View
240
Download
0
Tags:
Embed Size (px)
Citation preview
MMSS VV
1
Scattering of flexural wave in thin plate with multiple holes by using the null-field integral equation method
Wei-Ming Lee1, Jeng-Tzong Chen2
Ching-Lun Chien1, Yung-Cheng Wang1
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 UniversityMSVLAB ( 海大河工系 )
Department of Harbor and River Engineering
MMSS VV
2
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
MMSS VV
3
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
MMSS VV
4
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.
MMSS VV
5
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
MMSS VV
6
Overview of numerical methods
Finite Difference M ethod Finite Element M ethod Boundary Element M ethod
M esh M ethods M eshless M ethods
Numerical M ethods
6
PDE- variational IEDE
Domain
BoundaryMFS,Trefftz method MLS, EFG
開刀 把脈
針灸
MMSS VV
7
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.
MMSS VV
8
MotivationMotivation
Numerical methods for engineering problemsNumerical methods for engineering problems
FDM / FEM / BEM / BIEM / Meshless methodFDM / FEM / BEM / BIEM / Meshless method
BEM / BIEMBEM / BIEM
Treatment of siTreatment of singularity and hyngularity and hypersingularitypersingularity
Boundary-layer Boundary-layer effecteffect
Ill-posed modelIll-posed modelConvergence Convergence raterate
MMSS VV
9
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.
MMSS VV
10
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
MMSS VV
11
Flexural wave of plate
4 4( ) ( ),u x k u x xÑ = Î WGoverning Equation:
u is the out-of-plane displacement k is the wave number
4 is the biharmonic operator
is the domain of the thin plates
u(x)
24
3
12(1 )
hk
D
E hD
w r
n
=
=-
ω is the angular frequencyρ is the surface density
D is the flexural rigidityh is the plates thickness
E is the Young’s modulusν is the Poisson’s ratio
MMSS VV
12
Problem Statement
Problem statement for an infinite plate with multiple circular holes subject to an incident flexural wave
MMSS VV
13
The integral representation for the plate problem
MMSS VV
14
Kernel function
The kernel function is the fundamental solution which satisfies
MMSS VV
15
The slope, moment and effective shear operators
slope
moment
effective shear
MMSS VV
16
Kernel functions
In the polar coordinate of
MMSS VV
17
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
with respect to the field point x
with respect to the field point x
normal moment
effective shear force
MMSS VV
18
x
s
eU
O
iUr
qf
xr
Rf
Expansion
Degenerate kernel (separate form)
Fourier series expansions of boundary data
MMSS VV
19
Boundary contour integration in the adaptive observer system
MMSS VV
20
Vector decomposition
MMSS VV
21
Transformation of tensor components
MMSS VV
22
Linear system
where H denotes the number of circular boundaries
MMSS VV
23
MMSS VV
24
Techniques for solving scattering problems
MMSS VV
25
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
MMSS VV
26
Case 1: An infinite plate with one hole
Geometric data:a =1mthickness=0.002mBoundary condition:Inner edge : free
MMSS VV
27
MMSS VV
28
Distribution of DMCF on the circular boundary by using different
methods, the present method, analytical solution and FEM
MMSS VV
29
MMSS VV
30
MMSS VV
31
Case 2: An infinite plate with two holes
MMSS VV
32
MMSS VV
33
Distribution of DMCF on the circular boundary by using different methods, the present method and FEM
MMSS VV
34
MMSS VV
35
MMSS VV
36
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
MMSS VV
37
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 degenerat
e 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.
MMSS VV
38
Thanks for your kind attention
The End