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Nation Taiwan Ocean University Department of Harbor and River June 11, 2015 pp.1 Null-field Integral Equation Approach for Solving Stress Concentration.

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1 Nation Taiwan Ocean University Department of Harbor and River June 11, 2015 pp.1 Null-field Integral Equation Approach for Solving Stress Concentration Problems with Circular Boundaries 研 究 生 : 陳柏源 指導教授 : 陳正宗博士 日 期 : 2005/06/29 15:00-16:20 國立台灣海洋大學河海工程學系 結構組 碩士班論文口試

2 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.2 Outlines  Review the researches of three theses  Present method Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Image technique for solving scattering problems of half-plane  Numerical examples Laplace problem Helmholtz problem  Green’s function for annular Laplace problems  Conclusions

3 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.3 Outlines  Review the researches of three theses  Present method Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Image technique for solving scattering problems of half-plane  Numerical examples Laplace problem Helmholtz problem  Green’s function for annular Laplace problems  Conclusions

4 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.4 Review the researches of three theses MSVLAB Kao Jeng-Hong Wu An-Chien Chen Po-Yuan Regularized meshless method Null-field integral approach Laplace equation 1.Anti-plane shear problems 2.Anti-plane piezoelectricity problems Helmholtz equation 1.Acoustic eigenproblem --interior problem Laplace equation 1.Anti-plane piezoelectricity and in-plane electrostatic problems 2.Anti-plane elasticity problems Laplace equation 1.Torsion problems 2.Bending problems Helmholtz equation 1.Stress concentration factor of cavity problems 2.Surface amplitude of cavity or inclusion problems --exterior problem

5 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.5 Comparison with Chen and Wu Similar partDifferent part Chen 1.Formulation 2.Cavity and/or inclusion problem 1.Laplace and Helmholtz problem 2.Half-plane problem 3.Application of problem Wu 1.Only Laplace problem 2.Full-plane problem 3.Application of problem

6 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.6 Organization of the thesis Thesis Engineering problems Laplace problems Helmholtz problems Green’s function Chapter 5 Derivation of the Green’s function for annular Laplace problems Chapter 4 1.Half-plane problems with a cavity subject to the incident SH-wave 2.Half-plane problems with inclusions subject to the incident SH-wave Chapter 3 1.Torsion problem with circular holes 2.Bending problem with circular holes Semi-analytical approach Analytical approach Null-field approach

7 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.7 Goal  To develop a systematic approach approach in conjunction with Fourier series, degenerate kernels and adaptive observer system for solving Laplace and Helmholtz problems with multiple circular boundaries.  Advantages ： Mesh free. Well-posed model Free of CPV and HPV. Elimination of boundary-layer effect Exponential convergence

8 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.8 Outlines  Review the researches of three theses  Present method Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Image technique for solving scattering problems of half-plane  Numerical examples Laplace problem Helmholtz problem  Green’s function for annular Laplace problems  Conclusions

9 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.9 Present approach Degenerate kernel Fundamental solution CPV and HPV No principal value?

10 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.10 Present approach Exterior problem Interior problem Advantages of degenerate kernel 1.No principal value 2.Elimination of boundary-layer effect 3.Well-posed

11 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.11 Outlines  Review the researches of three theses  Present method Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Image technique for solving scattering problems of half-plane  Numerical examples Laplace problem Helmholtz problem  Green’s function for annular Laplace problems  Conclusions

12 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.12 Expansions of fundamental solution (2D)  Laplace problem--  Helmholtz problem-- O s x x

13 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.13 Limiting process of the Helmholtz problem  Fundamental solution  Degenerate kernel Rigid body term

14 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.14 U(s,x) T(s,x)

15 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.15 Jump behavior between domain with complementary domain

16 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.16 Outlines  Review the researches of three theses  Present method Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Image technique for solving scattering problems of half-plane  Numerical examples Laplace problem Helmholtz problem  Green’s function for annular Laplace problems  Conclusions

17 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.17 Image technique for solving half- plane problem Alluvial h a SH-Wave Matrix SH-Wave Free surface

18 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.18 Take free body Inclusion Matrix SH-Wave Matrix SH-Wave Matrix

19 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.19 Linear algebraic system  Matrix field  Inclusion field  Two constrains

20 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.20 Flowchart of present method Degenerate kernel Fourier series Collocation point and matching B.C. Adaptive observer system Linear algebraic equation Fourier coefficients Potential of domain point Surface amplitude Stress field Vector decomposition Numerical Analytical

21 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.21 Outlines  Review the researches of three theses  Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Image technique for solving scattering problems of half-plane  Numerical examples Laplace problem Helmholtz problem  Green’s function for annular Laplace problems  Conclusions

22 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.22 Numerical examples  Laplace problem Torsion problem for a bar Bending problem for a cantilever beam  Helmholtz problem Half-plane problems with a cavity subject to the incident SH-wave Half-plane problems with inclusions subject to the incident SH-wave

23 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.23 Torsion problem for a bar --CMES, Vol. 12(2)

24 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.24 Torsional rigidity (one hole) Exact solution [68]0.978720.951370.903120.824730.761680.744540.724460.699680.66555 Present method L=200.978720.951370.903120.824730.761680.744550.724510.699910.66705 L=100.978720.951370.903120.824760.762440.746030.727480.706160.68111 Caulk ’ s method (BIE) [14] 40 divisions0.978720.951370.903160.824970.762520.745690.726050.701780.66732 20 divisions0.978730.951400.903280.825740.765830.750570.733670.714730.69321 Torsional rigidity:

25 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.25 Torsional rigidity (equal angle) Caulk (First-order Approximate) [14] 0.86610.82240.7934 Caulk (BIE formulation) [14] 0.86570.82140.7893 Present method (L=10) 0.86570.82140.7893

26 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.26 Torsional rigidity (Ling’s problem) Caulk (First-order Approximate) [14] 0.87390.87410.7261 Caulk (BIE formulation) [14] 0.87130.87320.7261 Ling’s results0.88090.80930.7305 Present method (L=10) 0.87120.87320.7244 Because there is no apparent reason for the unusually large difference in the second example, Ling’s rather lengthy calculations are probably in error here. --ASME JAM 10%

27 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.27 Convergence

28 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.28 Bending problem for a cantilever beam

29 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.29 Stress concentration Point B Point C Stress concentration

30 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.30 Stress concentration at point B Present method Naghdi’s results Steele & Bird The two approaches disagree by as much 11%. The grounds for this discrepancy have not yet been identified. --ASME Applied Mechanics Review

31 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.31 Advantages of the present method Elimination of boundary-layer effect Convergence test of Fourier series

32 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.32 Two holes problem Present method Steele & Bird’s result [6] Point P

33 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.33 Contour of stress concentration Steele & Bird’s result [6] Present method

34 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.34 Numerical examples  Laplace problem Torsion problem for a bar Bending problem for a cantilever beam  Helmholtz problem Stress concentration factor of cavities problem subject to the incident SH-wave Half-plane problems with inclusions subject to the incident SH-wave

35 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.35 A full-plane problem with two cavities subject to the incident SH-wave. SH-wave Case 1 Case 2 BC of the Honein’s problem BC of the present problem

36 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.36 Shear stress ( ) around the smaller cavity Case 1 Case 2 k=0.001 k=0.001 Present method Honein’s results Shear stress

37 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.37 A half-plane problem with a circular cavity subject to incident SH-wave. h SH-wave a x y

38 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.38 Nondimensional stress ( ) around the cavity Lin and Liu’s results [89] Present method nondimensional stress

39 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.39 Limiting case of the half-plane problem Pao and Mow’s result [70] (only half). Limiting case of and A full-plane problem with a cavity subject to horizontally SH-wave A full-plane problem with a cavity subject to horizontally SH-wave.

40 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.40 A half-plane problem with a semi- circular alluvial valley subject to the SH- wave Alluvial Matrix h a SH-Wave x y Wave number Dimensionless frequency Governing equation Velocity of shear wave

41 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.41 Surface amplitudes of the alluvial valley problem Present method Manoogian’s results [60] verticalhorizontal

42 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.42 Limiting case of a canyon Present method Manoogian’s results [60] verticalhorizontal

43 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.43 Limiting case of a rigid alluvial valley Present method

44 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.44 Present method Soft-basin effect 14 18 3

45 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.45 A half-plane problem with two alluvial valleys subject to the incident SH-wave Canyon Matrix 3a SH-Wave 房 [93] 將正弦和餘弦函數的正交特性使用錯誤，以 至於推導出錯誤的聯立方程，求得錯誤的結果。 -- 亞太學報 曹 2004

46 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.46 Limiting case of two canyons Present method Tsaur et al.’s results [103] verticalhorizontal

47 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.47 Surface displacements of two alluvial valleys Present method

48 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.48 Inclusion Matrix h SH-Wave a x y A half-plane problem with a circular inclusion subject to the incident SH-wave

49 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.49 Surface displacements of a inclusion problem under the ground surface Present method Tsaur et al.’s results [102] Manoogian and Lee’s results [62] When I solved this problem I could find no published results for comparison. I also verified my results using the limiting cases. I did not have the benefit of published results for comparing the intermediate cases. I would note that due to precision limits in the Fortran compiler that I was using at the time. --Private communication

50 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.50 Limiting case of a cavity problem Present method Lee and Manoogian’s [53] for the cavity case. verticalhorizontal

51 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.51 Limiting case of a rigid alluvial valley Present method

52 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.52 A half-plane problem with two circular inclusions subject to the SH-wave Matrix Inclusion h SH-Wave a a D y x

53 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.53 Surface amplitudes of two-inclusions problem Present method

54 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.54 Limiting case of two-cavities problem Present method Jiang et al. result [95]

55 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.55 Outlines  Review the researches of three theses  Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Image technique for solving scattering problems of half-plane  Numerical examples Laplace problem Helmholtz problem  Green’s function for annular Laplace problems  Conclusions

56 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.56 Derivation of the Green’s function for annular Laplace problems Degenerate kernel Fourier series

57 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.57 Null-field integral equation

58 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.58 Contour plots for the annular Green’s function Analytical solution Analytical solution ( ) Semi-analytical solution Semi-analytical solution ( )

59 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.59 Two limiting cases of the annular Green’s function Chen & Wu’s[31] results Chen & Wu’s[31] results Exterior case Interior case

60 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.60 Two limiting cases of the annular Green’s function Limiting case of the annular Green’s function

61 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.61 Outlines  Review the researches of three theses  Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation Image technique for solving scattering problems of half-plane  Numerical examples Laplace problem Helmholtz problem  Green’s function for annular Laplace problems  Conclusions

62 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.62 Conclusions  A systematic way to solve the Laplace and Helmholtz problems with circular boundaries was proposed successfully in this thesis by using the null-field integral equation in conjunction with degenerate kernels and Fourier series.  The present method is more general for calculating the torsion and bending problems with arbitrary number of holes and various radii and positions than other approach.

63 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.63 Conclusions  When the wave number “k” approaches zero, the Helmholtz problem can be reduced to the Laplace problem. Laplace problem can be treated as a special case of the Helmholtz problem.  Our approach can deal with the cavity problem as a limiting of inclusion problem with zero shear modulus.

64 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.64 Some findings Laplace Helmholtz Ling 1947 Analytical solution Bird & Steele 1992 房營光 1995 Analytical solution Lee & Manoogian 1992 Caulk 1983 Naghdi 1991 Analytical solution Tsaur et al. 2004 Analytical solution Present method Present method (semi-analytical)Tsaur et al. ?

65 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.65 Further research  Following the successful experience of this thesis, extending to the problem of torsional rigidity of a bar with inclusions can be considered as a forum in the future.  The extension to hill scattering problem can be studied by using the present approach.  The Green’s function of eccentric case, mixed BC and multi-medium can be easily solved by using our approach.

66 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.66 Further research  The bi-observer expansion technique for the two point function of source and field systems may be suitable for the eccentric case in a more straightforward way free of adaptive observer system.  Our method can also be applied for problems with different boundaries. How to keep the orthogonal property is the main challenge.

67 Nation Taiwan Ocean University Department of Harbor and River June 11, 2015 pp.67 Thanks for your kind attentions. You can get more information from our website. http://msvlab.hre.ntou.edu.tw/

68 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.68 Separable form of fundamental solution (1D) Separable property continuous jump

69 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.69 Derivation of degenerate kernel  Graf’s addition theorem  Complex variable Real part If Bessel’s function

70 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.70 Fictitious frequencies

71 Mechanics Sound Vibration Laboratory HRE. NTOU http://ind.ntou.edu.tw/~msvlab/ June 11, 2015 pp.71 Derivation of the 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

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1 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. On the equivalence of the.

1 The 27 th Conference on Theoretical and Applied Mechanics --- 學生論文競賽 National Cheng-Kung University, Tainan, Taiwan, R. O. C. On the equivalence of the.

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