3.4 Stiness Reduction Method and In nite Elements
3.4.1 Analytical model
Before this approach can be implemented, an analytical solution must be established such that the performance of the layers can be quanti ed. Firstly, this is done for a boundary that contains only In nite Elements as discussed by Lysmer et al [95, 96]. Instead of the stress-free boundary conditions, the normal and tangential components of the stress at the model boundary are modi ed to include dimensionless parameters which when varied, alter the performance of the boundary (Equation 3.1.1 and Equation 3.1.2).
The proposed boundary conditions corresponds to a situation where the boundary is supported on in nitesimal dash-pots oriented normal and tangential to the boundary. By applying these conditions to the wave equation and using standard harmonic solutions from the Helmholtz decomposition, the amplitudes of the re ected wave modes from the boundary can be calculated (Equation 3.1.3, Equation 3.1.4, Equation 3.1.11 and Equation 3.1.12).
Figure 3.4.1 shows results of this analytical solution compared against results from a numerical simulation (using ABAQUS/Explicit [113]) for an incident compression wave across a range of incident angles.
0 15 30 45 60 75 90
−80
−60
−40
−20 0
Compression wave incidence angle (degrees)
Reflection coefficient (dB)
p wave (GMM) s wave (GMM) p wave (FE) s wave (FE)
Figure 3.4.1: Analytical and numerical re ection coe cients for an absorbing boundary composed of In nite Elements for an incident compression wave.
Figure 3.4.1 shows some disagreement between the two methods. The same trends are observed;
whereby increasing the angle of incidence results in an increase in the magnitude of the re ection coe cient. However, the predictions made using an analytical approach are far more optimistic that what has been measured numerically. This discrepancy can however be explained and accounted for.
From inspection of Equation 3.1.3, Equations 3.1.4, Equations 3.1.11 and Equations 3.1.12, it can be seen that the performance of In nite Elements is independent of inspection frequency. The only material constant that determines the magnitude of the re ection coe cients is the ratio of the com-pression and shear wavenumbers. For homogeneous, isotropic, elastic media this is almost always equal to 12, or small deviations away from this value.
The performance of the In nite Elements is however dependent upon the mesh density of the FE model. So far, the analytical solution does not account for any numerical re ections that result from the spatial discretisation of the mesh itself. This is unavoidable and is the dominant scattering mechanism associated with the application of In nite Elements to FE problems.
To understand this behaviour, the re ections caused by mesh scattering at the In nite Element boundary are quanti ed. This is done by calculating the re ection coe cient from a row of In nite Elements for a normal incidence compression wave. This case has been chosen because from Figure 3.4.1 it can be seen that the analytical model re ection coe cient is approaching zero, i.e. perfect performance at θinc = 0o. Therefore, any re ections that are observed for this case in a numerical model, must be a result of numerical mesh scattering. The re ection coe cient for this case is recorded for increasingly ner meshes, see Figure 3.4.2.
0 20 40 60 80
−60
−50
−40
−30
−20
−10 0
Nodes per wavelength, N (1/m)
Reflection coefficient (dB)
p wave θinc= 0
Figure 3.4.2: Numerical re ection from In nite Element boundary for increasingly re ned mesh.
From Figure 3.4.2 it can be seen that increasing the re nement of the mesh will decrease the mag-nitude the numerical mesh scattering. This however is not a linear relationship; to completely remove the component of the numerical re ection the mesh spacing between elements must tend to 0.
The introduction of mesh scattering is not so dissimilar to a phenomenon as reported by Rajagopal et al when investigating the performance of PML [92]. In theory, no re ection should occur at the interface with the PML, nor inside it. However, numerical results do not con rm this. Re ections from the PML are observed for normally incident compression waves. Again the magnitude of the numerical re ection is a function of mesh density, and therefore attributed to a mesh scattering phenomenon.
The numerical re ection caused by mesh scattering can be found by taking the re ection coe cient from a row of In nite Elements for a normal incidence compression wave. This component is then added to the prediction made using the analytical model, Figure 3.4.3.
By making this correction, there is now very good agreement between the two models. The behaviour of the In nite Elements can be predicted con dently and the performance assessed. The performance of the layers is aected by the variables a and b, with optimal performance achieved when a and b are equal to 1 [95].
Having established a means to predict the performance of a set of In nite Elements, a fully functional analytical model is required the can combine the performance of SRM containing a row of In nite Elements. This will allow for a means to validate the performance of the boundary and provide a quick method for calculating optimal model variables. To achieve this the Global Matrix method will be used [114]. Previously it has been used to calculate the re ection coe cient from a multi-layered system.
0 15 30 45 60 75 90
−80
−60
−40
−20 0
Compression wave incidence angle (degrees)
Reflection coefficient (dB)
p wave (GMM) s wave (GMM) p wave (FE) s wave (FE)
Figure 3.4.3: Analytical and numerical re ection coe cients for an absorbing boundary composed of In nite Elements for an incident compression wave with numerical mesh scattering correction.
The Global Matrix method establishes equilibrium for the displacements and stresses at each layer interface in both the forward and backward propagating directions. By generating an equation for each layer interface and inserting known model variables at the incident boundary, the system can be solved simultaneously. At the end of the nal layer, stress-free boundary conditions are applied.
σxx= 0 (3.4.1)
σxy = 0 (3.4.2)
Integration of In nite Elements in the Global Matrix method for this system (Equation 3.2.10), will require these boundary conditions to be changed to those given in Equation 3.1.1 and Equation 3.1.2.
The di culty with implementing the two techniques is caused by the fact that the In nite Element boundary conditions are non-zero and unknown. Referring to Equation 3.2.10, the layer properties and the RHS are all known, allowing for a matrix inversion techniques to be used to obtain the wave potentials traveling out of the layer, φN [1] and ψN [1].
To correct for this, the stress-free boundary conditions given by Equation 3.2.7 need to be moved to the LHS of the equation. This allows for the RHS to remain 0 and the In nite Element boundary conditions to be expressed as a function of wave potentials, which can then be solved using matrix inversion.
The equation within the nal layer of an absorbing boundary with In nite Elements becomes,
From Equation 3.1.1 and Equations 3.1.2 this becomes
Lb[end]
To solve the matrix all the variables are brought over to the LHS, giving
Lb[end]
The term ∂u∂t can now be expressed in terms of the wave potentials and becomes
Lb[end]
where M1[end]f and M2[end]f are the rst and second rows respectively of M[end]f given by Equation 3.2.4.
Finally, the common wave potentials can be factorised leaving
h
From inspection, in the instance where a and b are zero the In nite Elements are removed and the stress-free boundary conditions resumed.