A MULTILEVEL MULTIPOLE METHOD FOR MODELING
ELASTIC-WAVE MULTIPLE SCATTERING IN FIBERREINFORCED COMPOSITES
Arnaud Lange1, Anthony Harker2 and Nader Saffari1
Department of Mechanical Engineering, University College London, London
WC1E7JE,UK
Department of Physics & Astronomy, University College London, London
WC1E6BT,UK
ABSTRACT. A Multilevel Multipole method is presented for modeling elastic wave multiple
scattering in fiber-reinforced composites. Results are given for the case of an incident SH wave
impinging on a bounded Ti/SiC composite region, the plane of propagation being orthogonal to the
fibers' axis. The scattering effects of a square and hexagonal arrangement for the fibers are compared.
INTRODUCTION
Elastic-wave multiple scattering effects are significant in the NDE of fiberreinforced composites (FRCs). Indeed specific attenuation and dispersion effects are due to
multiple scattering, which in turn are related to the composite properties. These include the
elastic parameters, distribution, concentration and nature of the fiber reinforcements.
Varadan et al. [1] demonstrated the effects of multiple scattering on the coherent wave, in
particular the dependence on concentration at wavelengths comparable to scatterer size. At
very low concentrations (< 1% by volume) multiple scattering can be neglected and each
scatterer can be treated as independent [2], In many practical situations however, the
concentration can range from 1% to 40% where multiple scattering effects are important.
Considerable effort is being directed towards development of numerical models that can be
used as analysis tools for optimizing NDE techniques for these composites.
In general, available results for multiple scattering in FRCs are limited to the
Rayleigh scattering regime. The large number of fibers usually prevents the solution of
such large-scale scattering problems via standard numerical methods. Thus various
approximate theories have been developed to model the average field. Bose & Mal [3] and
Mal & Chatterjee [4] have used a statistical approach to study the multiple scattering of
elastic waves. This approach generally results in a hierarchy of equations for the average
fields with increasing order correlation functions. Various assumptions are proposed to
break this hierarchy [5]. In mixture theories [6] [7] [8], the composite constituents are
superimposed in space and may undergo individual deformations. In theory models of
arbitrary orders of accuracy may then be determined. The complexity of this approach
however, constrained the authors to applications with a first order theory. Because these
approaches are concerned with the average field, it is difficult to evaluate the sensitivity of
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/$20.00
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the actual field to local perturbations of the composite structure, which is of major interest
in practice.
To consider local perturbations, several deterministic multiple scattering
formulations have been considered. Twersky [9] introduced the concept of "ordered
scattering" for acoustic wave problems, where the overall multiple scattering problem is
decomposed into several single scattering problems in the frequency domain. Using this
approach, the solution is computed recursively. Cheng [10] extended this concept to the
case of elastic waves. Varadan, Varadan & Pao [11] derived an implicit form of the
multiple scattering solution. Although these formulations lead to the multiple scattering
solution, none of them can efficiently compute the solution for a large number of scatterers.
The convergence of the recursive forms of the solution is too slow, and the other forms are
limited by the computer memory.
Recent advances in multi-bodies problems however have allowed the numerical
cost to be reduced considerably by using multipole methods. Rokhlin & Greengard [12]
originally developed the Multilevel Multipole method for the rapid evaluation of the
potential field in systems involving a large number of particles whose interactions are
Coulombic or gravitational in nature. This approach was latter extended to acoustic [13]
and electromagnetic [14] scattering problems. Indeed, considering a 2D multiple scattering
problem with TV scatterers, using a multilevel technique may reduce the algorithm
complexity from O(N2) to O(N). It is proposed here to apply this approach to elastic-wave
scattering problems.
In this study Titanium composites reinforced with Silicon Carbide fibers are of
particular interest. The problem considered is that of an incident SH wave field propagating
in the cross-section plane and scattered by a bounded Titanium composite region. The
Titanium matrix and the various materials involved in the scatterers are taken to be
homogeneous and isotropic.
THEORY
Wave Fields Representations
Considering an incident SH wave, no mode conversion occurs during scattering,
and the out-of-plane displacement w can be expressed in the form:
w ( r , 0 = 0(r)e-ia"9
where
V 20 + k 2(j) = 0
(1)
is the scalar Helmholtz equation, ris the position vector, t is time, co is the angular
frequency, <f> is the displacement amplitude, k=co/ft is the shear wavenumber and ft the shear
velocity. For cylindrical scatterers only, the displacement amplitudes of the various fields
can be expanded in terms of the cylindrical solutions of the Helmholtz equation i.e. the
Bessel functions. For a single-scatterer problem, the incident wave amplitude (fnc is
expanded in terms of regular Bessel functions of the first kind :
* ={A}T{J(r,0»,
(2)
and the scattered wave amplitude 0s in terms of singular Hankel functions of the first kind :
=
BnH?(kr)eine ={B}T{H(r,0)},
(3)
where (r,9) are the polar coordinates of the observation point in the scatterer's local
coordinate system, whose origin lies within the scatterer.
By definition, the transition matrix [7] of the scatterer relates the wave expansion
coefficients of the incident and scattered waves as:
{B} = [T]{A}.
(4)
Multilevel Multipole Method
The method may be implemented in five steps: partition of the structure,
aggregation of scatterers, transfer-regularization, de-aggregation of fields and transition to
the next scattering order. Twersky's approach of orders of scattering is used [9].
Structure Partition
Considering a set of TV scatterers, the first step of the multilevel technique is to
partition this set in subsets, at different scales. This partition is usually done in quadrature
for convenience. Different scales or levels of partition are considered: a coarse level
corresponds to a partition in a small number of large cells, whereas a fine level partitions
the structure in a large number of small cells.
The procedure to build the partition is as follows: a cell that encompasses the whole
set of scatterers is defined first. This level is called Level 0. Then, following a 2D partition
in quadrature, the cell is partitioned in four identical cells. These correspond to the Level 1
of partition. The partition in quadrature is performed for each cell at level 1, thus defining
the Level 2 of partition, and this procedure is repeated until each cell contains a small
number of scatterers (Cf. Fig.l). A cell at level / is said to be the parent of the cells it
generates at level /+ 1 i.e. its offspring.
Aggregation of Scatterers
Once a partition has been defined, the first scattering order fields of the scatterers in
each cell are aggregated [12] at the finest level L:
with:
= J-kdei^
(6)
where Ns is small since the partition is chosen so that, at the finest level, each cell contains
a small number of scatterers. The aggregated fields of the cells at level l+l are then
themselves aggregated to define an approximation of the first scattering order field for the
parent cell at level /, for l=L-l,...,l (Cf. Fig.2) :
87
Offspring of Cell
0000000000000000
0000000000000000
0000000000000000
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oooooooooooooooo
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oooooooooooooooo
o o o o o o o o ooo p/6 ooo
o o o o o o o o o o o|*o ooo
oooooooo oooooooo
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Level 0
0000 0000 o o d/o p\o o o
o o o o 0000 00f/° «Hf O O
o o o o 0000 o oq o 6000
o o o oo o o oo o o oo o o o
0000 o o o o o oJ> o <Jp o o
0000 o o o o o o^ o OTO O O
o o o oo o o oo o o oo o o o
o o o oo o o oo o o oo o o o
0000 o o o o o o o o o o o o
o o o oo o o oo o o oo o o o
O O O 0 0 O O Oo o o o o o o o
o o o oo o o oo o o oo o o o
o o o oo o o oo o o oo o o o
0000 o o o o o o o o o o o o
o o o oo o o oo o o oo o o o
Level 2
Level 1
FIGURE 1. Structure partition in quadrature. Notion of parent cell and its offspring.
Thus, an approximation of the first scattering order field from the scatterers contained in
each cell at any level is obtained.
Transfer-Regularization of Fields
For each scatterer s, the contributions from the other scatterers need to be
transferred to scatterer s' local system of coordinates to account for the interactions. Using
the structure partition and the aggregated fields, this is done for each cell at different levels:
-
-
Considering a cell c at a coarse level of partition Lc, the set of cells whose contributions
may be transferred with a good accuracy is determined first. Such cells should be
distant enough, or well-separated, from cell c. The set of cells that are well-separated
from cell c is called the interaction list of cell c.
Once the interaction list of cell c is known, the contributions from this list is transferred
to cell c's local system of coordinates according to Graf series :
nlist
nlist
S=l
S=l
4)}-
(8)
Since the contributions are transferred from the singular (H(rs,^s)} basis functions to the
-
regular {J(rc,^c)J basis functions, it is said that the contributions are regularized.
All the cells that could not be transferred at level Lc are taken into account at the next
level Lc+l via their offspring (Cf. Fig.3).
De-aggregation of Fields
Once the contributions from the interaction list of cell c have been transferred to
cell c's system of coordinates, these contributions are further transferred to the offspring of
oe/oe/
1. Aggregation at
scatterer's level
2. Aggregated Cells at
level L
3. Aggregation at
level L
FIGURE 2. Aggregation from offspring to parents' level.
cell c at level Lc+l via Graf series :
nlist
:i)lr
y
^soffsprd)
Z
s=l
with :
offspr(i}
\nm = Jn-m (kds offspr(i} ) e
_]\J Voffspr(i}'
U
offspr(i)
i(n-m)6soffspr(i}
(9)
(10)
The interaction list of each offspring at level Lc+l is then taken into account via the same
procedure as in the previous section. Both steps Transfer-Regularization and Deaggregation are repeated until the finest level L is reached. Scatterers whose contributions
have not been taken into account are transferred at the scatterers' level.
Transition to next Scattering Order
At this stage, for each scatterer s, all the contributions from the other scatterers have
been transferred to the local coordinate system associated with scatterer s. The T-matrix of
scatterer s is then applied to compute the next scattering order field:
^' (2) ={C™}T{H(r.,e.)}, where: {cf>] = [r,]{C<'>},
The field 0 ' is the first scattering order field from all scatterers except scatterer s. The
whole procedure Aggregation;Transfer-Regularization;De-aggregation;Transition is
repeated to deduce higher scattering order fields. This is repeated until the order of
scattering is large enough for the corresponding field to be negligible compared to the total
scattered field:
ss
p
89
N
(13)
List of
IiittractMifi List of
Cell
Level /
Cell
Level l+l
FIGURE 3. Interaction list at different levels (white cells).
NUMERICAL RESULTS
In the following applications, an incident harmonic SH plane wave is propagating
along the horizontal x-axis in the positive direction in the cross-sectional plane. The wave
is scattered by a bounded composite region. The displacement amplitude of the total field
0totis computed in the neighborhood of the composite region. A composite structure
containing 1024 fibers is considered, and results are compared for two typical fiber
arrangements, square and hexagonal, respectively. The fiber diameter is 140um. Each fiber
has a carbon core of 30um diameter, and a thin carbon layer of lum at the fiber/matrix
interface is also taken into account in the models.
The displacement amplitude of the total field in the near-field region of the
composite structure is plotted on Fig.4 at frequency of 1 MHz. This corresponds to a
wavelength of about 23 times the fiber diameter. Major differences may be seen in the
fields for the different fiber arrangements. For a square fiber arrangement, little reflection
and attenuation occur along the x-axis in the backscattered and transmitted regions,
respectively. Higher reflection and transmission exist however for a hexagonal fiber
arrangement. The symmetry of the composite structure is apparent from the field obtained
for a square fiber arrangement, as opposed to the hexagonal one. In both cases, an effect
from the structure corners is visible at about the 40° direction. The far-field displacement
amplitude is plotted in Fig.5. Differences between both composite structures in reflection
and attenuation along the axis of propagation are still present. A slight effect due to the
structure corners is also visible.
CONCLUSIONS
The Multilevel Multipole method allows full-scale simulations of elastic wave
multiple scattering in fiber-reinforced composites. Deterministic models can thus be
developed to investigate the effects of various composite configurations on the total
scattered wave field. Major differences at low frequency have been detected in the nearfield as well as in the far-field between a square and a hexagonal fiber arrangement.
90
30
150
0
180
330
210
(a)
(b)
FIGURE 4. Near total displacement field for a composite structure with 1024 fibers: (a) square fiber
arrangement; (b) hexagonal fiber arrangement.
30
150
o iao
330
210
(a)
(b)
FIGURE 5. Far total displacement field for a composite structure with 1024 fibers: (a) square fiber
arrangement; (b) hexagonal fiber arrangement.
91
Higher frequency solutions are currently being studied. Although the method and
the results have been presented in the frequency domain for SH waves only, time-domain
solutions may be obtained using an inverse Fourier-Laplace transform. Moreover, the
extension to P/SV waves is straightforward [15]. The various materials involved in the
model have been taken to be homogeneous and isotropic, however a weak anisotropy for
the Titanium matrix can be taken into account by modifying the bases functions [16].
Cylindrical scatterers of various cross-sections may be implemented via a multiple
multipole method [17]. The results will be validated experimentally.
ACKNOWLEDGEMENTS
This work is supported by the Engineering and Physical Sciences Research Council
of the United Kingdom, grant no. GR/M83049.
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