The Finite Element Method: Theory, Implementation, and Applications

The Finite Element Method: Theory, Implementation, and Applications PDF

Author: Mats G. Larson

Publisher: Springer Science & Business Media

Published: 2013-01-13

Total Pages: 403

ISBN-13: 3642332870

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This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.​

The Finite Element Method: Theory, Implementation, and Applications

The Finite Element Method: Theory, Implementation, and Applications PDF

Author: Mats G. Larson

Publisher: Springer

Published: 2013-01-12

Total Pages: 385

ISBN-13: 9783642332869

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This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.​

Introduction to the Finite Element Method and Implementation with MATLAB®

Introduction to the Finite Element Method and Implementation with MATLAB® PDF

Author: Gang Li

Publisher: Cambridge University Press

Published: 2020-07-30

Total Pages: 525

ISBN-13: 110857386X

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Connecting theory with numerical techniques using MATLAB®, this practical textbook equips students with the tools required to solve finite element problems. This hands-on guide covers a wide range of engineering problems through nine well-structured chapters including solid mechanics, heat transfer and fluid dynamics; equilibrium, steady state and transient; and 1-D, 2-D and 3-D problems. Engineering problems are discussed using case study examples, which are solved using a systematic approach, both by examining the steps manually and by implementing a complete MATLAB®code. This topical coverage is supplemented by discourse on meshing with a detailed explanation and implementation of 2-D meshing algorithms. Introducing theory and numerical techniques alongside comprehensive examples this text increases engagement and provides students with the confidence needed to implement their own computer codes to solve given problems.

Understanding and Implementing the Finite Element Method

Understanding and Implementing the Finite Element Method PDF

Author: Mark S. Gockenbach

Publisher: SIAM

Published: 2006-01-01

Total Pages: 363

ISBN-13: 0898716144

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The ?nite element method is the most powerful general-purpose technique for comput­ing accurate solutions to partial differential equations. Understanding and Implementing the Finite Element Method is essential reading for those interested in understanding both the theory and the implementation of the ?nite element method for equilibrium problems. This book contains a thorough derivation of the finite element equations as well as sections on programming the necessary calculations, solving the finite element equations, and using a posteriori error estimates to produce validated solutions. Accessible introductions to advanced topics, such as multigrid solvers, the hierarchical basis conjugate gradient method, and adaptive mesh generation, are provided. Each chapter ends with exercises to help readers master these topics. Understanding and Implementing the Finite Element Method includes a carefully documented collection of MATLAB® programs implementing the ideas presented in the book. Readers will bene?t from a careful explanation of data structures and speci?c coding strategies and will learn how to write a ?nite element code from scratch. Students can use the MATLAB codes to experiment with the method and extend them in various ways to learn more about programming ?nite elements. This practical book should provide an excellent foundation for those who wish to delve into advanced texts on the subject, including advanced undergraduates and beginning graduate students in mathematics, engineering, and the physical sciences.Preface; Part I: The Basic Framework for Stationary Problems. Chapter 1: Some Model PDEs; Chapter 2: The weak form of a BVP; Chapter 3: The Galerkin method; Chapter 4: Piecewise polynomials and the finite element method; Chapter 5: Convergence of the finite element method; Part II Data Structures and Implementation. Chapter 6: The mesh data structure; Chapter 7: Programming the finite element method: Linear Lagrange triangles; Chapter 8: Lagrange triangles of arbitrary degree; Chapter 9: The finite element method for general BVPs; Part III: Solving the Finite Element Equations. Chapter 10: Direct solution of sparse linear systems; Chapter 11: Iterative methods: Conjugate gradients; Chapter 12: The classical stationary iterations; Chapter 13: The multigrid method; Part IV: Adaptive Methods. Chapter 14: Adaptive mesh generation; Chapter 15: Error estimators and indicators; Bibliography; Index.

The Finite Element Method

The Finite Element Method PDF

Author: Zhu

Publisher: John Wiley & Sons

Published: 2018-03-12

Total Pages: 872

ISBN-13: 1119107334

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A comprehensive review of the Finite Element Method (FEM), this book provides the fundamentals together with a wide range of applications in civil, mechanical and aeronautical engineering. It addresses both the theoretical and numerical implementation aspects of the FEM, providing examples in several important topics such as solid mechanics, fluid mechanics and heat transfer, appealing to a wide range of engineering disciplines. Written by a renowned author and academician with the Chinese Academy of Engineering, The Finite Element Method would appeal to researchers looking to understand how the fundamentals of the FEM can be applied in other disciplines. Researchers and graduate students studying hydraulic, mechanical and civil engineering will find it a practical reference text.

The Mathematical Theory of Finite Element Methods

The Mathematical Theory of Finite Element Methods PDF

Author: Susanne Brenner

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 369

ISBN-13: 1475736584

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A rigorous and thorough mathematical introduction to the subject; A clear and concise treatment of modern fast solution techniques such as multigrid and domain decomposition algorithms; Second edition contains two new chapters, as well as many new exercises; Previous edition sold over 3000 copies worldwide

Finite Elements

Finite Elements PDF

Author: Dietrich Braess

Publisher: Cambridge University Press

Published: 2007-04-12

Total Pages: 348

ISBN-13: 113946146X

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This definitive introduction to finite element methods was thoroughly updated for this 2007 third edition, which features important material for both research and application of the finite element method. The discussion of saddle-point problems is a highlight of the book and has been elaborated to include many more nonstandard applications. The chapter on applications in elasticity now contains a complete discussion of locking phenomena. The numerical solution of elliptic partial differential equations is an important application of finite elements and the author discusses this subject comprehensively. These equations are treated as variational problems for which the Sobolev spaces are the right framework. Graduate students who do not necessarily have any particular background in differential equations, but require an introduction to finite element methods will find this text invaluable. Specifically, the chapter on finite elements in solid mechanics provides a bridge between mathematics and engineering.

Extended Finite Element Method

Extended Finite Element Method PDF

Author: Amir R. Khoei

Publisher: John Wiley & Sons

Published: 2015-02-23

Total Pages: 600

ISBN-13: 1118457684

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Introduces the theory and applications of the extended finite element method (XFEM) in the linear and nonlinear problems of continua, structures and geomechanics Explores the concept of partition of unity, various enrichment functions, and fundamentals of XFEM formulation. Covers numerous applications of XFEM including fracture mechanics, large deformation, plasticity, multiphase flow, hydraulic fracturing and contact problems Accompanied by a website hosting source code and examples

Automated Solution of Differential Equations by the Finite Element Method

Automated Solution of Differential Equations by the Finite Element Method PDF

Author: Anders Logg

Publisher: Springer Science & Business Media

Published: 2012-02-24

Total Pages: 723

ISBN-13: 3642230997

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This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Following are chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

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