There are many parameters and processes that have to be adjusted to optimise the measurement setup, such as acquisition hardware, signal processing tools and thickness calculation approaches. Ideally, any optimisation process is carried out in
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simulations first, where uncertainties are minimal and are easily controlled compared to experimental data. Simulation tools are therefore introduced early on in this thesis. They will be used throughout the thesis as the basis of parametric studies or investigations.
One of the most popular simulation approaches is the Finite Element Method (FEM) [31, 32]. FEM relies on discretising the domain of the material into nodes and elements that make up a mesh. Because of the discrete mesh that covers the entire domain, it is very versatile and non-uniform material properties are straightforward to incorporate in the model. However, it is not numerically efficient in cases where wave propagation over large parts of bulk material has to be simulated. To some extent this can be mitigated by creating hybrid FEM models that mesh the feature of interest and the transducer only and use analytical formulations to propagate wavefields in the bulk material. [33–36]. Although such an approach does improve computational efficiency, it still relies on meshing parts of the domain, and its performance is limited.
As an alternative the DPSM (Distributed Point Source Method) as proposed by Placko and Kundu [37] is considered. The DPSM is a semi-analytical method originally developed to solve field equations for a wide array of engineering problems (e.g.: for ultrasonic, magnetic, eletromagnetic fields). DPSM is mesh-free, which allows for a potential performance improvement compared to methods relying on discrete meshes such as FEM. Instead, DPSM relies on point sources to simulate the behaviour of boundaries and interfaces.
This method has been successfully implemented to simulate signals for the waveguide sensor (also used in this thesis) by Jarvis et al. [11], which was verified against FEM.
In this thesis the same implementation is used, with some parametric adjustments as described here. This implementation relies on the assumption that SH waves can be modelled in 2D using an acoustic wave propagation model. This is because in a 2D model SH waves reflect from boundaries without mode conversion, since the direction of motion is perpendicular to the simulated plane. Hence the equation describing SH wave propagation is identical to that of acoustic wave propagation in 2D [12].
Furthermore, Jarvis et al. noted, that the cross-sectional width of the transducer (1 mm) is relatively small compared to the wavelength (λ ∼ 1.6 mm) whereas the
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cross-sectional length of the transducer is relatively large (15 mm). This conveniently allows for the simplifying assumption that the transducer can be modelled in two dimensions of the central plane of the transducer. Although the same 2D assumption about the internal wall surface is not as realistic, Jarvis et al. [38] concluded that such 2D simulations still capture the majority of the physical interactions on the backwall surface. With the 2D assumption the DPSM model for shear horizontal waves is equivalent to an acoustic model and implies that no mode conversion takes place at interfaces, as the simulated displacement is perpendicular to the plane in which modelling takes place. Phenomena such as beam spread from the transducer, multiple scattering and diffraction effects are fully simulated.
DPSM can be used to simulate monochromatic waves, therefore in order to implement a temporal domain simulation, the frequency components of the excitation signal using the Fast Fourier Transform (FFT) are first calculated. DPSM then simulates the response of the system for each and every frequency component. The response of the simulated system can then be reconstituted in the temporal domain by applying an Inverse Fast Fourier Transform (IFFT).
The simulation problem is therefore simplified to considering a single frequency component at one time. The central assumption of the DPSM model is that a large number of point sources placed at a small offset from an interface can be used to model the wave excitation, reflection and transmission behaviour of that interface. In DPSM, point sources can be active or passive. Active point sources are used to simulate areas of predefined pressure, such as the interface of a transmitter transducer. Passive point sources are used to simulate interactions on boundaries, such as the backwall surface of a sample. The excited wavefield of active and passive point sources are identical, and is a function only of wave number, distance, time and angular frequency as described by the Green’s function:
P (r, t) = AH0(2)(kr) · eiωt (2.17)
where the P is the pressure at time t separated by distance r from the point source, k is the wave number, ω is the angular frequency, H0(2) is the zero order Hankel function of the second kind and A is a complex constant relating to strength and phase of the wave excitation of the point source. It should be noted here that it is
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only A that is unique to a point source.
The logic of how DPSM then simulates the wavefield is the following: na active point sources are placed close to the interface of the transmitter transducer as shown in Figure 2.4. Boundary condition equations prescribing the source pressure are set up at the interface below each active point source, resulting in na number of boundary condition equations. Because of this, the sum of the wavefields excited by the active point sources at those boundary points are known. This can be formulated as a linear system of equations:
where PBC is the vector of prescribed pressures at the boundary points, AS is the amplitude constant associated with each active point source, QT S is the matrix of wave propagation equations (from Equation 2.17) from active point sources to the boundary points, N is the number of active point sources, M is the number of boundary locations and rab is the distance between point source a to boundary point b. Since only the vector AS is unknown, it can be calculated by:
AS = [QT S]−1PBC (2.20)
Once the values in vector AS are calculated, all variables associated with the active point sources are defined.
The same logic can be applied to compute the response of a reflecting surface. Namely a large number of passive point sources are placed at a small offset from the reflecting interface. The boundary condition of zero-pressure is then assumed on the backwall interface itself. Since the wavefield at the backwall boundary is equal to the sum of active and passive point sources, an equation similar to Equation 2.20 can be formulated, which in turn can be used to calculate the variables associated with the passive point sources. The total field response at any point can be calculated as a
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sum of contributions from all active and passive point sources.
In the implementation used in this thesis DPSM relies on placing point sources close to the interface of the transducer and the backwall surface. An example of the geometry of such a setup is shown in Figure 2.4. Here, the transmitter transducer is simulated by 100 active point source with a radius of 5µm offset from the transducer interface by 5 µm. The backwall surface is simulated by 800 passive point sources with a radius of 25 µm offset from the transducer interface by 25 µm. The receiver transducer is simulated by 50 receiver points that do not interact with the field placed on top of the interface.
The DPSM promises a simulation accuracy that has been shown to be as accurate as an equivalent FEM model but with a speed increase of an order of magnitude [11, 38],
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Figure 2.4: a) DPSM model of the permanently installed ultrasonic sensor on a flat backwall surface. Blue circles are the active point sources simulating the transmitter transducer. Red circles are passive point sources simulating the backwall geometry. Continuous black lines are shown where a zero pressure boundary condition has been applied. Dashed lines are purely for visual purposes only and therefore no boundary condition was applied to them. b) shows the simulated signal based on the model. The first wavepacket in the signal is the surface wavepacket. The second wavepacket is the backwall echo.
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and hence is used as the primary simulation tool in this thesis.
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