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2023 | Book

Introduction Read chapter Read first chapter

Enhanced Introduction to Finite Elements for Engineers

Author: Uwe Mühlich

Publisher: Springer International Publishing

Book Series : Solid Mechanics and Its Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

The book presents the fundamentals of the Galerkin Finite Element Method for linear boundary value problems from an engineering perspective. Emphasis is given to the theoretical foundation of the method rooted in Functional Analysis using a language accessible to engineers. The book discusses standard procedures for applying the method to time-dependent and nonlinear problems and addresses essential aspects of applying the method to non-linear dynamics and multi-physics problems. It also provides several hand-calculation exercises as well as specific computer exercises with didactic character. About one fourth of the exercises reveals common pitfalls and sources of errors when applying the method. Carefully selected literature recommendations for further studies are provided at the end of each chapter.

The reader is expected to have prior knowledge in engineering mathematics, in particular real analysis and linear algebra. The elements of algebra and analysis required in the main part of the book are presented in corresponding sections of the appendix. Students should already have an education in strength of materials or another engineering field, such as heat or mass transport, which discusses boundary value problems for simple geometries and boundary conditions.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
After defining the Finite Element Method concisely as a method for solving boundary value problems numerically based on a weak form approach, one dimensional heat transport is used to illustrate the meanings of “boundary value problem”, “initial boundary value problem” and “strong form”. The structure of the book is presented and contemporary challenges regarding the education of engineers in FEM are discussed emphasising on the importance of well synchronized theory lectures, hand-calculation exercises and carefully designed computer labs.
Uwe Mühlich
Chapter 2. Linear Boundary Value Problems
Abstract
The chapter describes in detail the Galerkin Finite Element Method (FEM) for time independent, linear boundary value problems. Such boundary value problems can be found in many areas like linear elasticity, linear stationary transport of heat or matter, respectively, or linear electrostatics, just to mention a few. The key ideas of the method are laid out in the first part of this chapter for the Poisson equation in one dimension. Starting from the strong form of the boundary value problem, weak derivative, variational form, and weak form of the boundary value problem are introduced. The latter provides a generic way to derive FEM solution schemes, together with the idea of piecewise defined trial and test functions, which is also rigorously adapted throughout the book. The transition to two or three spatial dimensions is illustrated by means of the Poisson equation in \( \mathbb {R}^N \). Necessary concepts from math are briefly revisited. Linear elasticity introduces new challenges due to the fact, that primary unknowns are no longer scalar fields but vector fields. Advanced continuity requirements are discussed for fourth order boundary value problems using the Euler-Bernoulli beam as illustrative example.
Uwe Mühlich
Chapter 3. Linear Initial Boundary Value Problems
Abstract
After illustrating briefly the new aspects introduced by a time dependence, common time integration methods such as Euler’s methods and Runge-Kutta methods are laid out. Stability of time integration schemes is discussed briefly. Main aspects of FEM solution schemes are illustrated for spatially one-dimensional problems, considering exemplary non-stationary linear transport and linear structural dynamics. In the context of the latter, Newmark’s method is presented in more detail due to its popularity.
Uwe Mühlich
Chapter 4. Non-linear Boundary Value Problems
Abstract
After discussing possible sources of non-linearities, the subject is discussed for the case of non-linear constitutive equations and spatially one-dimensional problems. Developing FEM for non-linear boundary value problems leads eventually to systems of non-linear algebraic equations and the use of Newton’s method for solving such systems is demonstrated. The commonly applied approach for controlling convergence of Newton’s method by employing a time or pseudo time incrementation procedure, respectively, with nested iteration loop is laid out. Numerical integration employing Gauss integration is discussed together with its benefits regarding the separation of spatial discretisation and material routines.
Uwe Mühlich
Chapter 5. A Primer on Non-linear Dynamics and Multiphysics
Abstract
This chapter combines the information presented in the two preceding chapters to present a solution procedure for non-linear dynamics. Phenomena or processes modelled by a system of coupled initial boundary value problems are usually classified as multi-physics problems. The development of FEM solution schemes for such cases is demonstrated for thermo-mechanical coupling. Since this chapter aims for a general understanding, only spatially one-dimensional problems are discussed.
Uwe Mühlich
Backmatter
Metadata
Title
Enhanced Introduction to Finite Elements for Engineers
Author
Uwe Mühlich
Copyright Year
2023
Electronic ISBN
978-3-031-30422-4
Print ISBN
978-3-031-30421-7
DOI
https://doi.org/10.1007/978-3-031-30422-4

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