main-content

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:

• Weighted residual methods and Galerkin approximations,

• A model problem for one-dimensional linear elastostatics,

• Weak formulations in one dimension,

• Minimum principles in one dimension,

• Error estimation in one dimension,

• Construction of Finite Element basis functions in one dimension,

• Iterative solvers and element by element data structures,

• A model problem for three-dimensional linear elastostatics,

• Weak formulations in three dimensions,

• Basic rules for element construction in three-dimensions,

• Assembly of the system and solution schemes,

• An introduction to time-dependent problems and

• An introduction to rapid computation based on domain decomposition

and basic parallel processing.

The approach is to introduce the basic concepts first in one-dimension, then move on to three-dimensions. A relatively informal style is adopted. This primer is intended to be a “starting point”, which can be later augmented by the large array of rigorous, detailed, books in the area of Finite Element analysis. In addition to overall improvements to the first edition, this second edition also adds several carefully selected in-class exam problems from exams given over the last 15 years at UC Berkeley, as well as a large number of take-home computer projects. These problems and projects are designed to be aligned to the theory provided in the main text of this primer.

### Chapter 1. Weighted Residuals and Galerkin’s Method for a Generic 1D Problem

Abstract
Let us start by considering a simple one-dimensional differential equation, written in abstract form.
Tarek I. Zohdi

### Chapter 2. A Model Problem: 1D Elastostatics

Abstract
In most problems of mathematical physics the true solutions are nonsmooth; i.e., they are not continuously differentiable.
Tarek I. Zohdi

### Chapter 3. A Finite Element Implementation in One Dimension

Abstract
Classical techniques construct approximations from globally kinematically admissible functions, which we define as functions that satisfy the displacement boundary condition beforehand.
Tarek I. Zohdi

### Chapter 4. Accuracy of the Finite Element Method in One Dimension

Abstract
As we have seen, the essential idea in the finite element method is to select a finite dimensional subspatial approximation of the true solution and to form the following weak boundary problem.
Tarek I. Zohdi

### Chapter 5. Iterative Solutions Schemes

Abstract
There are two main approaches to solving systems of equations resulting from numerical discretization of solid mechanics problems, direct and iterative.
Tarek I. Zohdi

### Chapter 6. Weak Formulations in Three Dimensions

Abstract
Albeit a bit repetitive, we follow similar constructions as done in one-dimensional analysis of the preceding chapters. This allows readers a chance to contrast and compare the differences between one-dimensional and three-dimensional formulations.
Tarek I. Zohdi

### Chapter 7. A Finite Element Implementation in Three Dimensions

Abstract
Generally, the ability to change the boundary data quickly is very important in finite element computations.
Tarek I. Zohdi

### Chapter 8. Accuracy of the Finite Element Method in Three Dimensions

Abstract
As we have seen in the one-dimensional analysis, the essential idea in the finite element method is to select a finite dimensional subspatial approximation of the true solution and form the following weak boundary problem
Tarek I. Zohdi

### Chapter 9. Time-Dependent Problems

Abstract
We now give a brief introduction to time-dependent problems through the equations of elastodynamics for infinitesimal deformations.
Tarek I. Zohdi

### Chapter 10. Summary and Advanced Topics

Abstract
The finite element method is a huge field of study. This set of notes was designed to give students only a brief introduction to the fundamentals of the method.
Tarek I. Zohdi