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S M V 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 2004/10/1 1 NTOU, MSVLAB Outlines S M V Introduction ODE Gaussian elimination Double Lapalce transform for Euler-Cauchy ODE. Poisson integral formula SVD technique Conclusions 2004/10/1 2 NTOU, MSVLAB Introduction S M V Students: Quantity (OK) Quality (?) 100% -> 30% -> 15% (Past) 26% -> 52% (Current) Attitude Interest (Tool and Method) Five demonstrative examples 2004/10/1 3 NTOU, MSVLAB ODE S M V Given a two order differential equation Q : y '' 2 y ' y 0 A: y et , tet . why t occurs ? (Wronskian, variation of parameters, L’Hospital rule……) Special case: Q : y '' 0 A : y e0t , te0t y 1, t 2004/10/1 4 NTOU, MSVLAB Gaussian elimination S M V . . . . . . . . 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 6 NASTRAN: DMAP (Direct Matrix Abstract Programming) Bathe: Substructure (Superelement, substructure, Guyan reduction, congruent transformation) 2004/10/1 5 NTOU, MSVLAB S M V Gaussian elimination (Cont.) . . . . 1 . . . . 1 . . . . 5 14 . . . . 7 6 8 7 2004/10/1 6 NTOU, MSVLAB S M V 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. 2004/10/1 7 NTOU, MSVLAB S Poisson integral formula M V 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) |xB 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( ) 2004/10/1 Degenerate kernel Poisson integral formula 8 NTOU, MSVLAB Searching the image point by using degenerate kernels S M V .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 m1 ln x s ln R m1 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 2004/10/1 9 NTOU, MSVLAB Free of image point - null-field integral equation in conjunction with degenerate kernels S M V 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, ) xB 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 2004/10/1 m 1 2 ( ) cos[ m ( )] a ( a cos( n ) b sin( n )) 0 n n d 0 m1 a n 1 1 2 1 2 ( ) m cos[m( )] f ( ) d m 1 a 2 0 u( , ) 1 2 2 10 NTOU, MSVLAB SVD technique S M V A matrix Amn, m is the number of function, n is the unknown number. We can get Am n xn 1 bm 1 SVD Amn mm mn Tnn n 0 0 0 , 1 0 m n mn where and are unitary matrices 2004/10/1 11 NTOU, MSVLAB S M V SVD for Continuum Mechanics F VR RU F : deformation gradient dx = F dX dX X x dx F VR RU 2004/10/1 12 NTOU, MSVLAB Principal directions S M V undeformed stretching rotation deformed F RU undeformed stretching rotation deformed F VR 2004/10/1 13 NTOU, MSVLAB Meaning of and S M V Spurious system Deformed system Degenerate system Fictitious system 2004/10/1 (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) 14 True system NTOU, MSVLAB Conclusions S M V 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. 2004/10/1 15 NTOU, MSVLAB S M V 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] 2004/10/1 16 NTOU, MSVLAB