A useful strategy to reduce drastically the computational requirements of magne-tostrictive models relies on the use of a hybrid analytical-numerical approach. This kind of simplification can only be used when analytical solutions to part of the prob-lem, either the electrodynamic or the elastic one, are available in closed form. It is seldom possible to find such analytical expressions, however, whenever they are available they can be employed to reduce the memory requirements and the
compu-4. Finite Element model
tational time greatly. A typical example is the case of those EMAT configurations whose coil is made of several long straight sections, like meander or race-track coils (Section 2.4). In these cases, the dynamic magnetic field induced in a ferromagnetic material can be approximated with the analytical expression found by Chari and Reece for an infinitely long wire above a half-space [73]. It is possible to exploit these equations, for example in the modelling of the SH wave EMAT described in the previous section, such that the first constitutive equation of (4.8) becomes:
e6 = s66eσ6+ d61He1?, (4.10)
where eH1? is not an unknown quantity to be computed numerically, but is given by a known closed form expression. Since the electrodynamic problem is solved via analytical means, the electromagnetic potentials A and V do not have to be computed and each node of the mesh requires four degrees of freedom less than those of the original model, that is, only a simple elastic model has to be solved.
This significantly reduces the computational time, as this is roughly a function of the square of the total number of degrees of freedom for harmonic simulations.
The use of analytical solutions introduces the pros and the cons of such methods.
Only special geometries can be addressed, effects such as the finiteness of the wires (edge effects) or their conductivity are neglected and mutual coupling between the generation and reception process is assumed to be insignificant. On the other hand, an appropriate choice of the simplifying model can lead to satisfactory approximate solutions with considerably smaller computational requirements. Figure 4.6 shows the results obtained for the same model of SH magnetostrictive EMAT described in the previous section. The magnitude of the displacement v is plotted along the x1 axis for a multiphysics model together with a model using an analytic solution for the dynamic magnetic field. While the discrepancy between the two simulations is limited to 5-10%, the model employing the analytic expression for eH1 is significantly faster and less demanding than the multiphysics model.
4. Finite Element model
0.8 1.2
mplitude, |v| [arb.]
Full Model Analytical EM
0 0.4
0 20 40 60 80 100
Displacement Am
Distance from Source, x1 [mm]
Figure 4.6: Magnitude of the particle displacement v plotted along the x1 axis (for x2= x3= 0) for a SH magnetostrictive EMAT as the one shown in Figure 4.2. The results from a full multiphysics model (continuous line) are shown together with those from a model using an analytic solution for the dynamic magnetic field (dashed line).
4.6 2D axisymmetric model
For some EMAT configurations, for example the pancake coil shear wave EMAT, the three dimensional problem can be approximated with a simplified two-dimensional axisymmetric model, in cylindrical coordinates {r, z, φ}. If the driving current flows in the azimuthal direction, i.e. perpendicular to the modelled plane {r, z}, the mag-netic vector potential has only one nonzero component Aφ(r, z) and the electro-dynamic problem reduces to finding the solution of a single scalar PDE, which is computationally simple. The mechanical part of the model can also be simplified by using COMSOL’s “Stress-Strain, Axial Symmetry” application mode [68]. A cylindrical coordinate system is employed and the elastic equations are solved for the displacements in the radial and axial directions. The assumption is that the circumferential component of displacement is zero, as well as the stress and strain components σrφ, σzφ, rφ, zφ, and that the loads act on the {r, z} plane only.
4. Finite Element model
Finally, assuming that the material is magnetized along the z direction, magne-tostriction constitutive equations reduce to:
er= s11eσr+ s13eσz+ d31Hez ez = s13eσr+ s33σez+ d33Hez
erz = 2 s44eσrz+ d15Her
(4.11)
and,
Ber = 2 d15erz+ µ11Her Bez = d31er+ d33ez + µ33Hez
. (4.12)
4.7 Conclusions
A description of the numerical methods used to simulate EMAT operation has been given. A SH wave magnetostrictive EMAT configuration has been taken as an ex-ample to illustrate how to effectively choose the size of the domains and of the mesh grid. The use of symmetries and analytical solutions to reduce the computational demands of the model has also been discussed. Several researchers have already proposed EMAT Finite Element models [7, 40–45, 56–59], even including magne-tostriction; however, the critical question is how well the predictions of the model correlate with real-world data. The experimental validation of the present FE model will be the topic of the next chapter.
Chapter 5
Validation of the model
5.1 Introduction
This chapter investigates the validity of the FE model presented in Chapter 4 by comparing its predictions with experimental tests. This is a multi-step process that starts with the assessment of the performance of the single COMSOL modules against analytical benchmarks. Then a full multiphysics model is validated: a 3D magnetostrictive EMAT on a nickel plate is used as a case study. First a qualitative validation is given, showing how the model is able to successfully compute the wave amplitude dependence on significant parameters: the static bias field, the driving current amplitude and the excitation frequency. A quantitative validation follows, where the wave amplitude per unit current is predicted without the use of any arbitrary adjustable parameter and is compared to experimental tests.