The Finite Element Method: Theory, Implementation, and Applications

Author: Mats G. Larson,Fredrik Bengzon

Publisher: Springer Science & Business Media

ISBN: 3642332870

Category: Computers

Page: 395

View: 6942

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.​

Automated Solution of Differential Equations by the Finite Element Method

The FEniCS Book

Author: Anders Logg,Kent-Andre Mardal,Garth Wells

Publisher: Springer Science & Business Media

ISBN: 3642230997

Category: Computers

Page: 731

View: 356

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.

Introduction to Finite Element Analysis Using MATLAB® and Abaqus

Author: Amar Khennane

Publisher: CRC Press

ISBN: 9781466580206

Category: Technology & Engineering

Page: 487

View: 5984

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.

Understanding and Implementing the Finite Element Method

Author: Mark S. Gockenbach

Publisher: SIAM

ISBN: 0898716144

Category: Mathematics

Page: 363

View: 1522

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.

An Introduction to the Mathematical Theory of Finite Elements

Author: J. T. Oden,J. N. Reddy

Publisher: Courier Corporation

ISBN: 0486142213

Category: Technology & Engineering

Page: 448

View: 7023

This introduction to the theory of Sobolev spaces and Hilbert space methods in partial differential equations is geared toward readers of modest mathematical backgrounds. It offers coherent, accessible demonstrations of the use of these techniques in developing the foundations of the theory of finite element approximations. J. T. Oden is Director of the Institute for Computational Engineering & Sciences (ICES) at the University of Texas at Austin, and J. N. Reddy is a Professor of Engineering at Texas A&M University. They developed this essentially self-contained text from their seminars and courses for students with diverse educational backgrounds. Their effective presentation begins with introductory accounts of the theory of distributions, Sobolev spaces, intermediate spaces and duality, the theory of elliptic equations, and variational boundary value problems. The second half of the text explores the theory of finite element interpolation, finite element methods for elliptic equations, and finite element methods for initial boundary value problems. Detailed proofs of the major theorems appear throughout the text, in addition to numerous examples.

Finite Element Method Electromagnetics

Antennas, Microwave Circuits, and Scattering Applications

Author: John L. Volakis,Arindam Chatterjee,Leo C. Kempel

Publisher: John Wiley & Sons

ISBN: 9780780334250

Category: Science

Page: 344

View: 2325

Employed in a large number of commercial electromagnetic simulation packages, the finite element method is one of the most popular and well-established numerical techniques in engineering. This book covers the theory, development, implementation, and application of the finite element method and its hybrid versions to electromagnetics. FINITE ELEMENT METHOD FOR ELECTROMAGNETICS begins with a step-by-step textbook presentation of the finite method and its variations then goes on to provide up-to-date coverage of three dimensional formulations and modern applications to open and closed domain problems. Worked out examples are included to aid the reader with the fine features of the method and the implementation of its hybridization with other techniques for a robust simulation of large scale radiation and scattering. The crucial treatment of local boundary conditions is carefully worked out in several stages in the book. Sponsored by: IEEE Antennas and Propagation Society.

Advanced Finite Element Methods and Applications

Author: Thomas Apel,Olaf Steinbach

Publisher: Springer Science & Business Media

ISBN: 3642303161

Category: Technology & Engineering

Page: 376

View: 2359

This volume on some recent aspects of finite element methods and their applications is dedicated to Ulrich Langer and Arnd Meyer on the occasion of their 60th birthdays in 2012. Their work combines the numerical analysis of finite element algorithms, their efficient implementation on state of the art hardware architectures, and the collaboration with engineers and practitioners. In this spirit, this volume contains contributions of former students and collaborators indicating the broad range of their interests in the theory and application of finite element methods. Topics cover the analysis of domain decomposition and multilevel methods, including hp finite elements, hybrid discontinuous Galerkin methods, and the coupling of finite and boundary element methods; the efficient solution of eigenvalue problems related to partial differential equations with applications in electrical engineering and optics; and the solution of direct and inverse field problems in solid mechanics.

Theory and Practice of Finite Elements

Author: Alexandre Ern,Jean-Luc Guermond

Publisher: Springer Science & Business Media

ISBN: 1475743556

Category: Mathematics

Page: 526

View: 2098

This text presenting the mathematical theory of finite elements is organized into three main sections. The first part develops the theoretical basis for the finite element methods, emphasizing inf-sup conditions over the more conventional Lax-Milgrim paradigm. The second and third parts address various applications and practical implementations of the method, respectively. It contains numerous examples and exercises.

Extended Finite Element Method

Theory and Applications

Author: Amir R. Khoei

Publisher: John Wiley & Sons

ISBN: 1118457684

Category: Science

Page: 584

View: 1344

Introduces the theory and applications of the extended finite element method (XFEM) in the linear and nonlinear problems of continua, structures and geomechanics Extended Finite Element Method: Theory and Applications introduces the theory and applications of the extended finite element method (XFEM) in the linear and nonlinear problems of continua, structures and geomechanics. The XFEM approach is based on an extension of standard finite element method based on the partition of unity method. Extended Finite Element Method: Theory and Applications begins by introducing the concept of partition of unity, various enrichment functions, and fundamentals of XFEM formulation. It then covers the theory and application of XFEM in large deformations, plasticity and contact problems. The implementation of XFEM in fracture mechanics, including the linear, cohesive, and ductile crack propagation is also covered. The theory and applications of the XFEM in multiphase fluid flow, including the hydraulic fracturing in soil saturated media and crack propagation in thermo-hydro-mechanical porous media, is also discussed in detail. 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

Hybrid and Incompatible Finite Element Methods

Author: Theodore H.H. Pian,Chang-Chun Wu

Publisher: CRC Press

ISBN: 0203487699

Category: Mathematics

Page: 400

View: 4539

While the theory and application of finite elements methods can be extended to incompatible, hybrid, and mixed element methods, important issues, such as determining the reliability of the solution of incompatible multivariable elements, along with a common perception of impracticality, have hindered the widespread implementation of these methods. Today, however, recent advances--many directly attributable to these authors--have allowed the development of the stability theory and abstract mathematics to useful tools. Hybrid and Incompatible Finite Element Methods introduces these advances in the theory and applications of incompatible and multivariable finite element methods. After an overview of the variation formulation of finite element methods in solid mechanics, the authors discuss the fundamental theory and systematically demonstrate the theoretical foundations of incompatible elements and their application to different problems in the theory of elasticity. They also introduce new ideas in the development of hybrid finite elements, study the numerical stability of the hybrid and mixed element, and establish the theory of zero energy deformation modes. The final chapters, explore applications to fracture problems, present a bound analysis for fracture parameters, and demonstrate an implementation of a finite element analysis program.

A Pragmatic Introduction to the Finite Element Method for Thermal and Stress Analysis

With the Matlab Toolkit SOFEA

Author: Petr Krysl

Publisher: World Scientific Publishing Company

ISBN: 9813101814

Category: Science

Page: 292

View: 6253

This textbook provides an accessible and self-contained description of the Galerkin finite element method for the two important models of continuum mechanics, transient heat conduction and elastodynamics, from formulation of the governing equations to implementation in Matlab. The coverage follows an intuitive approach: the salient features of each initial boundary value problem are reviewed, including a thorough description of the boundary conditions; the method of weighted residuals is applied to derive the discrete equations; and clear examples are introduced to illustrate the method.

The Scaled Boundary Finite Element Method

Introduction to Theory and Implementation

Author: Chongmin Song

Publisher: John Wiley & Sons

ISBN: 1119388457

Category: Science

Page: 504

View: 2020

An informative look at the theory, computer implementation, and application of the scaled boundary finite element method This reliable resource, complete with MATLAB, is an easy-to-understand introduction to the fundamental principles of the scaled boundary finite element method. It establishes the theory of the scaled boundary finite element method systematically as a general numerical procedure, providing the reader with a sound knowledge to expand the applications of this method to a broader scope. The book also presents the applications of the scaled boundary finite element to illustrate its salient features and potentials. The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation covers the static and dynamic stress analysis of solids in two and three dimensions. The relevant concepts, theory and modelling issues of the scaled boundary finite element method are discussed and the unique features of the method are highlighted. The applications in computational fracture mechanics are detailed with numerical examples. A unified mesh generation procedure based on quadtree/octree algorithm is described. It also presents examples of fully automatic stress analysis of geometric models in NURBS, STL and digital images. Written in lucid and easy to understand language by the co-inventor of the scaled boundary element method Provides MATLAB as an integral part of the book with the code cross-referenced in the text and the use of the code illustrated by examples Presents new developments in the scaled boundary finite element method with illustrative examples so that readers can appreciate the significant features and potentials of this novel method—especially in emerging technologies such as 3D printing, virtual reality, and digital image-based analysis The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation is an ideal book for researchers, software developers, numerical analysts, and postgraduate students in many fields of engineering and science.

Numerical Solution of Differential Equations

Author: Zhilin Li,Zhonghua Qiao,Tao Tang

Publisher: Cambridge University Press

ISBN: 1107163226

Category: Mathematics

Page: 293

View: 5016

A practical and concise guide to finite difference and finite element methods. Well-tested MATLAB® codes are available online.

p- and hp- Finite Element Methods

Theory and Applications in Solid and Fluid Mechanics

Author: C. Schwab

Publisher: Clarendon Press

ISBN: 9780198503903

Category: Computers

Page: 386

View: 872

The finite element method (FEM) is a numerical procedure for solving differential equations. Ever-increasing computing power means that engineers and applied mathematicians are seeking more complicated and sophisticated numerical methods to obtain progressively more accurate answers to problems in solid and fluid mechanics. The p- and hp- finite element methods are just such methods, and are therefore of great current interest. This book is the first to cover comprehensively the mathematical underpinnings of hp-FEM in one and two dimensions and pays particular attention to its applications in engineering.

Partial Differential Equations and the Finite Element Method

Author: Pavel Ŝolín

Publisher: John Wiley & Sons

ISBN: 0471764094

Category: Mathematics

Page: 512

View: 848

A systematic introduction to partial differential equations and modern finite element methods for their efficient numerical solution Partial Differential Equations and the Finite Element Method provides a much-needed, clear, and systematic introduction to modern theory of partial differential equations (PDEs) and finite element methods (FEM). Both nodal and hierachic concepts of the FEM are examined. Reflecting the growing complexity and multiscale nature of current engineering and scientific problems, the author emphasizes higher-order finite element methods such as the spectral or hp-FEM. A solid introduction to the theory of PDEs and FEM contained in Chapters 1-4 serves as the core and foundation of the publication. Chapter 5 is devoted to modern higher-order methods for the numerical solution of ordinary differential equations (ODEs) that arise in the semidiscretization of time-dependent PDEs by the Method of Lines (MOL). Chapter 6 discusses fourth-order PDEs rooted in the bending of elastic beams and plates and approximates their solution by means of higher-order Hermite and Argyris elements. Finally, Chapter 7 introduces the reader to various PDEs governing computational electromagnetics and describes their finite element approximation, including modern higher-order edge elements for Maxwell's equations. The understanding of many theoretical and practical aspects of both PDEs and FEM requires a solid knowledge of linear algebra and elementary functional analysis, such as functions and linear operators in the Lebesgue, Hilbert, and Sobolev spaces. These topics are discussed with the help of many illustrative examples in Appendix A, which is provided as a service for those readers who need to gain the necessary background or require a refresher tutorial. Appendix B presents several finite element computations rooted in practical engineering problems and demonstrates the benefits of using higher-order FEM. Numerous finite element algorithms are written out in detail alongside implementation discussions. Exercises, including many that involve programming the FEM, are designed to assist the reader in solving typical problems in engineering and science. Specifically designed as a coursebook, this student-tested publication is geared to upper-level undergraduates and graduate students in all disciplines of computational engineeringand science. It is also a practical problem-solving reference for researchers, engineers, and physicists.

The Mathematical Theory of Finite Element Methods

Author: Susanne C Brenner,L. Ridgway Scott

Publisher: Springer Science & Business Media

ISBN: 1475736584

Category: Mathematics

Page: 363

View: 6433

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 Element Analysis with Error Estimators

An Introduction to the FEM and Adaptive Error Analysis for Engineering Students

Author: J. E. Akin

Publisher: Elsevier

ISBN: 9780080472751

Category: Technology & Engineering

Page: 512

View: 4873

This key text is written for senior undergraduate and graduate engineering students. It delivers a complete introduction to finite element methods and to automatic adaptation (error estimation) that will enable students to understand and use FEA as a true engineering tool. It has been specifically developed to be accessible to non-mathematics students and provides the only complete text for FEA with error estimators for non-mathematicians. Error estimation is taught on nearly half of all FEM courses for engineers at senior undergraduate and postgraduate level; no other existing textbook for this market covers this topic. The only introductory FEA text with error estimation for students of engineering, scientific computing and applied mathematics Includes source code for creating and proving FEA error estimators

Finite Element Method Simulation of 3D Deformable Solids

Author: Eftychios Sifakis,Jernej Barbič

Publisher: Morgan & Claypool Publishers

ISBN: 162705443X

Category: Computers

Page: 69

View: 3558

This book serves as a practical guide to simulation of 3D deformable solids using the Finite Element Method (FEM). It reviews a number of topics related to the theory and implementation of FEM approaches: measures of deformation, constitutive laws of nonlinear materials, tetrahedral discretizations, and model reduction techniques for real-time simulation. Simulations of deformable solids are important in many applications in computer graphics, including film special effects, computer games, and virtual surgery. The Finite Element Method has become a popular tool in many such applications. Variants of FEM catering to both offline and real-time simulation have had a mature presence in computer graphics literature. This book is designed for readers familiar with numerical simulation in computer graphics, who would like to obtain a cohesive picture of the various FEM simulation methods available, their strengths and weaknesses, and their applicability in various simulation scenarios. The book is also a practical implementation guide for the visual effects developer, offering a lean yet adequate synopsis of the underlying mathematical theory. Chapter 1 introduces the quantitative descriptions used to capture the deformation of elastic solids, the concept of strain energy, and discusses how force and stress result as a response to deformation. Chapter 2 reviews a number of constitutive models, i.e., analytical laws linking deformation to the resulting force that has successfully been used in various graphics-oriented simulation tasks. Chapter 3 summarizes how deformation and force can be computed discretely on a tetrahedral mesh, and how an implicit integrator can be structured around this discretization. Finally, chapter 4 presents the state of the art in model reduction techniques for real-time FEM solid simulation and discusses which techniques are suitable for which applications. Topics discussed in this chapter include linear modal analysis, modal warping, subspace simulation, and domain decomposition.

Finite element analysis

from concepts to applications

Author: David S. Burnett

Publisher: Prentice Hall


Category: Technology & Engineering

Page: 844

View: 9235

The emphasis is on theory, programming and appilications to show exactly how Finite Element Method can be applied to quantum mechanics, heat transfer and fluid dynamics. For engineers, physicists and mathematicians with some mathematical sophistication.