![Page 1: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/1.jpg)
MMSS VV
1
Scattering of flexural wave in a thin plate with multiple circular inclusions by using the
multipole Trefftz method
Wei-Ming Lee1, Jeng-Tzong Chen2, Hung-Ho Hsu1
1 Department of Mechanical Engineering, China University of Science and Technology, Taipei, Taiwan
2 Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan
2009年 11月 14日國立聯合大學
![Page 2: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/2.jpg)
MMSS VV
2
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
![Page 3: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/3.jpg)
MMSS VV
3
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
![Page 4: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/4.jpg)
MMSS VV
4
Introduction
Inclusions, or inhomogeneous materials, usually take place in shapes of discontinuity such as thickness reduction or strength degradation .
The deformation and corresponding stresses caused by the dynamic force are induced throughout the structure by means of wave propagation.
![Page 5: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/5.jpg)
MMSS VV
5
Scattering
At the irregular interface of different media, stress wave reflects in all directions scattering
The near field scattering flexural wave results in the dynamic stress concentration which will reduce loading capacity and induce fatigue failure.
Certain applications of the far field scattering flexural response can be obtained in vibration analysis or structural health-monitoring system.
![Page 6: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/6.jpg)
MMSS VV
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 inclusion.
In 1932, Trefftz presented the Trefftz method categorized as the boundary-type solution such as the BEM or BIEM which can reduce the dimension of the original problem by one.
The Trefftz formulation is regular and free of calculating improper boundary integrals.
![Page 7: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/7.jpg)
MMSS VV
7
Objective
The concept of multipole method to solve multiply scattering problems was firstly devised by Zavika in 1913.
Our objective is to develop an analytical approach to solve the scattering problem of flexural waves in an infinite thin plate with multiple circular inclusions by combining the Trefftz method and the multipole method .
![Page 8: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/8.jpg)
MMSS VV
8
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
![Page 9: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/9.jpg)
MMSS VV
9
For time-harmonic motion, the governing equation of motion for the plate
w is the out-of-plane displacement k is the wave number
4 is the biharmonic operator
eΩ is the unbounded exterior plate
w(x)
ω is the angular frequency
D is the flexural rigidityh is the plates thickness
E is the Young’s modulus
μ is the Poisson’s ratio
ρ is the surface density
![Page 10: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/10.jpg)
MMSS VV
10
Problem Statement
Problem statement for an infinite plate containing multiple circular
inclusions subject to an incident flexural wave
![Page 11: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/11.jpg)
MMSS VV
11
The integral representation for the plate problem
![Page 12: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/12.jpg)
MMSS VV
12
![Page 13: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/13.jpg)
MMSS VV
13
The slope, moment and effective shear operators
slope
Normal moment
effective shear
Tangential moment
![Page 14: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/14.jpg)
MMSS VV
14
Analytical derivations for flexural wave scattered by multiple circular inclusions
The scattered field of plate can be expressed as an infinite sum of multipole
The displacement field of the plate is
![Page 15: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/15.jpg)
MMSS VV
15
The displacement field of the pth inclusion is expressed as
![Page 16: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/16.jpg)
MMSS VV
16
The continuity conditions
![Page 17: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/17.jpg)
MMSS VV
17
For the pth circular displacement continuity
(1) The higher order derivatives (2) Several different variables
![Page 18: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/18.jpg)
MMSS VV
18
,k k kx
p
p
pΟ
,p p p x
,p p pk k kr r
x
p pk k rx x
( )( )e ( )e ( )p pk kim i m n inm p m n pk n k
n
J J r J e
( )( )e ( )e ( )p pk kim i m n inm p m n pk n k
n
I I r I e
( )
( )
( )e ( ) ,
( )e
( )e ( ) ,
pk k
p
pk k
i m n inm n pk n k k pk
im nm p
i m n inm n pk n k k pk
n
Y r J e r
Y
J r Y e r
( )
( )
( 1) ( )e ( ) ,
( )e
( 1) ( )e ( ) ,
pk k
p
pk k
i m n innm n pk n k k pk
im nm p
i m n inm nm n pk n k k pk
n
K r I e r
K
I r K e r
k
k
kO
Addition Theorem
kpkpr
![Page 19: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/19.jpg)
MMSS VV
19
![Page 20: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/20.jpg)
MMSS VV
20
The interface displacement continuity
![Page 21: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/21.jpg)
MMSS VV
21
The interface slope continuity
![Page 22: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/22.jpg)
MMSS VV
22
The interface bending moment continuity
![Page 23: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/23.jpg)
MMSS VV
23
The interface shear force continuity
![Page 24: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/24.jpg)
MMSS VV
24
The analytical model for the flexural scattering of an infinite plate containing multiple circular inclusions
![Page 25: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/25.jpg)
MMSS VV
25
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
![Page 26: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/26.jpg)
MMSS VV
26
Case 1: An infinite plate with one inclusion
Geometric data:R1=1mThicknessPlate:0.002m
Distribution of DMCF on the circular boundary (ka=0.005, h/h0=0.0005)
1.8514
![Page 27: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/27.jpg)
MMSS VV
27
Distribution of DMCF on the circular boundary
ka=0.5 ka=1.0 ka=3.0
![Page 28: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/28.jpg)
MMSS VV
28
Far-field backscattering amplitude
![Page 29: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/29.jpg)
MMSS VV
29
Case 2: An infinite plate with two inclusions
![Page 30: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/30.jpg)
MMSS VV
30
Distribution of DMCF on the circular boundary (L/a=2.1)
ka=0.5 ka=1.0 ka=3.0
![Page 31: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/31.jpg)
MMSS VV
31
Far-field backscattering amplitude (L/a=2.1)
![Page 32: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/32.jpg)
MMSS VV
32
Distribution of DMCF on the circular boundary (L/a=4.0)
ka=0.5 ka=1.0 ka=3.0
![Page 33: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/33.jpg)
MMSS VV
33
Far-field backscattering amplitude (L/a=4.0)
![Page 34: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/34.jpg)
MMSS VV
34
Outlines
4. Concluding remarks
3. Illustrated examples
2. Methods of solution
1. Introduction
![Page 35: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/35.jpg)
MMSS VV
35
Concluding remarks
An analytical approach to solve the scattering problem of flexural waves
and to determine the DMCF and the far field scattering amplitude in an infinite thin plate with multiple circular inclusions was proposed By using the addition theorem, the Trefftz method can be extended to
deal with multiply scattering problems.
The magnitude of DMCF of two inclusions is larger than that of one when the space of inclusions is small .
The proposed algorithm is general and easily applicable to problems with multiple inclusions which are not easily solved by using the traditional analytical method.
1.
2.
3.
4.
5. The effect of the space between inclusions on the near-field DMCF is
different from that on the far-field scattering amplitude.
![Page 36: M M S S V V 0 Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole Trefftz method Wei-Ming Lee 1, Jeng-Tzong](https://reader030.vdocuments.mx/reader030/viewer/2022032704/56649d6b5503460f94a4a412/html5/thumbnails/36.jpg)
MMSS VV
36
Thanks for your kind attention
The End