Solving The Westervelt Equation With Losses Using First And Second Order Finite Element Method

Abstract

In this Bachelor thesis we researched the second order finite element method as a method to solve the one-dimensional Westervelt wave equation. The Westervelt equation is a wave equation that describes the propagation of a nonlinear plane wave. The goal was to use a first order finite element method and a second order finite element method to solve the Westervelt equation, research the difference between these methods and known solutions and simulate the effect of certain changes in parameters. First we developed an understanding of the nonlinear wave propagation by analyzing the Burgers equation, which is an approximation to the Westervelt equation without attenuation term in another coordinate system. This analysis explains the influence of nonlinearity on the frequency spectrum of the solution. By looking at the derivation of the Westervelt equation changes in wave form during propagation are explained. We used our own finite element method to make implementations in Matlab to solve the linear wave equation, the Westervelt equation without attenuation term and the complete Westervelt equation. To deduce the implementations, we assumed that our unknown solution was a linear combination of first order basis functions. We then wrote the term of the equations into these linear combination and paraphrased them as matrix vector multiplications. Finally, a backwards differential method was used to solves these matrix vector multiplications iteratively in time.We compared the solutions of the linear wave equation using FEM to the known analytic solution and compared the solutions of the Westervelt equation without attenuation term to the solution of the Burgers equation. The finite element method cannot not be completely accurate because it depends on a mesh size that is not infinitely small. This means we always use an approximation in space. There is also an error in time, because we use backwards difference scheme to solve the equation iteratively in time. To approve accuracy we should put research into an adaptive mesh, where we choose a mesh with a lot more points around the peaks. The difference between the first and second basis functions of the finite element method was for the linear wave equation only caused by the small phase shift of the solutions. For the Westervelt equation with and without attenuation term the difference was larger, because of the effect of the nonlinear term on the Westervelt equation. The nonlinearity caused the slope of the wave to steepen when going from a maximum to a minimum. This effect eventually caused the formation of shock waves. The attenuation term inhibited the effect of the nonlinearity on the form of the wave during propagation and thus made the slope of the wave less steep. As a result, no shock waves were formed. The formation of shock waves was immediately clear, when we looked at the solutions at two times the shock wave distance. We concluded from this that the finite element method is not usable after shock waves are formed. We have also shown the frequency spectrum of the solutions and saw the transfer of energy to higher harmonics for the solutions that depended on nonlinearity. The attenuation term damped the transfer of energy and indeed in the spectra we saw less peaks in the higher harmonics. Lastly, we discussed a sudden change in velocity of the wave and calculated the amplitude of the reflected and transmitted wave. We simulated these waves with our implementation in Matlab and concluded that the theory about reflection and transmission agrees with our simulations