One-dimensional Finite Element simulations

To understand the behaviour of the system, an explicit one-dimensional FE model has been created that is governed by Equation 2.3.19. The system is composed of a string of 2-node rod or Truss elements each with a single degree of freedom being the displacement parallel to the direction of propagation. A length of undamped elements forming the AoS is terminated by an absorbing region. An illustration of this system is given in Figure 3.2.1.

Absorbing Layer

X(x) = 0

z

y x Element

Node u_x

Fixed Point AoS

X(x) = 1

Figure 3.2.1: Illustration of an explicit one-dimensional FE model used to investigate absorbing layer performance.

A short pulse is generated in the AoS and is incident at the absorbing region. The performance of the absorbing region is assessed by examining the unwanted reection returning from it. The FE model is non-dimensional with material properties that produce a propagation wavelength of 1 unit.

The total length of the absorbing region is 1.5 units (1.5λincof the system) and the model is discretised using 20 nodes per wavelength with a Courant or CFL number of 1 [65].

SRM variables that need to be optimised are the maximum Rayleigh damping CM max, the power to which successive layer properties are raised p, and the maximum attenuation factor in the nal absorbing layer αmax. To achieve this an optimisation function is used which calculates the optimal values for input variables that will render the lowest possible reection coecient for the absorbing boundary (see Section 3.3.3 for an explanation of the optimisation function).

The SRM and ALID are directly compared for the one-dimensional case, with their respective layer properties optimised for maximum absorption. All variables between the two models remain constant except for the Young's modulus in the absorbing region of the SRM. Figure 3.2.2a) and Figure 3.2.2b) show the performance of the two techniques. The time history of a single excited node is recorded showing the incident 5 cycle tone burst and the observed reection from the absorbing region later in time.

Figures 3.2.2a) and Figure 3.2.2b) show that for the example under consideration, the SRM behaves signicantly better than ALID. The maximum amplitude of the reected signal for the ALID is -35.2 dB whereas the SRM gives -45.0 dB.

a) .

Figure 3.2.2: Reected signals from absorbing layer regions for a) ALID and SRM, full time record b)

Due to the likened approach to a Rayleigh/Caughey damping formulation proposed by Semblat et al [90], a further comparison is made between SRM and CALM. To incorporate Rayleigh/Caughey damping, the damping matrix is dened by Equation 3.1.29, where CM and CK are non-zero. Again C_M and CK are varied gradually across the absorbing layer, as in Equation 3.1.36, but with the introduction of an additional variable CKmax, the maximum stiness proportional damping coecient.

The denition of SRM remains unchanged.

For a fair comparison to be made with SRM, an optimisation function (see Section 3.3.3) is used which calculates the optimal values for CM maxand CKmaxthat will render the lowest possible reection coecient for the CALM absorbing boundary. The SRM and CALM are directly compared for the one-dimensional case, with their respective layer properties optimised for maximum absorption. All variables between the two models remain constant except for the Young's modulus in the absorbing region of the SRM. Like the comparison made with ALID and SRM in Figure 3.2.2b), Figure 3.2.2c) shows the time history of the observed reection from the absorbing regions.

CALM absorbing boundaries have marginally outperformed the ALID in this example, with a max-imum amplitude of the reected signal being -36.2 dB. This increase in performance over conventional ALID has been achieved by the inclusion of stiness proportional damping to the damping matrix, with the same ease of implementation as mass proportional damping. However, the increase in performance has come at the expense of a signicant decrease in the stable time increment that is associated with having a non-zero CK term. Despite this improvement Rayleigh/Caughey damping does not oer the same performance that has been achieved with SRM.

Figure 3.2.3 shows how the change of matrix variables changes the value of the wavenumber. Both the real and imaginary components have been plotted. The rst observation is that within the SRM, k_imag increases dramatically in comparison to the ALID. This is as expected and is in agreement with the predictions made using Equation 3.1.35. It is the increase in this value that produces such successful decay of any incident waves whilst inside each layer. The desired increase in kimag is also accompanied by the undesired increase in kreal. Again the increase observed inside the SRM is considerably greater than that in the ALID, which should suggest sizable reections at successive layer boundaries. However, dramatic changes only occur once the wave has propagated a signicant distance into the absorbing region.

Here, the thickness of each layer remains constant and is xed to a value of one-element-thickness.

Work carried out by Rajagopal et al [92], has stated that it is preferable to minimise the change in any material properties between adjacent layers so that impedance changes are gradual and thus the thinner each layer is the better. However, for one-dimensional FE models, increasing element thickness could potentially oset impedance mismatches. The stiness of a single Truss element is the product of the Young's modulus multiplied by the element thickness, thereby allowing for another parameter by which to control material properties. This is not however suitable for two and three-dimensional FE

0 0.2 0.4 0.6 0.8 1.0 0

5 10 15 20 25

Distance across absorbing region (m)

Wavenumber (1/m)

ALID k

real

ALID k_imag SRM k_real SRM k_imag

Figure 3.2.3: Increase of real and imaginary components of the wavenumber with distance.

models so has not been pursued. In this instance a variation in element size must be performed across a rectangular or quadrilateral element, greatly distorting the structured arrangement of the mesh.

Furthermore, variation of the mesh density will result in scattering from the mesh itself, producing an additional mechanism whereby unwanted reections are radiated back into the AoS.