001. 276 pp. ISBN 1-85312-839-2. $149.00. Reviewed by DE Beskos (Dept of Civil Eng, Univ of Patras, Patras, GR-26500, Greece).This is a very well written introductory textbook on the foundations of the direct Boundary Element Method (BEM). It is very useful to both teachers and their undergraduate students in applied mathematics and engineering, as well as those interested in learning the basics of the method. The emphasis is on the principles and the mathematical derivations of the BEM and not on its numerical implementation. In that sense, the book is unique since most of the existing books emphasize the numerical implementation of the method. Detailed mathematical derivations are provided and solved problems are presented in detail in each chapter to help the student understand the subject matter of the book. Applications are described for one-, two- and three-dimensional problems of potential theory and elastostatics in a unified manner. Only constant elements are considered here for which the computation of singular integrals can be done analytically (in closed form). There are two aspects of the book this reviewer considers very important and worth mentioning: i) The concepts of the Green’s function and the fundamental solution are both discussed in detail. It is further shown how one can obtain the latter as a combination of Green’s functions defined for different boundary conditions. In many other books these two concepts are used one for the other and this creates confusion. ii) The boundary integral equation for internal points is derived here through the method of weighted residuals, which is a very powerful and general method for formulating boundary element and finite element methods. The whole book consists of seven chapters, one appendix, a bibliography, and a subject index. More specifically, Chapter 1 deals with a discussion on the derivation of the basic field equations (governing equations) of the scalar or vector type and of the second or fourth order in one or more dimensions. Chapter 2 discusses Green’s functions, their properties, derivation, and use in solving boundary value problems. The concept of the fundamental solution, its relation to Green’s functions, and its derivation are taken up in Chapter 3. The method of weighted residuals is described in Chapter 4, and its use in the derivation of the direct boundary integral equation for one-dimensional, potential, and elastostatic problems are described in Chapters 5, 6, and 7, respectively. In those last three chapters, related problems are solved to illustrate the method and demonstrate its advantages. Finally, the last chapter contains various appendices dealing with details of various mathematical derivations, which were presented in Chapters 3 and 5–7. The book concludes with a list of 20 reference books on the BEM and a short subject index. The author has certainly succeeded in fulfilling his stated aim of providing an introductory textbook emphasizing the underlying principles of the BEM. Underlying Principles of the Boundary Element Method should be purchased by teachers, undergraduate and graduate students, researchers who would like to start working in the field, and certainly by libraries.

We consider a boundary element (BE) Algorithm for solving linear diffusion desorption problems with localized nonlinear reactions. The proposed BE algorithm provides an elegant representation of the effect of localized nonlinear reactions, which enables the effects of arbitrarily oriented defect structures to be incorporated into BE models without having to perform severe mesh deformations.We propose a one‐step recursion procedure to advance the BE solution of linear diffusion localized nonlinear reaction problems and investigate its convergence properties. The separation of the linear and nonlinear effects by the boundary integral formulation enables us to consider the convergence properties of approximations to the linear terms and nonlinear terms of the boundary integral equation separately.For the linear terms we investigate how the degree of piecewise polynomial collocation in space and the size of the spatial mesh relative to the time step affects the accumulation of errors in the one‐step recursion scheme. We develop a novel convergence analysis that combines asymptotic methods with Lax's Equivalence Theorem. We identify a dimensionless meshing parameter θ whose magnitudé governs the performance of the one‐step BE schemes. In particular, we show that piecewise constant (PWC) and piecewise linear (PWL) BE schemes are conditionally convergent, have lower asymptotic bounds placed on the size of time steps, and which display excess numerical diffusion when small time steps are used. There is no asymptotic bound on how large the tie steps can be–this allows the solution to be advanced in fewer, larger time steps. The piecewise quadratic (PWQ) BE scheme is shown to be unconditionally convergent; there is no asymptotic restriction on the relative sizes of the time and spatial meshing and no numerical diffusion. We verify the theoretical convergence properties in numerical examples. This analysis provides useful information about the appropriate degree of spatial piecewise polynomial and the meshing strategy for a given problem.For the nonlinear terms we investigate the convergence of an explicit algorithm to advance the solution at an active site forward in time by means of Caratheodory iteration combined with piecewise linear interpolation. We consider a model problem comprising a singular nonlinear Volterra equation that represents the effect of the term in the BE formulation that is due to a single defect. We prove the convergence of the piecewise linear Caratheodory iteration algorithm to a solution of the model problem for as long as such a solution can be shown to exist. This analysis provides a theoretical justification for the use of piecewise linear Caratheodory iterates for advancing the effects of localized reactions.

An efficient 2D non‐linear numerical wave tank called LONGTANK has been developed based on a multi‐subdomain (MSD) approach combined with the conventional boundary element method (BEM). The multi‐subdomain approach aims at optimized matrix diagonalization, thus minimizing the computing time and reserved storage. The CPU per time step in LONGTANK simulation is found to increase only linearly with the number of surface nodes, which makes LONGTANK highly efficient especially when simulating long‐time wave evolutions in space.Appropriate treatment of special points on the boundary ensures high resolution in LONGTANK simulation beyond initial deformation and breaking, which allows detailed study of breaking criterion, breaker morphology, breaking dissipation, vorticity generation, etc.Detailed numerical implementation has been given with demonstration of LONGTANK simulations.

A massive volume of biological sequence data is available in over 36 different databases worldwide, including the sequence data generated by the Human Genome project. These databases, which also contain biological and bibliographical information, are growing at an exponential rate. Consequently, the computational demands needed to explore and analyze the data contained in these databases is quickly becoming a great concern. To meet these demands, we must use high performance computing systems, such as parallel computers and distributed networks of workstations. We present two parallel computational methods for analyzing these biological sequences. The first method is used to retrieve sequences that are homologous to a query sequence. The biological information associated with the homologous sequences found in the database may provide important clues to the structure and function of the query sequence. The second method, which helps in the prediction of the function, structure, and evolutionary history of biological sequences, is used to align a number of homologous sequences with each other. These two parallel computational methods were implemented and evaluated on an Intel IPSC/860 parallel computer. The resulting performance demonstrates that parallel computational methods can significantly reduce the computational time needed to analyze the sequences contained in large databases.

A Monte Carlo solution method for linear elasticity