es;(Nx × Nz), operating over the whole system.

The application of the Newmark Method in this 2-D scheme is not as simple as in 1-D, since the elements are still one-dimensional and there are right and left matrix multiplications involved. The following expression is an adaptation of the Newmark Method to the 2-D implementation proposed:

. (15)

In this expression, t+Δtu appears in both sides of the equation, thus requiring the application of an iterative method for its calculation. This procedure not only increases the computational effort significantly but can also cause instability for large time steps.

4. Finite Element Formulation

Assuming that displacement u is approximated by a series of interpolating scale functions, the following may be written:

. (16)

Stiffness and mass matrices can be obtained by solving the wave propagation PDE using the FEM. Dimensionless coordinates (ξ) within the interval [0 1] are used in wavelet space, which leads to the subsequent expressions:

(17)

The so-called connection coefficients Γ appear in the expressions above. Wavelet dilation and translation properties allow the calculation of connection coefficients to be summarized by the solution of an eigenvalue problem based only on filter coefficients [14] .

(18)

Since the expression above leads to an infinite number of solutions, there is the need for a normalization rule that provides a unique eigenvector. This unique solution comes with the inclusion of the so-called moment equation, derived from the wavelet property of exact polynomial representation, which is given originally in [15] and adapted for Interpolets by [8] :

(19)

The dimensionless expressions for the stiffness and mass matrices are in wavelet space and need to be transformed to physical space, using a transformation matrix T obtained by evaluating the wavelet basis at the element node coordinates using Equation (16). It can be noticed that some terms related to the length (L) of the element emerge from coordinate changes.

(20)

5. Examples

To validate the formulation of the interpolet-based finite element, a 1-D example was tested. It consists in applying a forced displacement at the free end of a pinned, unit length rod (Figure 2). The propagation was modeled by the FDM using 601 points and Δt = 0.1 ms. For the FEM based on the IN6 interpolet, the discretization was performed with 91 points and Δt = 1 ms. The rod’s middle point time response for both methods is shown in Figure 3, which shows that the FEM result is acceptably close to that of the FDM, especially considering that FEM generates over 60 times less data per simulated second.

In order to validate the 2-D formulation, a simple example was proposed by analyzing the propagation in a 2 km × 2 km model with a constant velocity of 3000 m/s. Finite Element model was sampled every 6.17 m in both directions with a time step Δt of 0.46 ms, obtained by the generalized eigenvalue problem already described. Non-integer values appear due to the 11 degrees-of-freedom present in the element implemented with IN6. A Newmark scheme was also implemented using the same spatial sampling and Δt = 0.6 ms. The results were compared to the FDM using 5 m spacing in both directions and Δt = 0.2 ms. Finite Element mesh has 325 × 325 points and Finite Difference mesh has 401 × 401. Results are shown in Figure 4. Even with a less refined mesh and a more than twice bigger time step, the IN6 element was able to identify characteristic peaks of the wave propagation. The results of the proposed adaptation to the Newmark Method were very similar to the Central Difference ones using an even bigger time step.

6. Conclusions

This work presented the formulation and validation of an interpolet-based finite element. Newmark’s method for time discretization appears promising, although its application in 2-D problems with 1-D elements remains a challenge. The main improvement in the presented formulation was the possibility of using a bigger time step

Figure 2. Unit length rod under forced displacement.

Figure 3. FEM results using IN6 interpolet and newmark method (Δt = 1 ms) compared to FDM (Δt = 0.1 ms) for a one-dimensional wave propagation.

(a) (b)(c) (d)

Figure 4. Propagation snapshots at time t = 0.3 s for 2-D example using: (a) FEM and Central Difference Method, (b) FEM and Newmark Method, (c) FDM and Central Difference Method; (d) comparison of amplitudes along the indicated segment at depth 1000 m.

than the one required by the FDM. In future works, models with greater complexity will be analyzed and different families of wavelets will be explored.

The number of DOFs present in interpolets leads to non-dyadic divisions within the element, requiring an additional approximation which introduces errors in the transformation matrix. The inaccuracy in some values can be attributed mostly to these approximation errors.

As done in the traditional FEM, all matrices involved can be stored and operated in a sparse form, since most of their components are null, thus saving computer resources.

Acknowledgements

Authors would like to thank CNPq and FAPERJ for their financial support.

Cite this paper
Burgos, R. and Santos, M. (2016) Finite Elements Based on Deslauriers-Dubuc Wavelets for Wave Propagation Problems. Applied Mathematics, 7, 1490-1497. doi: 10.4236/am.2016.714128.
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