Lecture Notes: Introduction to Finite Element Method |
Chapter 1. Introduction |
Examples: |
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(1) |
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Checking, |
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1 −1 |
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(2) |
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Checking, |
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1 −1 |
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If det A = 0 (i.e., A is singular), then A-1 does not exist!
The solution of the linear system of equations (Eq.(1)) can be expressed as (assuming the coefficient matrix A is nonsingular)
x = A −1b
Thus, the main task in solving a linear system of equations is to found the inverse of the coefficient matrix.
© 1997-2003 Yijun Liu, University of Cincinnati |
11 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 1. Introduction |
Solution Techniques for Linear Systems of Equations
•Gauss elimination methods
•Iterative methods
Positive Definite Matrix
A square (n×n) matrix A is said to be positive definite, if for all nonzero vector x of dimension n,
xT Ax > 0
Note that positive definite matrices are nonsingular.
Differentiation and Integration of a Matrix
Let
A(t) = [aij (t)]
then the differentiation is defined by
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daij (t) |
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A(t) = |
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dt |
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and the integration by
∫A(t)dt = ∫a (t)dt
ij
© 1997-2003 Yijun Liu, University of Cincinnati |
12 |
Lecture Notes: Introduction to Finite Element Method |
Chapter 1. Introduction |
Types of Finite Elements
1-D (Line) Element
(Spring, truss, beam, pipe, etc.)
2-D (Plane) Element
(Membrane, plate, shell, etc.)
3-D (Solid) Element
(3-D fields - temperature, displacement, stress, flow velocity)
© 1997-2003 Yijun Liu, University of Cincinnati |
13 |