Finite Element Method for two dimensional Westervelt Equation
L.J.F. Hendriks (TU Delft - Electrical Engineering, Mathematics and Computer Science)
M.D. Verweij – Mentor (TU Delft - ImPhys/Medical Imaging)
Domenico Lahaye – Mentor (TU Delft - Mathematical Physics)
Arnold Heemink – Graduation committee member (TU Delft - Mathematical Physics)
D.J. Verschuur – Graduation committee member (TU Delft - ImPhys/Computational Imaging)
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Abstract
This essay shows a two dimensional implementation of the finite element method for the Westervelt equation. To do this the finite element method is first applied to the linear wave equation, then to non-linear diffusion and finally to the Westervelt equation. Both an element by element and a faster vectorized implementation are given for the finite element method. To verify the numerical solution two analytical solutions are used. The first is a one dimensional wave and the second a circularly symmetric wave.
We found that the two-dimensional implementation was successful in computing the Westervelt equation. The error of the solution scales with the mesh size with a power of around 1.7. It was also found that the time step used to compute the solution needs to be small enough for the implementation to converge.