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About this book

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,

• Gaussian Quadrature,

• 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.

Table of Contents


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

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

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

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

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

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

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

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

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

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

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


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