Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods

695 Pages·2002·369.32 MB·English

by  Victor N. Kaliakin

Introduction to Approximate
Solution Techniques,
Numerical Modeling, and
Finite Element Methods
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Victor N. Kaliakin
Additional Volumes in Production
Introduction to Approximate
Solution Techniques,
Numerical Modeling, and
Finite Element Methods
Victor N. Kaliakin
University of Delaware
Newark, Delaware
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Preface
This book has evolved from notes compiled while teaching the senior/first-
year graduate level course CIEG 401/601 (Introduction to the Finite El-
ement Method) offered at the University of Delaware. The goals of this
course are to: (1) Introduce students to the general subject of approximate
solutions; (2) Introduce and emphasize the finite element method; (3) Help
students understand the strengths and limitations of various approximation
solution techniques; and (4) Develop in the students the ability to assess
the correctness of approximate solutions.
Quite often in the past, introductory finite element texts were written on
the assumption that students possessed a background in structural mechan-
ics. This is evidenced by the common use of simple bar and beam examples
to illustrate the development of element equations. The present reality is,
however, that a wider range of students is striving to learn the finite el-
ement method. For example, here at the University of Delaware, CIEG
401/601, though populated largely by students from Mechanical and Civil
Engineering, is also taken by students from the College of Marine Studies.
Water Resources, and Health and Exercise Sciences. Consequently, struc-
tural mechanics concepts, though well understood by engineers, represent
potential stumbling blocks for students with non-engineering backgrounds.
In order to address the changing make-up of the student body enrolled
in introductory finite element classes, this book emphasizes fairly generic
forms of differential equations. The approximate solution techniques stud-
ied are applied to these generic problems; once introduced and "digested,"
these techniques are then applied to specific physical problems such as heat
conduction, elastostatics, flow through porous media, etc.
Many of the chapters have been designed to be quite modular. As such.
certain sections or chapters can be omitted from a course plan without
detrimentally affecting the understanding of material in subsequent chap-
ters. A brief overview of the respective chapters is presented at the end of
Chapter 1.
Victor N.Kaliakin
Acknowledgements
Very rarely do individuals work successfully in a vacuum. The prepara-
tion and refinement of this book over the past ten years have been greatly
aided by the comments of students who have taken the introductory course
in approximate solution techniques. Their help is thankfully acknowledged.
The author is very fortunate to have studied under H L. Taylor (Uni-
versity of California, Berkeley) and L. R. Herrmann (University of Califor-
nia, Davis). These pioneers in the finite element method have significantly
influenced the structure and content of these notes.
The author's philosophy towards programming, however peculiar, has
been influenced by past collaborations with Dr. K. J. Perano.
If one phrase is chosen to host summarize approximate solution tech-
niques, it would be:
"When considering a specific numerical procedure, you don't
get something for nothing."
cheers,
V.N.K.
VI
Glossary of Notations and Units
Units
E : denotes units of energy (e.g., Joules, cal, BTU, ft-lb)
F : denotes units of force (e.g., Newtons, dynes, pounds, etc.)
L : denotes units of length (e.g., meters, centimeters, inches, etc.)
m : denotes units of mass (e.g., kilograms, grams, slugs, etc.)
Q : denotes units of charge (e.g., Coulombs)
t : denotes units of time (e.g., seconds, minutes, etc.)
T : denotes units of temperature (e.g., degrees Kelvin, Fahrenheit, etc.
Sets
U : union (.OR.)
n : intersection (.AND.)
0 : empty set
£ : is member of (a set)
£ : is not a member of (a set)
C : is subset of (contained in)
<£_ : is not a subset of (not contained in)
=> : implies
<=X if and only if
V : for all
3 : there exists
B : such that
Miscellaneous Integers
Neddoj = number of element displacement degrees of freedom.
-\ n — total number of nodes in an element.
N t = total number of points used to describe the element geometry.
p
Nrowb = number of rows in the strain-displacement matrix B.
Nsdim = spatial dimension of the analysis (1, 2 or 3).