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Weighted Residual Methods

Weighted Residual Methods
Principles, Modifications and Applications

by Mohammad Hatami

  • Publisher : Academic Press
  • Release : 2017-11-07
  • Pages : 384
  • ISBN : 0128132191
  • Language : En, Es, Fr & De
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Weighted Residual Methods: Principles, Modifications and Applications introduces a range of WRMs, providing examples that show how they can be used to solve complex engineering problems with greater accuracy and computational efficiency. Examples focus on non-linear problems, including the motion of a spherical particle, nanofluid flow and heat transfer, magnetohydrodynamic flow and heat transfer, and micropolar fluid flow and heat transfer. These are important factors in understanding processes, such as filtration, combustion, air and water pollution and micro contamination. In addition to the applications, the reader is provided with full derivations of equations and summaries of important field research. Includes the basic code for each method, giving readers a head start in using WRMs for computational modeling Provides full derivations of important governing equations in a number of emerging fields of study Offers numerous, detailed examples of a range of applications in heat transfer, nanotechnology, medicine, and more

The Method of Weighted Residuals and Variational Principles

The Method of Weighted Residuals and Variational Principles

by Bruce A. Finlayson

  • Publisher : SIAM
  • Release : 2013-12-30
  • Pages : 412
  • ISBN : 1611973244
  • Language : En, Es, Fr & De
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This classic book covers the solution of differential equations in science and engineering in such as way as to provide an introduction for novices before progressing toward increasingly more difficult problems. The Method of Weighted Residuals and Variational Principles describes variational principles, including how to find them and how to use them to construct error bounds and create stationary principles. The book also illustrates how to use simple methods to find approximate solutions, shows how to use the finite element method for more complex problems, and provides detailed information on error bounds. Problem sets make this book ideal for self-study or as a course text.

Minimum Weighted Residual Methods in Endogenous Growth Models

Minimum Weighted Residual Methods in Endogenous Growth Models
A Book

by Michael Kejak

  • Publisher : Unknown Publisher
  • Release : 2000
  • Pages : 10
  • ISBN : 9788086288291
  • Language : En, Es, Fr & De
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The Finite Element Method

The Finite Element Method
An Introduction with Partial Differential Equations

by A. J. Davies

  • Publisher : OUP Oxford
  • Release : 2011-09-08
  • Pages : 312
  • ISBN : 0191630349
  • Language : En, Es, Fr & De
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The finite element method is a technique for solving problems in applied science and engineering. The essence of this book is the application of the finite element method to the solution of boundary and initial-value problems posed in terms of partial differential equations. The method is developed for the solution of Poisson's equation, in a weighted-residual context, and then proceeds to time-dependent and nonlinear problems. The relationship with the variational approach is also explained. This book is written at an introductory level, developing all the necessary concepts where required. Consequently, it is well-placed to be used as a textbook for a course in finite elements for final year undergraduates, the usual place for studying finite elements. There are worked examples throughout and each chapter has a set of exercises with detailed solutions.

The Finite Element Method

The Finite Element Method
Basic Concepts and Applications

by Darrell W. Pepper,Juan C. Heinrich

  • Publisher : Taylor & Francis
  • Release : 2005-10-31
  • Pages : 328
  • ISBN : 0203942353
  • Language : En, Es, Fr & De
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This much-anticipated second edition introduces the fundamentals of the finite element method featuring clear-cut examples and an applications-oriented approach. Using the transport equation for heat transfer as the foundation for the governing equations, this new edition demonstrates the versatility of the method for a wide range of applications, including structural analysis and fluid flow. Much attention is given to the development of the discrete set of algebraic equations, beginning with simple one-dimensional problems that can be solved by inspection, continuing to two- and three-dimensional elements, and ending with three chapters describing applications. The increased number of example problems per chapter helps build an understanding of the method to define and organize required initial and boundary condition data for specific problems. In addition to exercises that can be worked out manually, this new edition refers to user-friendly computer codes for solving one-, two-, and three-dimensional problems. Among the first FEM textbooks to include finite element software, the book contains a website with access to an even more comprehensive list of finite element software written in FEMLAB, MAPLE, MathCad, MATLAB, FORTRAN, C++, and JAVA - the most popular programming languages. This textbook is valuable for senior level undergraduates in mechanical, aeronautical, electrical, chemical, and civil engineering. Useful for short courses and home-study learning, the book can also serve as an introduction for first-year graduate students new to finite element coursework and as a refresher for industry professionals. The book is a perfect lead-in to Intermediate Finite Element Method: Fluid Flow and Heat and Transfer Applications (Taylor & Francis, 1999, Hb 1560323094).

Finite Elements and Approximation

Finite Elements and Approximation
A Book

by O. C. Zienkiewicz,K. Morgan

  • Publisher : Courier Corporation
  • Release : 2013-04-22
  • Pages : 352
  • ISBN : 048631801X
  • Language : En, Es, Fr & De
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A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises. Starting with continuum boundary value problems and the need for numerical discretization, the text examines finite difference methods, weighted residual methods in the context of continuous trial functions, and piecewise defined trial functions and the finite element method. Additional topics include higher order finite element approximation, mapping and numerical integration, variational methods, and partial discretization and time-dependent problems. A survey of generalized finite elements and error estimates concludes the text.

Numerical Methods for Engineers and Scientists, Second Edition,

Numerical Methods for Engineers and Scientists, Second Edition,
A Book

by Joe D. Hoffman,Steven Frankel

  • Publisher : CRC Press
  • Release : 2001-05-31
  • Pages : 840
  • ISBN : 9780824704438
  • Language : En, Es, Fr & De
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Emphasizing the finite difference approach for solving differential equations, the second edition of Numerical Methods for Engineers and Scientists presents a methodology for systematically constructing individual computer programs. Providing easy access to accurate solutions to complex scientific and engineering problems, each chapter begins with objectives, a discussion of a representative application, and an outline of special features, summing up with a list of tasks students should be able to complete after reading the chapter- perfect for use as a study guide or for review. The AIAA Journal calls the book "...a good, solid instructional text on the basic tools of numerical analysis."

The Finite Element Method for Boundary Value Problems

The Finite Element Method for Boundary Value Problems
Mathematics and Computations

by Karan S. Surana,J. N. Reddy

  • Publisher : CRC Press
  • Release : 2016-11-17
  • Pages : 824
  • ISBN : 1498780512
  • Language : En, Es, Fr & De
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Written by two well-respected experts in the field, The Finite Element Method for Boundary Value Problems: Mathematics and Computations bridges the gap between applied mathematics and application-oriented computational studies using FEM. Mathematically rigorous, the FEM is presented as a method of approximation for differential operators that are mathematically classified as self-adjoint, non-self-adjoint, and non-linear, thus addressing totality of all BVPs in various areas of engineering, applied mathematics, and physical sciences. These classes of operators are utilized in various methods of approximation: Galerkin method, Petrov-Galerkin Method, weighted residual method, Galerkin method with weak form, least squares method based on residual functional, etc. to establish unconditionally stable finite element computational processes using calculus of variations. Readers are able to grasp the mathematical foundation of finite element method as well as its versatility of applications. h-, p-, and k-versions of finite element method, hierarchical approximations, convergence, error estimation, error computation, and adaptivity are additional significant aspects of this book.

Introduction to Finite Element Analysis Using MATLAB® and Abaqus

Introduction to Finite Element Analysis Using MATLAB® and Abaqus
A Book

by Amar Khennane

  • Publisher : CRC Press
  • Release : 2013-06-10
  • Pages : 487
  • ISBN : 1466580208
  • Language : En, Es, Fr & De
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There are some books that target the theory of the finite element, while others focus on the programming side of things. Introduction to Finite Element Analysis Using MATLAB® and Abaqus accomplishes both. This book teaches the first principles of the finite element method. It presents the theory of the finite element method while maintaining a balance between its mathematical formulation, programming implementation, and application using commercial software. The computer implementation is carried out using MATLAB, while the practical applications are carried out in both MATLAB and Abaqus. MATLAB is a high-level language specially designed for dealing with matrices, making it particularly suited for programming the finite element method, while Abaqus is a suite of commercial finite element software. Includes more than 100 tables, photographs, and figures Provides MATLAB codes to generate contour plots for sample results Introduction to Finite Element Analysis Using MATLAB and Abaqus introduces and explains theory in each chapter, and provides corresponding examples. It offers introductory notes and provides matrix structural analysis for trusses, beams, and frames. The book examines the theories of stress and strain and the relationships between them. The author then covers weighted residual methods and finite element approximation and numerical integration. He presents the finite element formulation for plane stress/strain problems, introduces axisymmetric problems, and highlights the theory of plates. The text supplies step-by-step procedures for solving problems with Abaqus interactive and keyword editions. The described procedures are implemented as MATLAB codes and Abaqus files can be found on the CRC Press website.

A Finite Element Primer for Beginners

A Finite Element Primer for Beginners
The Basics

by Tarek I. Zohdi

  • Publisher : Springer
  • Release : 2014-08-12
  • Pages : 108
  • ISBN : 3319090364
  • Language : En, Es, Fr & De
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The purpose of this primer is to provide the basics of the Finite Element Method, primarily illustrated through a classical model problem, linearized elasticity. The topics covered are: (1) Weighted residual methods and Galerkin approximations, (2) A model problem for one-dimensional linear elastostatics, (3) Weak formulations in one dimension, (4) Minimum principles in one dimension, (5) Error estimation in one dimension, (5) Construction of Finite Element basis functions in one dimension, (6) Gaussian Quadrature, (7) Iterative solvers and element by element data structures, (8) A model problem for three-dimensional linear elastostatics, (9) Weak formulations in three dimensions, (10) Basic rules for element construction in three-dimensions, (11) Assembly of the system and solution schemes, (12) Assembly of the system and solution schemes, (13) An introduction to time-dependent problems and (14) A brief introduction to rapid computation based on domain decomposition and basic parallel processing.

TEXTBOOK OF FINITE ELEMENT ANALYSIS

TEXTBOOK OF FINITE ELEMENT ANALYSIS
A Book

by P. SESHU

  • Publisher : PHI Learning Pvt. Ltd.
  • Release : 2003-01-01
  • Pages : 340
  • ISBN : 8120323157
  • Language : En, Es, Fr & De
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Designed for a one-semester course in Finite Element Method, this compact and well-organized text presents FEM as a tool to find approximate solutions to differential equations. This provides the student a better perspective on the technique and its wide range of applications. This approach reflects the current trend as the present-day applications range from structures to biomechanics to electromagnetics, unlike in conventional texts that view FEM primarily as an extension of matrix methods of structural analysis. After an introduction and a review of mathematical preliminaries, the book gives a detailed discussion on FEM as a technique for solving differential equations and variational formulation of FEM. This is followed by a lucid presentation of one-dimensional and two-dimensional finite elements and finite element formulation for dynamics. The book concludes with some case studies that focus on industrial problems and Appendices that include mini-project topics based on near-real-life problems. Postgraduate/Senior undergraduate students of civil, mechanical and aeronautical engineering will find this text extremely useful; it will also appeal to the practising engineers and the teaching community.

The Finite Element Method for Engineers

The Finite Element Method for Engineers
A Book

by Kenneth H. Huebner,Donald L. Dewhirst,Douglas E. Smith,Ted G. Byrom

  • Publisher : John Wiley & Sons
  • Release : 2001-09-07
  • Pages : 744
  • ISBN : 9780471370789
  • Language : En, Es, Fr & De
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A useful balance of theory, applications, and real-world examples The Finite Element Method for Engineers, Fourth Edition presents a clear, easy-to-understand explanation of finite element fundamentals and enables readers to use the method in research and in solving practical, real-life problems. It develops the basic finite element method mathematical formulation, beginning with physical considerations, proceeding to the well-established variation approach, and placing a strong emphasis on the versatile method of weighted residuals, which has shown itself to be important in nonstructural applications. The authors demonstrate the tremendous power of the finite element method to solve problems that classical methods cannot handle, including elasticity problems, general field problems, heat transfer problems, and fluid mechanics problems. They supply practical information on boundary conditions and mesh generation, and they offer a fresh perspective on finite element analysis with an overview of the current state of finite element optimal design. Supplemented with numerous real-world problems and examples taken directly from the authors' experience in industry and research, The Finite Element Method for Engineers, Fourth Edition gives readers the real insight needed to apply the method to challenging problems and to reason out solutions that cannot be found in any textbook.

Finite Elements Using Maple

Finite Elements Using Maple
A Symbolic Programming Approach

by Artur Portela,A. Charafi

  • Publisher : Springer Science & Business Media
  • Release : 2012-12-06
  • Pages : 326
  • ISBN : 3642559360
  • Language : En, Es, Fr & De
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This text provides the reader with a unique insight into the finite element method, along with symbolic programing that fundamentally changes the way applications can be developed. It is an essential tool for undergraduate or early postgraduate courses as well as an excellent reference book for engineers and scientists who want to quickly develop finite-element programs. The use of symbolic computation in Maple system delivers new benefits in the analysis and understanding of the finite element method.

Adaptive Finite Element Methods for Differential Equations

Adaptive Finite Element Methods for Differential Equations
A Book

by Wolfgang Bangerth,Rolf Rannacher

  • Publisher : Birkhäuser
  • Release : 2013-11-11
  • Pages : 208
  • ISBN : 303487605X
  • Language : En, Es, Fr & De
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These Lecture Notes have been compiled from the material presented by the second author in a lecture series ('Nachdiplomvorlesung') at the Department of Mathematics of the ETH Zurich during the summer term 2002. Concepts of 'self adaptivity' in the numerical solution of differential equations are discussed with emphasis on Galerkin finite element methods. The key issues are a posteriori er ror estimation and automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method (or shortly D WR method) for goal-oriented error estimation is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. The basics of the DWR method and various of its applications are described in the following survey articles: R. Rannacher [114], Error control in finite element computations. In: Proc. of Summer School Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds), pp. 247-278. Kluwer Academic Publishers, 1998. M. Braack and R. Rannacher [42], Adaptive finite element methods for low Mach-number flows with chemical reactions.

Advanced Numerical and Semi-Analytical Methods for Differential Equations

Advanced Numerical and Semi-Analytical Methods for Differential Equations
A Book

by Snehashish Chakraverty,Nisha Mahato,Perumandla Karunakar,Tharasi Dilleswar Rao

  • Publisher : Wiley
  • Release : 2019-04-30
  • Pages : 256
  • ISBN : 1119423422
  • Language : En, Es, Fr & De
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Examines numerical and semi-analytical methods for differential equations that can be used for solving practical ODEs and PDEs This student-friendly book deals with various approaches for solving differential equations numerically or semi-analytically depending on the type of equations and offers simple example problems to help readers along. Featuring both traditional and recent methods, Advanced Numerical and Semi Analytical Methods for Differential Equations begins with a review of basic numerical methods. It then looks at Laplace, Fourier, and weighted residual methods for solving differential equations. A new challenging method of Boundary Characteristics Orthogonal Polynomials (BCOPs) is introduced next. The book then discusses Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), and Boundary Element Method (BEM). Following that, analytical/semi analytic methods like Akbari Ganji's Method (AGM) and Exp-function are used to solve nonlinear differential equations. Nonlinear differential equations using semi-analytical methods are also addressed, namely Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), and Homotopy Analysis Method (HAM). Other topics covered include: emerging areas of research related to the solution of differential equations based on differential quadrature and wavelet approach; combined and hybrid methods for solving differential equations; as well as an overview of fractal differential equations. Further, uncertainty in term of intervals and fuzzy numbers have also been included, along with the interval finite element method. This book: Discusses various methods for solving linear and nonlinear ODEs and PDEs Covers basic numerical techniques for solving differential equations along with various discretization methods Investigates nonlinear differential equations using semi-analytical methods Examines differential equations in an uncertain environment Includes a new scenario in which uncertainty (in term of intervals and fuzzy numbers) has been included in differential equations Contains solved example problems, as well as some unsolved problems for self-validation of the topics covered Advanced Numerical and Semi Analytical Methods for Differential Equations is an excellent text for graduate as well as post graduate students and researchers studying various methods for solving differential equations, numerically and semi-analytically.

Fundamentals of the Finite Element Method

Fundamentals of the Finite Element Method
A Book

by Hartley Grandin

  • Publisher : Waveland PressInc
  • Release : 1991
  • Pages : 528
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Applied Functional Analysis and Variational Methods in Engineering

Applied Functional Analysis and Variational Methods in Engineering
A Book

by Junuthula Narasimha Reddy

  • Publisher : Unknown Publisher
  • Release : 1991
  • Pages : 546
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Boundary Element Techniques in Engineering

Boundary Element Techniques in Engineering
A Book

by C. A. Brebbia,S. Walker

  • Publisher : Elsevier
  • Release : 2016-01-29
  • Pages : 220
  • ISBN : 1483102564
  • Language : En, Es, Fr & De
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Boundary Element Techniques in Engineering deals with solutions of two- and three-dimensional problems in elasticity and the potential theory where finite elements are inefficient. The book discusses approximate methods, higher-order elements, elastostatics, time-dependent problems, non-linear problems, and combination of regions. Approximate methods include weighted residual techniques, weak formulations, the inverse formulation, and boundary methods. The text also explains Laplace's equation, indirect formulation, matrix formulation, Poisson's equation, and the Helmholtz equation. It describes how elements with linear variations of u and q (i.e. linear elements) can be developed for two dimensional problems, as well as for quadratic and higher order elements for two-dimensional problems. The text investigates the Dirac delta function as a sum of Eigen functions, including some methods to determine the explicit form of fundamental solutions for recurrent problems. The book also tackles the application of boundary elements to problems with both material and certain types of geometric non-linearities, and also the applications of boundary elements to plasticity problems. The text is ideal for mathematicians, students, and professor of calculus or advanced mathematics.

Finite Element Simulation in Surface and Subsurface Hydrology

Finite Element Simulation in Surface and Subsurface Hydrology
A Book

by George F. Pinder,William G. Gray

  • Publisher : Elsevier
  • Release : 2013-09-03
  • Pages : 308
  • ISBN : 1483270424
  • Language : En, Es, Fr & De
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Finite Element Simulation in Surface and Subsurface Hydrology provides an introduction to the finite element method and how the method is applied to problems in surface and subsurface hydrology. The book presents the basic concepts of the numerical methods and the finite element approach; applications to problems on groundwater flow and mass and energy transport; and applications to problems that involve surface water dynamics. Computational methods for the solution of differential equations; classification of partial differential equations; finite difference and weighted residual integral techniques; and The Galerkin finite element method are discussed as well. The text will be of value to engineers, hydrologists, and students in the field of engineering.

Computational Techniques for Fluid Dynamics 1

Computational Techniques for Fluid Dynamics 1
Fundamental and General Techniques

by Clive A.J. Fletcher

  • Publisher : Springer Science & Business Media
  • Release : 2013-03-14
  • Pages : 409
  • ISBN : 3642970354
  • Language : En, Es, Fr & De
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The purpose of this two-volume textbook is to provide students of engineer ing, science and applied mathematics with the specific techniques, and the framework to develop skill in using them, that have proven effective in the various branches of computational fluid dynamics (CFD). Volume 1 de scribes both fundamental and general techniques that are relevant to all branches of fluid flow. Volume 2 provides specific techniques, applicable to the different categories of engineering flow behaviour, many of which are also appropriate to convective heat transfer. An underlying theme of the text ist that the competing formulations which are suitable for computational fluid dynamics, e.g. the finite differ ence, finite element, finite volume and spectral methods, are closely related and can be interpreted as part of a unified structure. Classroom experience indicates that this approach assists, considerably, the student in acquiring a deeper understanding of the strengths and weaknesses of the alternative computational methods. Through the provision of 24 computer programs and associated exam ples and problems, the present text is also suitable for established research workers and practitioners who wish to acquire computational skills without the benefit of formal instruction. The text includes the most up-to-date techniques and is supported by more than 300 figures and 500 references.