Preliminaries
Motivation
Computational Modeling
Idealization
Introduction to the Finite Element Method
Lecture 01
P.S. Koutsourelakis
pk285@cornell.edu
369 Hollister Hall
August 31 2009
Last Updated: August 31, 2009
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Organization
Two lectures each week (MW 1:25-2:40) at Olin 218
The course webpage will be the place where all releveant
information will be posted, i.e. lectures slides, homework
assignments, deadlines, readings etc
Prerequisites- Corequisites:
Proficiency in Linear algebra and calculus
Introductory PDE
Programming (in any language)
Introductory continuum meachnics is desirable but not
necessary
Books:
The main book we are going to be following is:
J. Fish and T. Belytschko, A First Course in Finite Elements
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Organization
Other useful books:
T.J.R. Hughes, The Finite Element Method: Linear static
and dynamic finite element analysis, Prentice Hall. 1987.
R.D. Cook, D.S. Malkus, M.E. Plesha, and R.D.Witt,
Concepts and Applications of Finite Element Analysis, 4th
Edition, John Wiley & Sons , 2001.
K.J. Bathe, Finite Element Procedures,Prentice Hall, 1996.
B. Szabó, I. Babuska, Finite Element Analysis, John Wiley
& Sons , 1991.
Final grades will be based on Homeworks (30%), Midterm
(30%) and Final Project (40%).
Disclaimer: In the event of a major campus emergency like an
H1N1 flu outbreak, course requirements, deadlines, and grading
percentages are subject to changes that may be necessitated by
a revised semester calendar or other circumstances. Additional
information will be posted on the course website as needed.
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Organization
CEE6075 Practicum in Finite Element (Prof. Aquino)
This course consists of weekly laboratories on proper use of
finite element software for solving modern engineering
problems.
Registration for CEE6075 is obligatory for CEE Structures
MEng students
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Objectives
The objective of the course is to introduce the basic
mathematical and computational aspects of the finite element
method. Students will get acquainted with:
the fundamental mathematical principles employed,
the linear-algebraic issues involved,
structure of a finite element code,
use of existing finite element software.
modeling considerations for model verification, error
estimation and interpretation of results.
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
What, Who, When, Why
What is the Finite Element Method?
It is a numerical technique for finding approximate solutions to
partial differential equations (PDE).
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
What, Who, When, Why
Who came up with it?
Depends on whether you are talking to a mathematician, an applied
mathematician, an engineer, a computational physicist. In
engineering circles it first appeared in the mid-50’s, picked up steam
in the 1960’s and has been developing hand-in-hand with the
progress in hardware.
(a)
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Lecture 01
John Argyris (1913-2004)
(b)
Ray Clough (1920- )
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
What, Who, When, Why
Why is it important?
Because PDEs are the standard mathematical models in a
wide range of physical systems and mathematical problems.
Heat equation:
∂ 2 T (x, t)
∂T (x, t)
=κ
∂t
∂x 2
where T (x, t) is temparature at location x at time t
Single particle Shröndinger equation:
ρc
∂Ψ(x, t)
1 2
=−
∇ Ψ(x, t) + V (x) Ψ(x, t)
∂t
2m
where |Ψ(x, t)|2 is the the probability a particle is at
location x at time t.
i
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
What, Who, When, Why
Why is it important?
Because PDEs are the standard mathematical models in a
wide range of physical systems and mathematical problems.
Wave equation of a stretched string:
2
∂2u
2 ∂ u(x, t)
=
c
∂t 2
∂x 2
where u(x, t) is the displacement at location x at time t
Elastodynamics
Ci,j,k ,l
∂ 2 ui (x, t)
∂ 2 uk (x, t)
+ ρbi = ρ
∂xj ∂xl
∂t 2
where uk (x, t) is the dispacement in the k direction at time
t of a point at x
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Cornell University
Maxwell’s equation in electromagnetism
Lecture 01
Einstein’s equation in general relativity
Preliminaries
Motivation
Computational Modeling
Idealization
The four steps in computational modeling
1
2
3
4
Mathematical Idealization: What are the important
quantities (functions) and what are their mathematical
relations.
Discretization: Can we express or even approximate the
solution of the governing PDEs with respect to a finite
number of variables. - Converting differential equations to
algebraic A (x) = b.
Solution: How can we solve the resulting system of
algebraic equations A (x) = b
Verification/Validation: Did we corerctly solve the
mathemqatical equations? Does the mathematical model
correspond to the actual physical problem?
(In this class we are mostly going to focus on Discretization
and Solution)
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Idealization
Example
Consider cars travelling on a straight road [0, L] and let ρ(x, t) denote
their density at x ∈ [0, L] and at time t. So if N(t) cars are in [0, L] at
RL
time t then N(t) = 0 ρ(x, t) dx. Let v (x, t) be the the velocity of the
cars at location x and at time t.
1
Conservation Law (conservation of “mass”):
∂ρ(x, t)
∂ (ρ(x, t)v (x, t))
=−
∂t
∂x
2
Constitutive Law: v (x, t) = vmax e−ρ(x,t)/ρ0
3
Boundary Conditions : ρ(0, t) = ρ1 ,
4
Initial Conditions t = 0: ρ(x, 0) = ρ̂0 (x)
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Lecture 01
v (L, t) = vL
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Idealization
Example
Consider cars travelling on a straight road L and let ρ(x, t) denote the
density of cars
R at x ∈ L and at time t. So if N(t) cars are in L at time t
then N(t) = L ρ(x, t) dx. Let v (x, t) be the the velocity of the cars at
location x and at time t.
Initial Boundary Value Problem (IBVP)
∂ρ
ρ
−ρ(x,t)/ρ0 ∂ρ
= −vmax e
1−
∀x ∈ (a, b), t > 0
∂t
∂x
ρ0
with the following initial/boundary conditions:
vL
ρ(0, t) = ρ1 , ρ(b, t) = −ρ ln
, ρ(x, 0) = ρ̂0 (x)
vmax
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Idealization
Example
Consider a bar of length L, cross-sectional area A which is held fixed
on the left end and pulled with a force F on the right end and
stretched with a distributed force b(x) along its length. Whats will be
the deformation of the bar u(x) at each point x?
b(x)
F
A
u(x)
L
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Idealization
Example
Consider a bar of length L, cross-sectional area A which is held fixed
on the left end and pulled with a force F on the right end and
stretched with a distributed force b(x) along its length. Whats will be
the deformation of the bar u(x) at each point x?
1
Conservation Law (conservation of linear momentum)
dσ(x)
+ b(x) = 0, σ(x) : stress
dx
Constitutive Law
du(x)
E : elastic modulus
σ(x) = E
dx
Boundary Conditions [0, L]
u(0) = 0,
A σ(L) = F
A
2
3
pk285@cornell.edu
Lecture 01
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Idealization
Example
Consider a bar of length L, cross-sectional area A which is held fixed
on the left end and pulled with a force F on the right end and
stretched with a distributed force b(x) along its length. Whats will be
the deformation of the bar u(x) at each point x?
Boundary Value Problem (BVP)
EA
d 2 u(x)
+ b(x) = 0 ∀x ∈ (0, L)
dx 2
with the following boundary conditions:
u(0) = 0,
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Lecture 01
EA
du
|x=L = F
dx
Cornell University
Preliminaries
Motivation
Computational Modeling
Idealization
Success Stories
Rapid, accurate, reliable solution of governing PDEs with
the Finite Element method has allowed us to perform
virtual experiments.
understand and control complex multi-physics phenomena
analyze and design multi-component engineering
systems.
Areas that have seen dramatic growth:
Materials
Industrial and defense applications
Medicine
Energy and Environment
pk285@cornell.edu
Lecture 01
Cornell University