Lesson 32
Element Matrices
The 2-D isoparametric element (continued)
Element Matrices
A scalar variable, , can represent unknown temperature, concentration, or velocity. The
shape functions are obtained from the previous relations discussed in Lesson 30.
Lesson 7
Steady-state Conduction
The 1-D element (continued)
Steady-state conduction
Steady-state conduction of heat in a solid can be
expressed by the equation
d dT
K
Q
dx dx
dT
-K
q
dx
T TL
0<x<L
x=0
x=L
Weighted residual form gives
d dT
WR W
K
Q
Lesson 4
Review of Vectors and Matrices
Vectors
A vector is normally expressed as
uuv uuv
V V x1, x 2 , x 3
or in terms of unit vectors
uuv
V V1$i V2j V3k
likewise
a ijx j a i1x1 a i2 x 2 a i3x 3
i, j1,2,3
Vectors (continued)
Dot product
v
uuvuu
AgB A1i
The Finite Element Method
A self-study course designed for
engineering students
Course Outline
Week 1
Lesson 1: Introduction and overview
Lesson 2: Brief history of the FEM
Lesson 3: Basic concepts
Week 2
Lesson 4: Review of numerical methods
Vecto
Lesson 2
A Brief History of the FEM
Beginnings
Began in earnest in the 1950s to solve
elasticity problems
First person to use the term finite element
method in a publication was Clough in
1960
Early efforts attributed to structural
problems in aircraft
Lesson 5
Method of Weighted Residuals
Classical Solution Technique
The fundamental problem in calculus of variations is to
obtain a function f(x) such that small variations in the
function f(x) will not change the original function
The variational functio
Lesson 12
Solution Methods
The 1-D element (continued)
Matrix solution methods
Once the discretized equation has been established, and the various individual
element operations performed (or assembled), the overall global matrix
must be solved. Since the
Lesson 3
Basic Concepts
Fundamentals
Any continuous quantity (temperature,
displacement, etc.) can be approximated by a
discrete model composed of a set of piecewise
continuous functions
Functions defined using values of continuous
quantities at a finit
Introduction and overview
Introductory Finite Elements
1
Introduction
Hello my name is Darrell Pepper and I am the
instructor for this course
The name of this course is The Finite Element Method
for Mechanical Engineers MEG 704
This course will be taug
Lesson 16
Area Coordinates
The 2-D triangular element (continued)
Area Coordinates Triangular Elements
When using triangular elements, it is generally more advantageous to employ
area coordinates. The Area coordinate system is analogous to a natural
coord
Lesson 10
Time Dependence
The 1-D element (continued)
Time-dependent Problems
In many instances, problems are not steady state. In such cases, the inclusion of
the transient term is required. This leads to the need to also provide initial
boundary conditi
Lesson 9
Natural Coordinate System
The 1-D element (continued)
Natural Coordinate System
It is prudent to use a natural coordinate system. This permits values to be
bounded between 1 (or 0) and 1 and reduces computational round off
and truncation errors w
Lesson 1- Introduction and
overview
Introductory Finite Elements
The beginning
Overview
What to expect
What you want
The course will cover:
1-D, 2-D, and 3-D Elements
Mesh generation
Shape functions
Assembly
Free computer codes (source and executable)
H
Lesson 11
Matrix Formulation
Matrix Equivalent Equations
In most cases, the integral expressions can be easily expressed in
terms of matrix equivalent equations. One need only to glance
at the matrix equation, and it becomes readily apparent what the
coef
Lesson 18
Steady-State Diffusion Equation
The 2-D triangular element (continued)
Steady-state Diffusion Problems
There are many instances where a steady-state conduction solution is needed.
Assume a domain, , exists with boundary, as shown below
with Neum
Lesson 8
Using Variable Coefficients and
Axisymmetric Coordinates
The 1-D element (continued)
Variable coefficients
Values for K, Q, and other parameters may vary as a function of space
(and time). In this case we treat them juse like the unknown variable
Lesson 14
The Mesh - Triangles
The Mesh 2-D Triangles
The element of choice is the linear triangle consisting of 3 vertex nodes
Knowing where to optimally place and size elements is more of an art than a science.
There is a rule of thumb to follow:
place
Lesson 21
Exercises
The 2-D triangular element (continued)
Exercises & Home Work Problems
It is important to apply what you have now learned about 2-D triangular elements. The following
problems are assigned and should be helpful in reinforcing the princi
Lesson 17
Numerical Integration
The 2-D triangular element (continued)
Numerical Integration - Triangles
Using area coordinates allows one to evaluate the integral equations from
integration formulae, as in the one-dimensional case. For a two-dimensional
Lesson 27
Steady-State Diffusion Equation
The 2-D quadrilateral element (continued)
Steady-state Diffusion
We direct our attention once more to determining the steady-state conduction
of heat within a two-dimensional domain. We will generalize the heat
co
Lesson 20
Time Dependence & Bandwidth
The 2-D triangular element (continued)
Time-dependence
Like the 1-D time dependent case, 2-D time dependence involves only the
extension of the coordinate system to a second dimension. The governing
equation for 2-D,
Lesson 24
Shape Functions
The 2-D quadrilateral element (continued)
Shape functions (continued)
As in the two-dimensional triangular element, the two-dimensional quadrilateral
element is based on linear, quadratic, cubic, or higher approximations. The
lin