1. M M S S V V 0 Scattering of flexural wave in thin plate with multiple holes by using the null-field integral equation method Wei-Ming Lee 1, Jeng-Tzong Chen 2 Ching-Lun Chien 1, Yung-Cheng Wang 1 1 Department of Mechanical Engineering, China Institute of Technology, Taipei, Taiwan 2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan 2008 年 05 月 14 日 台北科技大學 National Taiwan Ocean University MSVLAB ( 海大河工系 ) Department of Harbor and River Engineering

2. M M S S V V 1 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

3. M M S S V V 2 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

4. M M S S V V 3 Introduction Circular holes can reduce the weight of the whole structure or to increase the range of inspection. Geometric discontinuities result in the stress concentration, which reduce the load carrying capacity. The deformation and corresponding stresses produced by the dynamic force are propagated through the structure in the form of waves.

5. M M S S V V 4 Scattering At the irregular interface of different media, stress wave reflects in all directions scattering The scattering of the stress wave results in the dynamic stress concentration

6. M M S S V V 5 Overview of numerical methods 5 PDE- variational IE DE Domain Boundary MFS,Trefftz method MLS, EFG 開刀 把脈把脈 針灸針灸

7. M M S S V V 6 Literature review From literature reviews, few papers have been published to date reporting the scattering of flexural wave in plate with more than one hole. Kobayashi and Nishimura pointed out that the integral equation method (BIEM) seems to be most effective for two-dimensional steady-state flexural wave. Improper integrals on the boundary should be handled particularly when the BEM or BIEM is used.

8. M M S S V V 7 Motivation Numerical methods for engineering problems FDM / FEM / BEM / BIEM / Meshless method BEM / BIEM Treatment of singularity and hypersingularity Boundary-layer effect Ill-posed model Convergence rate

9. M M S S V V 8 Objective For the plate problem, it is more difficult to calculate the principal values Our objective is to develop a semi-analytical approach to solve the scattering problem of flexural waves and dynamic moment concentration factors in an infinite thin plate with multiple circular holes by using the null-field integral formulation in conjunction with degenerate kernels and Fourier series.

10. M M S S V V 9 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

11. M M S S V V 10 Flexural wave of plate Governing Equation: is the out-of-plane displacement is the wave number is the biharmonic operator is the domain of the thin plates u(x)u(x) ω is the angular frequency ρ is the surface density D is the flexural rigidity h is the plates thickness E is the Young’s modulus ν is the Poisson’s ratio

12. M M S S V V 11 Problem Statement Problem statement for an infinite plate with multiple circular holes subject to an incident flexural wave

13. M M S S V V 12 The integral representation for the plate problem

14. M M S S V V 13 Kernel function The kernel function is the fundamental solution which satisfies

15. M M S S V V 14 The slope, moment and effective shear operators slope moment effective shear

16. M M S S V V 15 Kernel functions In the polar coordinate of

17. M M S S V V 16 Direct boundary integral equations Among four equations, any two equations can be adopted to solve the problem. displacement slope with respect to the field point x normal moment effective shear force

18. M M S S V V 17 x s O x Expansion Degenerate kernel (separate form) Fourier series expansions of boundary data

19. M M S S V V 18 Boundary contour integration in the adaptive observer system

20. M M S S V V 19 Vector decomposition

21. M M S S V V 20 Transformation of tensor components

22. M M S S V V 21 Linear system where H denotes the number of circular boundaries

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24. M M S S V V 23 Techniques for solving scattering problems

25. M M S S V V 24 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

26. M M S S V V 25 Case 1: An infinite plate with one hole Geometric data: a =1m thickness=0.002m Boundary condition: Inner edge : free

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28. M M S S V V 27 Distribution of DMCF on the circular boundary by using different methods, the present method, analytical solution and FEM

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31. M M S S V V 30 Case 2: An infinite plate with two holes

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33. M M S S V V 32 Distribution of DMCF on the circular boundary by using different methods, the present method and FEM

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36. M M S S V V 35 Outlines 4. Concluding remarks 3. Illustrated examples 2. Methods of solution 1. Introduction

37. M M S S V V 36 Concluding remarks A semi-analytical approach to solve the scattering problem of flexural waves and to determine DMCF in an infinite thin plate with multiple circular holes was proposed The present method used the null BIEs in conjugation with the degenerate kernels, and the Fourier series in the adaptive observer system. The improper integrals in the direct BIEs were avoided by employing the degenerate kernels and were easily calculated through the series sum. The DMCFs have been solved by using the present method in comparison with the available exact solutions and FEM results using ABAQUS. 1. 2. 3. 4. 5. Numerical results show that the closer the central distance is, the larger the DMCF is.

38. M M S S V V 37 Thanks for your kind attention The End