Simulation Efficiency

The separation between the transmitting and receiving waveguides of the sensor is 2mm and the transmitted pulse is an SH wave 5 cycle Hanning windowed toneburst with a centre frequency of 2MHz and a wave speed of 3260ms⁻¹. To ensure sufficient temporal separation between the surface skimming wave travelling directly from the transmitter to the receiver and the backwall reflection, a wall thickness greater than 5mm is required. A thickness of 10mm was specified as it was within the operational range of the sensor and provided good quality experimental signals. In order to include the majority of the reflected signal up to the start of the second backwall reflection a simulated signal length of 10µs was defined. Using trigonometry this value resulted in an x-axis length of the reflecting surface of 28mm (17λ). The z-axis surface length was specified by inspection of the beam profile in the yz plane and by incrementally increasing it to observe changes in the simulated signal; a value of 16mm (10λ) ensured that the result would closely match that of an infinite surface.

The generalized minimum residual (GMRes [83]) method was used to solve equation 3.10 to decrease simulation time as much as possible. Running different frequency components in parallel on 12 processors also improved speed; however when using a point source separation of 100µm it was found that memory requirements exceeded what was available on the machine which had 256GB random access memory and 4 quad-core 2.7GHz processors (AMD Opteron, Sunnyvale, CA). Therefore a method

5. Three Dimensional Simulation and Size Considerations

simulation time compared to 3D (%)

0 10 20 30 40 500

simulation time compared to full 3D (%)

Figure 5.4: (a) Change in reflected pulse shape from a flat surface as subdivided section areas decrease. (b) Average absolute difference of amplitudes within a 20dB bandwidth of the centre frequency as section area decreases compared with original surface, also showing a comparison of simulation time with that of original surface.

was sought which could improve computational efficiency with minimum impact on simulated result accuracy.

The majority of the time taken is during the solution of equation 3.10 when large matrices are being considered. Similar in theory to the beam superposition tech-nique [84] a method of subdividing the singular large surface into many sections was developed which could decrease memory allocation requirements. During solution the incident field remained constant and the reflected field from each section of the surface was calculated and summed in the frequency domain at all of the receiv-ing target points. Subsequent conversion to the time domain results in a signal which in theory would have reflected from the original surface, minus any interac-tions between secinterac-tions such as multiple scattering. In reality the major source of error originates from large calculated amplitudes along edges caused by disregard-ing reflected contributions from other sections of the surface. The edge sources for each section were made to coincide with the same locations along the joins between sections to mitigate these effects as much as possible.

Figure 5.4 (a) shows how the shape of the reflected pulse from a flat reflector with a passive point source separation of 200µm changes as the number of sections is

5. Three Dimensional Simulation and Size Considerations

increased, and Fig. 5.4 (b) compares the time taken and the average absolute difference in amplitude. The impact on simulation time is considerable; decreasing to 5% of the time that would be required for the original surface simply by splitting it into two sections. The associated difference in pulse shape is also very slight for section areas below 20% being on average below 0.15dB for all frequency components within a 20dB bandwidth of the centre frequency. Above this point differences start to increase rapidly; therefore during all subsequent work presented in this thesis this method was used to split original surfaces into 5 sections, incurring minimal error while decreasing simulation times by approximately 95%. It is understood that rewriting the simulation in alternative programming languages such as c++ or fortran could improve efficiency still further; however the simulation was developed as a proof of concept and built to be as flexible as possible. Using the efficiency enhancing methods described a simulation time of around 1500 seconds per signal from a three dimensional rough surface was achieved, making statistical analysis of rough surface signals a viable option for future investigations into three dimensional scattering.

5.3 Simulation Validation

Numerical methods such as the FEM are unsuitable for large three dimensional simulations at the roughness levels and surface sizes being considered since the mesh would require so many nodes that the simulation would become unfeasible, if not impossible to solve. Considering the inaccuracy of approximate analytical methods in the near field as described in Chapter 3, it was decided that DSPM simulation validation could only be carried out by comparison to experimental results. The added advantage of this approach is that not only would it provide an example of how well the simulation could predict signal shape, it would also give an indication of how well the scalar wave approximation works when predicting SH wave scattering in three dimensions. The same experimental equipment and signal post-processing was used as detailed in section 3.4 and for the sake of brevity will not be repeated here. Further details of this experiment have been submitted for publication in [P6];

as such the relevant excerpts are reproduced in sections 5.3.1 and 5.3.2.

5. Three Dimensional Simulation and Size Considerations