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This book describes the mathematical background and reviews the techniques for solving problems, including those that require large computations such as transonic flows for compressible fluids and the Navier-Stokes equations for incompressible viscous fluids.

This book provides an introduction to the fundamental theory, practical implementation, and core and emerging applications of the material point method (MPM) and its variants. The MPM combines the advantages of both finite element analysis (FEM) and meshless/meshfree methods (MMs) by representing the material by a set of particles overlaid on a background mesh that serves as a computational scratchpad.The book shows how MPM allows a robust, accurate, and efficient simulation of a wide variety of material behaviors without requiring overly complex implementations. MPM and its variants have been shown to be successful in simulating a large number of high deformation and complicated engineering problems such as densification of foam, sea ice dynamics, landslides, and energetic device explosions, to name a few, and have recently found applications in the movie industry. It is hoped that this comprehensive exposition on MPM variants and their applications will not only provide anopportunity to re-examine previous contributions, but also to re-organize them in a coherent fashion and in anticipation of new advances.Sample algorithms for the solutions of benchmark problems are provided online so that researchers and graduate students can modify these algorithms and develop their own solution algorithms for specific problems. The goal of this book is to provide students and researchers with a theoretical and practical knowledge of the material point method to analyze engineering problems, and it may help initiate and promote further in-depth studies on the subjects discussed.

In the second part of the book numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear, are constructed.

This book is about computational methods based on operator splitting.

This book is primarily intended as a research monograph that could also be used in graduate courses for the design of parallel algorithms in matrix computations.It assumes general but not extensive knowledge of numerical linear algebra, parallel architectures, and parallel programming paradigms. The book consists of four parts: (I) Basics; (II) Dense and Special Matrix Computations; (III) Sparse Matrix Computations; and (IV) Matrix functions and characteristics. Part I deals with parallel programming paradigms and fundamental kernels, including reordering schemes for sparse matrices. Part II is devoted to dense matrix computations such as parallel algorithms for solving linear systems, linear least squares, the symmetric algebraic eigenvalue problem, and the singular-value decomposition. It also deals with the development of parallel algorithms for special linear systems such as banded ,Vandermonde ,Toeplitz ,and block Toeplitz systems. Part III addresses sparsematrix computations: (a) the development of parallel iterative linear system solvers with emphasis on scalable preconditioners, (b) parallel schemes for obtaining a few of the extreme eigenpairs or those contained in a given interval in the spectrum of a standard or generalized symmetric eigenvalue problem, and (c) parallel methods for computing a few of the extreme singular triplets. Part IV focuses on the development of parallel algorithms for matrix functions and special characteristics such as the matrix pseudospectrum and the determinant. The book also reviews the theoretical and practical background necessary when designing these algorithms and includes an extensive bibliography that will be useful to researchers and students alike. The book brings together many existing algorithms for the fundamental matrix computations that have a proven track record of efficient implementation in terms of data locality and data transfer on state-of-the-art systems, as well as several algorithms that are presented for the first time, focusing on the opportunities for parallelism and algorithm robustness.

The process of breaking up a physical domain into smaller sub-domains, known as meshing, facilitates the numerical solution of partial differential equations used to simulate physical systems.

Spectral Methods in Chemistry and Physics

One of the main purposes of a monograph, in my opinion, is to fit together in a comprehensive way scattered contributions that provide fragmented information to the readers.

The study of optimal shape design can be arrived at by asking the following question: "What is the best shape for a physical system?" Each of these fields is reviewed briefly: PDEs (Chapter 1), optimization (Chapter 4), optimal control (Chapter 5), and numerical methods (Chapters 1 and 4).

The purpose and organisation of this book are described in the preface to the first edition (1988). I would like to take this opportunity to thank the many readers for their usually generous comments about the first edition and particularly those readers who went to the trouble of drawing specific errors to my attention.

The second half of the book covers the treatment of a variety of steady flow problems, including effects of both viscosity and compressibi lity, by the Method of Integral Relations, Telenin's Method, and the Method of Lines.

Contains unique recent developments of various finite elements such as nonconforming, mixed, discontinuous, characteristic, and adaptive finite elements, along with their applications.

The LES-method is rapidly developing in many practical applications in engineeringThe mathematical background is presented here for the first time in book form by one of the leaders in the field

A comprehensive description of hybrid plasma simulation models providing a very useful summary and guide to the vast literature on this topic.

This is the first monograph on the subject, providing a comprehensive introduction to the LSFEM method for numerical solution of PDEs. LSFEM is simple, efficient and robust, and can solve a wide range of problems in fluid dynamics and electromagnetics.

The book gives the reader the basis for understanding the way numerical schemes achieve accurate and stable simulations of physical phenomena. The book builds on simple model equations and, pedagogically, on a host of problems given together with their solutions.

From the reviews: "A unique feature of this book is the nice blend of engineering vividness and mathematical rigour. [...] The authors are to be congratulated for their valuable contribution to the literature in the area of theoretical thermoelasticity and vibration of plates." Journal of Sound and Vibration

From the reviews of the first edition: "This book is directed to graduate students and research workers interested in the numerical solution of problems of fluid dynamics, primarily those arising in high speed flow.

The chosen semi-discrete approach of a reduction procedure of partial differential equations to ordinary differential equations and finally to difference equations gives the book its distinctiveness and provides a sound basis for a deep understanding of the fundamental concepts in computational fluid dynamics.

This well-known 2-volume textbook provides senior undergraduate and postgraduate engineers, scientists and applied mathematicians with the specific techniques, and the framework to develop skills in using the techniques in the various branches of computational fluid dynamics. A solutions manual to the exercises is in preparation.