Finite Element Methods in Civil and Mechanical Engineering

A Mathematical Introduction

Arzhang Angoshtari, Ali Gerami Matin

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9781138335165
• eBook ISBN: 9780429809279
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Mechanics
• Index: Technology & Engineering

The finite element method is widely employed for numerical simulations in engineering and science due to its accuracy and efficiency. This concise introduction to the mathematical theory of the finite element method presents a selection of applications in civil and mechanical engineering including beams, elastic membranes, the wave equation, heat transfer, seepage in embankment, soil consolidation, incompressible fluids, and linear elasticity. Jupyter notebooks containing all Python programs of each chapter can be downloaded from the book's companion website.

Arzhang Angoshtari is an assistant professor and Ali Gerami Matin is a graduate student, both in the department of Civil and Environmental Engineering at the George Washington University, USA. Their research interests cover theoretical and computational mechanics and finite element methods.

1 Overview

In this introductory chapter, we briefly discuss the basic aspects of the finite element method for solving differential equations. The goal is to give an overview of the book and to motivate the topics that will be discussed in the following chapters. Only main ideas are discussed and the readers are expected to focus only on the big picture and not on details. Technical details of this chapter are the subject of the following ones. This chapter is not a prerequisite for the rest of the book and the readers may skip this chapter if they wish.

Roughly speaking, one may consider the following four stages for the approximation of responses of a system using the finite element method: (i) Identifying the governing equations, which are assumed to be differential equations in this book; (ii) Deriving weak forms of the governing equations; (iii) Discretizing weak forms by using the Galerkin method to obtain a discrete problem; and (iv) Employing piecewise polynomial finite element spaces to solve the discrete problem.

To be more specific, let us consider the approximation of the equilibrium position of an elastic membrane Ω shown in Figure 1.1, which is fixed at its boundary ∂⁡Ω and is under a vertical load f.

Figure 1.1An elastic membrane under a vertical load: (Left) Undeformed configuration Ω and the load f (dashed arrows); (Right) The vertical displacement u⁢(x,y) at the point (x,y).

Governing Equations

The laws of physics imply that the vertical displacement u of the membrane Ω satisfies the boundary value problem

{−Δu=f, in Ω,u=0, on ∂Ω,

1.1

where Δ is the Laplacian, i.e. Δu=∂xxu+∂yyu. This governing equation is also called Poisson’s equation.

Weak Forms

The equation (1.1) is called the strong form of the governing equation. To solve the strong form, one should find a function which is at least twice differentiable and satisfies (1.1). Usually, it is hard to directly find such a function. Alternatively, it may be easier to find a suitable solution candidate first and then showing that the solution candidate is a real solution by establishing its differentiability.

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Table of contents

• Cover
• Half Title
• Title
• Copyright
• Dedication
• Contents
• Preface
• Chapter 1 Overview
• Chapter 2 Mathematical Preliminaries
• Chapter 3 Finite Element Interpolation
• Chapter 4 Conforming Finite Element Methods for PDEs
• Chapter 5 Applications
• Appendix A FEniCS Installation
• Appendix B Introduction to Python
• References
• Index