mech345 introduction to finite element methods chapter 1 numerical methods - introduction
TRANSCRIPT
MECH345 Introduction to Finite Element Methods
Chapter 1
Numerical Methods
- Introduction
Numerical Methods - Introduction
Numerical Methods - Introduction
Definition: - Methods that seek quantitative approximations to the solutions of mathematic problems.
Applications: - structural and stress analysis - thermal analysis - dynamic analysis - acoustic analysis - manufacturing process modeling - fluid mechanics - ………….
Numerical Integration Calculate: dxxfI
b
a
• Newton – Cotes integration
• Trapezoidal rule – 1st order Newton-Cotes integration
• Trapezoidal rule – multiple application
)()()(
)()()( 1 axab
afbfafxfxf
b
a
b
a
bfafabdxxfdxxfI
2
)()()()()( 1
n
n
n x
x
x
x
b
a
x
x
x
x
n dxxfdxxfdxxfdxxfdxxfI1
2
10
1
0
)()()( )()(
1
1
)()(2)(2
n
ii bfxfaf
hI
Numerical Integration Calculate: dxxfI
b
a
• Newton – Cotes integration
• Simpson 1/3 rule – 2nd order Newton-Cotes integration
)())((
))(()(
))((
))(()(
))((
))(()()( 2
1202
101
2101
200
2010
212 xf
xxxx
xxxxxf
xxxx
xxxxxf
xxxx
xxxxxfxf
b
a
b
a
xfxfxfxxdxxfdxxfI
6
)()(4)()()()( 210
022
Numerical Integration Calculate: dxxfI
b
a
• Gauss Quadrature
)(2
)()(
2
)(2
)()()(
bfab
afab
bfafabI
)()( 1100 xfcxfcI
Trapezoidal Rule: Gauss Quadrature:
Choose according to certain criteria1010 ,,, xxcc
Numerical Integration Calculate: dxxfI
b
a
• Gauss Quadrature
• 2pt Gauss Quadrature
• 3pt Gauss Quadrature
3
1
3
11
1
ffdxxfI
77.055.0089.077.055.01
1
fffdxxfI
111100
1
1
nn xfcxfcxfcdxxfI
ab
axx
)(2
1~Let:
xdxabbafabdxxfb
a
~~)(2
1)(
2
1)(
2
1)(
1
1
Numerical Integration - Example Calculate: dxxeI x
1
0
sin
• Trapezoidal rule
• Simpson 1/3 rule
• 2pt Gauss quadrature
• Exact solution90933.0
2
cossinsin
1
0
1
0
xexe
dxxeIxx
x
Linear System Solver Solve: bAx
• Gauss Elimination: forward elimination + back substitution
Example:
2
3
0
610
960
621
3
2
1
x
x
x
2
3
0
031
322
621
3
2
1
x
x
x
23
3
0
21500
960
621
3
2
1
x
x
x
Linear System Solver Solve: bAx
• Gauss Elimination: forward elimination
Example:
Linear System Solver Solve: bAx
• Gauss Elimination: back substitution
Example:
Computer
sum
Linear System Solver Solve: bAx
• Gauss Elimination: forward elimination + back substitution
Pseudo code:
Forward elimination: Back substitution:
Do k = 1, n-1Do i = k+1,n
Do j = k+1, nkk
ik
a
ac
kii
kjijij
cbbb
caaa
Do ii = 1, n-1i = n – iisum = 0Do j = i+1, nsum = sum +
ii
ii a
sumbb
jijba
nn
nn
bb
a
Finite Difference Method
Example 1: 0 (0)du
u u udt
t
u 1 0 0 0 00
1t
duu u t u u t u t
dt
Find in 0,1u
1
2
2 1 1 1 0 1t t
duu u t u u t u t
dt
0 1
1
1 0 1N
N
N Nt t
duu u t u t
dt
Selection of : t
1 0 1t
N N
Finite Difference Method
Example 2: -02 5 (0)tdu
u e u udt
1
1
01 0 0 0 0 0
0
2 1 1 1 1 1
2 5 2 5
2 5 2 5
t
t t
t t
duu u t u u e t u u t
dt
duu u t u u e t u u e t
dt
Find in 0,1u
Finite Difference Method
Example 3: 1)0( 2 utudt
du
1
21 0 0 0 0
0
22 1 1 1 1
1 1t
t t
duu u t u u t t t t
dt
duu u t u u t t
dt
in 0,1uFind
1 i
it t i t
N N
0 1
1
21 1 1 1
i
i i i i it t
duu u t u u t t
dt