M S V 2004/10/1 NTOU, MSVLAB 1 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

M S V 2004/10/1 NTOU, MSVLAB 2 Outlines Introduction ODE Gaussian elimination Double Lapalce transform for Euler-Cauchy ODE. Poisson integral formula SVD technique Conclusions

M S V 2004/10/1 NTOU, MSVLAB 3 Introduction Students: Quantity (OK) Quality (?) 100% -> 30% -> 15% (Past) 26% -> 52% (Current) Attitude Interest (Tool and Method) Five demonstrative examples

M S V 2004/10/1 NTOU, MSVLAB 4 ODE Given a two order differential equation why t occurs ? (Wronskian, variation of parameters, L ’ Hospital rule …… ) Special case:

M S V 2004/10/1 NTOU, MSVLAB 5 Gaussian elimination Solve linear algebraic equation Matrix operation for Guassian elimination NASTRAN: DMAP (Direct Matrix Abstract Programming) Bathe: Substructure (Superelement, substructure, Guyan reduction, congruent transformation).. 1 1

M S V 2004/10/1 NTOU, MSVLAB 6 Gaussian elimination (Cont.).. 1 1

M S V 2004/10/1 NTOU, MSVLAB 7 Double Lapalce transform for Euler-Cauchy ODE FF HH 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

M S V 2004/10/1 NTOU, MSVLAB 8 Poisson integral formula G. E.: B. C. : a Traditional method 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

M S V 2004/10/1 NTOU, MSVLAB 9 Searching the image point by using degenerate kernels Fundamental solution: B.

M S V 2004/10/1 NTOU, MSVLAB 10 Free of image point - null-field integral equation in conjunction with degenerate kernels B Degenerate kernel Unknown coefficients unknown specified Fundamental solution Green’s identity

M S V 2004/10/1 NTOU, MSVLAB 11 SVD technique ， A matrix, m is the number of function, n is the unknown number. We can get SVD and are unitary matrices where

M S V 2004/10/1 NTOU, MSVLAB 12 SVD for Continuum Mechanics dx dX dx = F dX X x F : deformation gradient

M S V 2004/10/1 NTOU, MSVLAB 13 Principal directions stretching rotation undeformed stretching rotation undeformed deformed

M S V 2004/10/1 NTOU, MSVLAB 14 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

M S V 2004/10/1 NTOU, MSVLAB 15 Conclusions 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.

M S V 2004/10/1 NTOU, MSVLAB 16 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: