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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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Outlines
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Introduction
ODE
Gaussian elimination
Double Lapalce transform for Euler-Cauchy
ODE.
Poisson integral formula
SVD technique
Conclusions
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Introduction
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Students: Quantity (OK) Quality (?)
100% -> 30% -> 15% (Past)
26% -> 52% (Current)
Attitude
Interest (Tool and Method)
Five demonstrative examples
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ODE
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Given a two order differential equation
Q : y '' 2 y ' y  0
A: y  et , tet .
why t occurs ? (Wronskian, variation of parameters,
L’Hospital rule……)
Special case:
Q : y ''  0
A : y  e0t , te0t  y  1, t
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Gaussian elimination
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. . . .
. . . .
1
Solve linear algebraic equation
0   p  0 
 5 4 1
 4 6 4 1   q  1 

     
 1 4 6 4   r  0 


 0 1 4 5   s  0 
1
Matrix operation for Guassian elimination
 5 4 1 0   p 
 4 6 4 1   q 
 a    a T
aT 
 1 4 6 4   r 


 0 1 4 5   s 
0 
1 
 
 
0 
0 
. . . .
. . . .
8
7
5
14
7
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NASTRAN: DMAP (Direct Matrix Abstract Programming)
Bathe: Substructure (Superelement, substructure, Guyan
reduction, congruent transformation)
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Gaussian elimination (Cont.)
. . . .
1
. . . .
1
. . . .
5
14
. . . .
7
6
8
7
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Double Lapalce transform for
Euler-Cauchy ODE
Eurler-Cauchy ODE
at 2 y '' (t )  bty ' (t )  cy (t )  0
a, b, c are constants, y is the function of t
Higher order Eurler-Cauchy ODE
ant n y ( n ) (t )  an 1t n 1 y ( n 1) (t ) 
 a1ty ' (t )  a0 y (t )  0
LL (Euler-Cauchy ODE) = origin Euler-Cauchy ODE
L : Laplace transform
FF ( f (t ))  2 f ( t )
HH ( f (t ))   f (t )
F and H are Fourier and Hilbert transforms, respectively.
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Poisson integral formula
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Methods
G. E.:  2 u ( x)  0, x  
B. C. : u  f ( )
Image concept
Free of image concept
 2u ( x )  0
a

R
u( x) |xB  f ( )

2
0
Null-field integral
equation method
R'
Image source
1
u(  , ) 
2
Searching the
image point
Reciprocal radii
method
Traditional method


a2   2
 2
 f ( )d
2
a



2
a

cos(



)


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Degenerate
kernel
Poisson integral formula
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Searching the image point by using degenerate
kernels
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.x
Fundamental solution:
 1 
 I
U F ( x, s )  ln R  
( ) m cos[m(   )],


m 1 m R
U F ( x, s )  ln( r )  ln x  s  
 1 R
U FE ( x, s )  ln   
( ) m cos[m(   )],

m

m

1

x

ln x  s  ln   
1 R m
( ) cos[m(   )],   R
m 

1  m
( ) cos[m(    )],   R 
m R
m1
ln x  s   ln R   
m1
R



R
 R 
2
R

  R,
  R,
a
x
s
.
B
 s   a
s
2
R
2UG ( x; s, s)   ( x  s)   ( x  s), x  
UG ( x; s, s)  0, x  B
a2
R
U G ( x; s, s)  ln | x  s |  ln | x  s |  ln a  ln R
1  m
( ) cos[m(   )]}
s
m 1 m R

 {ln R  
{ln(

a2
1  R
)   ( 2 ) m cos[m(   )]}  ln a  ln R
R
a
m 1 m

R
1 
 R
 ln( )   [( ) m  ( 2 ) m ]cos[m(   )], 0    R.
a
R
a
m 1 m
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Free of image point - null-field integral equation in
conjunction with degenerate kernels
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U F ( x, s)  ln r ,
2 u( x) 
T
B
0

B
c
Green’s
identity
Fundamental
solution
I
F ( s, x) u ( s) dB( s) 
U
B

I
F ( s, x) t ( s) dB( s),
x 
s  ( R,  )
xB
TFE ( s, x) u ( s) dB( s)
 U FE ( s, x) t ( s) dB(s),

u (s)  f ( )  a0 

t (s)  p0 

 (a
n
x 
B
c
cos(n )  bn sin(n ))
a
x  (,  )  B
specified
n 1
( pn cos(n )  qn sin(n )). unknown
n 1
Degenerate kernel
Unknown
coefficients
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
 



m
1

2
(
)
cos[
m
(



)]
a

(
a
cos(
n

)

b
sin(
n

))




0
n
n

 d
0  m1 a

n 1

 
1 2 


1

2
( ) m cos[m(   )] f ( ) d



m 1 a
2 0 

u(  , ) 
1
2
2
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SVD technique
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A matrix Amn, m is the number of function, n is the unknown number.
We can get
Am  n xn 1  bm 1
SVD
Amn  mm mn  Tnn
 n
 

   0

 
 0
 0
，
  
 1

 
 0  m  n
mn
where  and  are unitary matrices
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SVD for Continuum Mechanics
F  VR  RU
F : deformation gradient
dx = F dX
dX
X
x
dx
F  VR  RU
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Principal directions
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undeformed
stretching
rotation
deformed
F  RU
undeformed
stretching
rotation
deformed
F  VR
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Meaning of  and 
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 
Spurious system
Deformed system
Degenerate system
Fictitious system
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(Chen et. al. Royal Society, 2001)
True system
(Chen et. al. IJCNAA, 2002)
Undeformed system
(Chen et. al. IJNME, 2004)
Normal system
(Chen et. al. JCA, Rev., 2004)
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Conclusions
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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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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]
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