Application of various forward and inverse scattering techniques
to non-destructive testing
Department of civil and environmental engineering, Gunma university
Associate Prof. Takahiro SAITOH
Keywords: FDM, FEM, BEM, wave propagation

1. Introduction
The ultrasonic non-destructive testing (UT) has attracted
attention over the past few decades. The final goal of UT
is to identify the position, shape, size of a defect. The
ultrasonic waves satisfy the elastic wave equations inelastic materials. Numerical simulation tools are useful tounderstand the elastic wave propagation behavior.

(a)

(b)

Fig.1: Elastic wave propagation in L-shaped CFRP (a)L-shaped CFRP model (b) a snapshot of elastic wave
propagation in L-shaped CFRP.

Therefore, in this presentation, some forward and inverse
scattering techniques for UT application are introducedwith the numerical results.
2．Forward analysis techniques
The main forward numerical techniques are the finite
difference method (FDM), finite element method (FEM),
and boundary element method (BEM). FDM and FEM
are used to analyze wave propagation in a finite andheterogeneous domain. On the other hand, BEM can deal

(b)

(a)

Fig.2: Reconstruction of a delamination in L-shaped CFRP (a)time-reversal result (b) reconstruction result obtained bycross-spectrum.

with infinite region without any modifications. However,
BEM has a lot of trouble simulating the wave propagation

SAFT algorithm. Therefore, there is still room forin a heterogeneous domain. Therefore, the author uses heimprovement to achieve high accuracy for the defectthese forward numerical techniques differently in theshapeproblem settings. Especially, some numerical results

application of the other innovative inverse scatteringobtained by the elastodynamic finite integration technique

techniques, such as the inverse scattering based on the(EFIT), which is an improved FDM, the voxel based

Born

FEM, and the convolution quadrature time-domain BEM

time-reversal method, has been done to improve the(CQBEM) [1] are demonstrated in this presentation.

accuracy of the defect shape reconstruction in UT. Some

3．Inverse scattering techniques

defect shape reconstruction results obtained by theSeveral inverse scattering techniques have been applied

time-reversal

to UT to reconstruct a defect in materials. The syntheticpresentation.

aperture focusing technique (SAFT) is the most widely

4．Numerical examples

used technique for the defect reconstruction in UT. The

Some numerical examples are shown in this section.

principle of SAFT is very simple, and is based on theFigure 1(a) shows a forward analysis model of L-shaped

superposition of received waveforms obtained by

CFRP whose elastic constants vary in the carbon fiber

ultrasonic transducers. The phase information included indirection [2]. A matrix array transducer is considered andthe received waveforms cannot be used in the general

the incident spherical wave is given from one element ofreconstruction.

andKirchhoff

methodFromthisperspective,approximations,aredemonstrated

and

inthethethisfor wave analysis with an infinite space, is used for thiscomputation.

TheCQBEM

canproduce

stabletime-domain solutions better than the classical BEM. It
can be confirmed that the scattered waves are generated
by the interaction between the cavity surface and theincident plane wave coming from the bottom. The
scattered waves are received at the observation points(a)(b)

shown in Fig.3(a). The received scattered waves are used
for solving the corresponding adjoint problem. The
numerical solutions obtained from the forward analysis
results such as shown in Fig.3(b), and adjoint analysis
results are used for the calculation of the topologicalsensitivity [3]. Figure 3(c) and (d) show the topologicalsensitivity in the front and obliquely upward view,

(c)

(d)

Fig.3: Shape reconstruction of cavities in an elastic solid (a)analysis model (b) a snapshot of elastic wave scattering (c) front
view (d) obliquely upward view of the defect shapereconstruction result.

the transducer through the intermediary of a wedge.
Figure 1(b) shows a snapshot of elastic wave propagationin L-shaped CFRP, which is obtained by the voxel based

respectively, of analysis domain. The topologicalsensitivity shows large values at the vicinity of 5 cavities.
Therefore,

thetime-reversal

technique

with

the

topological sensitivity is effective for the defect position
detection in UT.
5．Conclusions
Application of some numerical forward and inverse

FEM. It can be confirmed that the qp-wave (quasiscattering techniques to NDT was introduced. NumericalP-wave) propagates according to the carbon fiberresults show the potential of the present methods fordirection. The carbon fiber direction of the L-shaped

practical engineering UT applications. The author hopes

CFRP causes the anisotropic property and effects theourwave propagation behavior. Scattered waves generated bymathematics and engineering applications of UT.

the delamination in L-shaped CFRP are received by thearray transducer, time-reversed, and sent back to theL-shaped CFRP. Figure 2(a) shows a snapshot of thetime-reversal wave propagation in L-shaped CFRP. Thegroupresearchbecomes

abridge

between

References[1] T. Maruyama, T. Saitoh, T. Q. Bui and S. Hirose:
Transient elastic wave analysis of 3-D large-scalecavities by fast multipole BEM using implicit

superposition of time-reversed waves from the array

Runge-Kutta convolution quadrature, Computertransducer shows large values at the delamination of theMethods in Applied Mechanics and Engineering,

L-shaped CFRP. In addition, Fig. 2(b) indicates thevol.303, pp.231-259, (2016)

cross-spectrum between the time-reversed waves and the[2] Y. Maehara and T. Saitoh: 3-D forward andincident waves. The cross-spectrum shows large value attime-reversal analysis for defect in bent CFRP using

the vicinity of the delamination. Next, the time-reversal
technique is applied to the reconstruction of 5 cavities in
an infinite elastic solid, as shown in Fig.3(a). In this case,the topological sensitivity is utilized to determine thecavity position. Figure 3(b) shows a total elastic wave
field around the 5 cavities at a certain time step. The
CQBEM for 3-D elastodynamics, which is the suitable

finite element method, Journal of JASCOME, vol.
19, pp.103-108, (2019) (in Japanese)[3] M.

Bonnet:Topologicalsensitivity

for3D

elastodynamic and acoustic inverse scattering in thetime domain, Computer Methods in Applied
Mechanics and Engineering, 195(45), pp.94, (2014)