Applied case

To illustrate the performance of the SRM, the absorbing layer technique is applied to an example realistic simulation. An embedded rough defect is acting as a scatterer of incident ultrasonic waves.

The system is surrounded by innite elastic space and the scattered signal is monitored. Both the SRM and ALID absorbing boundary methods are compared for increasing thickness of absorbing region, as shown in Figure 3.2.10.

U₂

U₁

Monitoring Line

Absorbing Layer - Variable Thickness

5λ_p

5λ_p

λ_p λ_p

λ_p

λp

AoS

Excitation Node

Monitoring Nodes

Single Monitoring Node

p s

Figure 3.2.10: FE model used to test performance of SRM and ALID as a function of region thickness.

The material is dened by a Young's modulus of 70 GPa, Poisson's ratio of 0.31 and density of 2700 kgm⁻³. The compression wavelength of the incident wave, λp, is 3.00 mm and the system is spatially discretised at 30 nodes per compression wavelength. A 2 MHz, 5 cycle incident wave is applied to a single excitation node which is excited in a direction that is inclined at 45^o to the U1 axis. Exciting the node in this manner generates a spherically propagating compression wave and a perpendicular spherically propagating shear wave. This excitation ensures that both wave modes interact with the complex multifaceted scatterer, fully testing the performance of the absorbing layers.

The defect of choice is a two-dimensional rough surface with a rms σ = 0.100λp, away from the

mean plane. The rough surface is generated using a moving average process as described by Ogilvy, [116]. The method represents a rough surface by a set of discrete heights at discretised points along the defect surface. Initially the heights are uncorrelated, having a normal distribution governed by the rms height. To correlate the heights, a moving average is taken, eectively smoothing out the height prole, thereby correlating the surface.

The depth and width of the defect is 5λpand the distance separating the excitation node/defect from the absorbing region boundary is λp. Again this ensures a complex scattering pattern that thoroughly tests the performance of the layers, see Figure 3.2.10.

The scattering pattern is monitored along a line parallel to the rough surface at the same separating distance as the excitation node/defect of 5λp. Again the extremities of the monitoring line are separated from the absorbing layer by λp. The line consists of a series of nodes that monitor displacements in the U1 and U2 directions. The response is monitored using two techniques. Firstly displacement in the U1 direction is averaged along the monitoring line for all time producing a single time history response from the defect. The second method measures displacement at a single, arbitrarily selected, node (denoted by a cross) within the monitoring line to ensure that averaging time histories has not eliminated any scattering behaviour.

The performance of the SRM and ALID are assessed. CM max is set to ω for both cases and the Young's modulus within the nal layer for the SRM is 1% of the AoS. Layer thickness is varied in increments of 0.2λp to a value of 4.0λp.

Figure 3.2.11a) and Figure 3.2.11b) compare two of the time histories directly for the case where the SRM thickness is set to 0.2λp and 4.0λp respectively at a single node within the motoring line. Four signals can be identied corresponding to the incident and reected waves for each mode.

From the results it can be seen that an insuciently thick absorbing region of 0.2λp performs very poorly in comparison to the 4λp case. Reections from the smaller absorbing region pollute the time signal almost immediately, with artifacts continuing to reverberate within the model for the duration.

The characteristic features of the incident shear wave have been completely lost and both reected wave modes have been heavily distorted.

The contribution of the absorbing layers pollutants to the time history can be quantied by calcu-lating the convergence of the time history to that obtained from the case where the model boundaries have no contribution to the scattered signal. This time history is obtained by compiling a model with a signicantly larger spatial domain, such that the model boundaries have no interaction with the incident or scattered elds. The convergence of the time histories is calculated using a correlation function given by Equation 3.2.12.

a)

Figure 3.2.11: Comparison of time history from a single node for SRM with a) 0.2λ SRM thickness and b) 4λ SRM thickness. The 0.2λ SRM shows pollutants to time series from unwanted absorbing layer reections.

a)

0 1.0 2.0 3.0 4.0

0.80 0.85 0.90 0.95 1.00

R (no unit)

Absorbing region thickness (λ

p)

SRM ALID

b)

0 1.0 2.0 3.0 4.0

0.80 0.85 0.90 0.95 1.00

R (no unit)

Absorbing region thickness (λ

p)

SRM ALID

Figure 3.2.12: Correlation function for a) average displacement over a monitoring line and b) at an ar-bitrary chosen node, for performance of SRM and ALID as a function of region thickness.

R =

PN

i=1(xi− x)(y_i− y)

 PN

i=1(xi− x)²PN

i=1(yi− y)²

¹₂ (3.2.12)

where R is the correlation function ranging between 0 and 1, i is an increment in the discretised time series of length N, x is the response from the classic case with mean value x and y is the response for each simulation with mean value y. The value R = 1 corresponds to a case of perfect correlation between two compared time histories; the signicance being that there is no inuence of the absorbing layers on the scattered response. The results are shown in Figure 3.2.12.

It can be seen that the SRM tends to a value of R = 1 for an absorbing layer thickness of λp, after which there are diminishing returns in performance. The more varied response observed at a single node can be explained by the lack of averaging, which would lter out any spurious reections detected. Comparing time series suggests that there is little to gain by having an excessively large SRM since any pollutants that add to the desired time series are of negligible value. In comparison the ALID performance is less eective in reduced spatial domains. Again there are diminishing returns for excessively large thicknesses, however this trend does not begin until thicknesses of 2.5λp.

These results show that a proposed SRM thickness of 1.5λ would be sensible. This is suitably larger that the thickness λ at which reasonable convergence has occurred giving a suitable margin of safety, but not so large as to unnecessarily increase the spatial domain.