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2004/10/1 NTOU, MSVLAB1
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工程數學教學經驗談
陳正宗海洋大學 特聘教授
河海工程學系Oct. 1, 2004, NTU, 13:30~13:50
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2004/10/1 NTOU, MSVLAB2
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VOutlines
Introduction ODE Gaussian elimination Double Lapalce transform for Euler-Cauchy
ODE. Poisson integral formula SVD technique Conclusions
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2004/10/1 NTOU, MSVLAB3
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VIntroduction
Students: Quantity (OK) Quality (?) 100% -> 30% -> 15% (Past) 26% -> 52% (Current) Attitude Interest (Tool and Method) Five demonstrative examples
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2004/10/1 NTOU, MSVLAB4
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VODE
: '' 2 ' 0
: , .t t
Q y y y
A y e te
0 0
: '' 0
: , 1,t t
Q y
A y e te y t
Given a two order differential equation
why t occurs ?
(Wronskian, variation of parameters, L’Hospital rule……)
Special case:
M
2004/10/1 NTOU, MSVLAB5
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VGaussian elimination
5 4 1 0 0
4 6 4 1 1
1 4 6 4 0
0 1 4 5 0
p
q
r
s
5 4 1 0 0
4 6 4 1 1
1 4 6 4 0
0 1 4 5 0
T T
p
qa a a
r
s
Solve linear algebraic equation
Matrix operation for Guassian elimination
NASTRAN: DMAP (Direct Matrix Abstract Programming)
Bathe: Substructure (Superelement, substructure, Guyan
reduction, congruent transformation)
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. . . .1
. . . .8
75
14
. . . .7
6
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2004/10/1 NTOU, MSVLAB6
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Gaussian elimination (Cont.)
. . . .1
. . . .1
. . . .8
75
14
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6
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2004/10/1 NTOU, MSVLAB7
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Double Lapalce transform for Euler-Cauchy ODE
2 '' '( ) ( ) ( ) 0at y t bty t cy t
( ) 1 ( 1) '1 1 0( ) ( ) ( ) ( ) 0n n n n
n na t y t a t y t a ty t a y t
FF ( ( )) 2 ( )f t f t
HH ( ( )) ( )f t f t
Eurler-Cauchy ODE
a, b, c are constants, y is the function of t
Higher order Eurler-Cauchy ODE
F and H are Fourier and Hilbert transforms, respectively.
LL (Euler-Cauchy ODE) = origin Euler-Cauchy ODE
L : Laplace transform
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2004/10/1 NTOU, MSVLAB8
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VPoisson integral formula
( ) | ( )x Bu x f
G. E.: xxu ,0)(2
B. C. :
)(fu
a
Traditional method
R 'R
Image source
Null-field integral equation method
Reciprocal radii method
Poisson integral formula
Image concept
Methods
Free of image concept
Searching the image point
Degenerate kernel
2 22
2 20
1( , ) ( )
2 2 cos( )
au f d
a a
2 ( ) 0u x
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2004/10/1 NTOU, MSVLAB9
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Searching the image point by using degenerate kernels
x
s
,)],(cos[)(1
ln),(
,)],(cos[)(1
ln),(ln)ln(),(
1
1
m
mEF
m
mIF
FRm
R
msxU
RmRm
RsxUsxrsxU
Fundamental solution:
RmR
msx
m
m
,)](cos[)(
1lnln
1
RmRm
Rsxm
m
,)](cos[)(1
lnln1
R
a
RR
R
R 22
1
2
21
21
( ; , ) ln | | ln | | ln ln
1{ln ( ) cos[ ( )]}
1{ln( ) ( ) cos[ ( )]} ln ln
1ln( ) [( ) ( ) ]cos[ ( )], 0 .
G
m
m
m
m
m m
m
U x s s x s x s a R
R mm R
a Rm a R
R m a
R Rm R
a m R a
sB
xsxsxssxUG ),()(),;(2 BxssxUG ,0),;(
xs
a
s
2a
R
.
x.
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2004/10/1 NTOU, MSVLAB10
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Free of image point - null-field integral equation in conjunction with degenerate kernels
xsdBstxsUsdBsuxsTxuB
IF
B
IF ,)()(),()()(),()(2 a
Bx ),( ),( Rs Bx
BcEF
B
EF xsdBstxsUsdBsuxsT ),()(),()()(),(0
1
0 ))sin()cos(()()(n
nn nbnaafsu
.))sin()cos(()(1
0
n
nn nqnppst
Degenerate kernel
Unknown coefficients
2
00 1 1
2
0 1
1( , ) 1 2 ( ) cos[ ( )] ( cos( ) sin( ))
2
11 2 ( ) cos[ ( )] ( )
2
mn n
m n
m
m
u m a a n b n da
m f da
c
unknown
specified
Fundamental solution
Green’s identity
,ln),( rsxU F
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2004/10/1 NTOU, MSVLAB11
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VSVD technique
nmA
11 mnnm bxA
nm
T nnnmmmnmA
nm
n
00
0
0
1
，
A matrix , m is the number of function, n is the unknown number.
We can get
SVD
and are unitary matriceswhere
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2004/10/1 NTOU, MSVLAB12
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SVD for Continuum Mechanics
RUVRF
dx
dX
dx = F dX
Xx
F VR RU F : deformation gradient
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2004/10/1 NTOU, MSVLAB13
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VPrincipal directions
stretching rotation
undeformed
stretching
rotation
undeformeddeformed
deformed
RUF
VRF
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2004/10/1 NTOU, MSVLAB14
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V Meaning of and
Spurious system (Chen et. al. Royal Society, 2001) True system
Deformed system (Chen et. al. IJCNAA, 2002) Undeformed system
Degenerate system (Chen et. al. IJNME, 2004) Normal system
Fictitious system (Chen et. al. JCA, Rev., 2004) True system
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2004/10/1 NTOU, MSVLAB15
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VConclusions
Five examples were demonstrated for the teaching of engineering mathematics.
Teaching and research merge may have the opportunity to merge together.
How to teach eng. math. for current students is a challenge to us.
Not only tools but also technique should be considered to strengthen our teaching.
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2004/10/1 NTOU, MSVLAB16
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歡迎參觀海洋大學力學聲響振動實驗室
烘培雞及捎來伊妹兒
URL: http://ind.ntou.edu.tw/~msvlab/
Email: [email protected]
