Finite Element Method Applied to the One-dimensional Westervelt Equation

In this thesis we researched the applicability, properties and efficiency of the finite element method to solve the one-dimensional Westervelt equation, which describes nonlinear plane wave propagation. The goal was to investigate whether this lesser-known solution method has advantages or disadvantages compared to more commonly used solution techniques. We developed an understanding of nonlinear wave propagation by analyzing the Burgers equation, which we used to benchmark solutions. We used the commercial finite element software package COMSOL to calculate first solutions, where we found that numerical errors occur as the wave propagates through the shock wave formation distance. We examined the effect of several numerical parameters and concluded that reducing the element size decreases the overall error of the solution, both near the shock wave front and elsewhere. This also helps reduce numerical oscillations if present. Increasing the element order also improved the solution. The time stepping algorithm was found to have a strong connection to the element size. The maximum time step depends strongly on the minimum element size. Reducing physical parameters such as the amplitude of the source, or adding damping, were also researched but were shown to have little effect on reducing the numerical error around the shock wave front. The finite element method can solve inhomogeneous domains with relative ease compared to homogeneous domains, which may be an advantage over other methods. We then developed our own Matlab implementation of Galerkin's finite element method for the Westervelt equation to get more insight into the algorithms behind this method and get a better understanding of the effect of numerical parameters. We implemented two different time solvers and we concluded that our specific choice of backward differential formulas was producing more accurate results than more general build-in time solvers that come with COMSOL or Matlab. Furthermore we saw that the accuracy of the solution does not only depend on spatial numerical parameters, but also on the time solving parameters. Different time solving techniques can yield different degrees of accuracy and efficiency, and must therefore be chosen with care. We finally turned to adaptive finite element method techniques in order to improve overall accuracy and efficiency. We have shown that a simple form of adaptiveness can help improve the accuracy of the solution, but its efficiency depends on the implementation and the number of spatial dimensions in which the equation is solved. The finite element method provides different types of adaptiveness, such as local refinement/coarsening, node movement and local change of the order of the basis functions, which may be combined together. We showed the advantages and disadvantages of a node movement implementation based on the MMPDE-6 algorithm. We concluded that more research can be put in incorporating (combined types of) adaptiveness to solve the Westervelt equation.

Subject

Finite Element Method
Adaptive Mesh
Westervelt equation
Nonlinear Wave Propagation

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http://resolver.tudelft.nl/uuid:033dda97-4ed3-4962-95d4-fc583d1c2f64

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bachelor thesis

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