Numerical Methods on Solving Helmholtz Equation

It is a well known fact that the time harmonic acoustic problems governed by the Helmholtz equation face a major challenge in the non-coercive nature associated with extreme high frequencies [96]. Either its conditioning or its complexity will lead to intolerable computational costs. Many techniques and numerical methods have been developed and devised to tackle this issue for decades, such as least square finite element methods [166, 247, 313], discontinuous Galerkin method [146], the partition of unity methods [46], ultra-weak variational formulations [96, 178, 175, 97], and the higher-order finite element method which will be discussed in following chapters [183]. However, many of the methods suffer from ill-conditioning of the problem with fine mesh, instability of the system or the high computational costs arisen

from the complexity of the formulation.

In 2008, a two dimensional Helmholtz problem was solved on square domain near the scatterer by a novel locally enriched finite element method [301]. The idea behind is to enrich the local approximation basis by oscillatory functions. The pro- posed method proved the planar wave basis and provided accurate results near the scatterer, while the Bessel functions perform better in far fields. Furthermore, the required degrees of freedom per wavelength for accurate results with the proposed method are far less than the rule of thumb (10 nodal points per wavelength). Con- sequently, a hybrid combination of Bessel and plane wave basis for local enrichment are expected to deliver efficient calculation for the Helmholtz equation solved on large computational domain [207]. This type of method is categorised as the Trefftz type method, and originally introduced by Melenk and Babuˇska [232].

Generally speaking, the Trefftz methods exploit the a priori knowledge about the local behaviour of the solution function, and thus enrich the approximation spaces by well selected enrichment functions for the Helmholtz operator (e.g. spherical plane wave functions or Bessel functions) [207, 232]. The conditioning of the prob- lem will be ultimately dependent on the chosen enrichment function space [43]. The efficiency of the partition of unity finite element method (PUFEM) method over the conventional FEM has been identified by Laghrouche et al in [203], through the specified angle of incidence for plane wave basis. Further in 2009, Huttunen

et al. [176] have compared PUFEM with the ultra weak variational formulation

(UWVF) on singular problem of L shaped domain. In later years, the partition of unity method has been further developed in two and three dimensional prob- lems [204, 203, 201]. On the other hand, Lagrange multipliers applied on edges of elements enhanced the continuity of the solution. This technique can be cou- pled with wave solution based shape functions which leads to the reduction of the number of degrees of freedom (DOFs) compared to conventional Lagrange finite element methods[146, 147, 311]. In [205], the method is applied to solve Helmholtz problems with piecewise constant wavenumbers in the computational domain.

Alternatively, higher order finite element methods are proved to be efficient for solving the wave problems governed by the Helmholtz equation if accurate solutions are sought [103, 135, 72, 253, 288]. In essence, the hierarchical finite element basis can minimizes the computational costs associated with conventional higher order approximation basis, while further increasing the flexibility and scalability of ap- proximation spaces. It should be kept in mind that the hierarchical higher order FEM can be coupled with various Galerkin methods and Trefftz methods. Since the higher order finite element method involves localised p enrichments, a goal ori- ented adaptivity scheme based on sufficient local error estimates can be devised to mitigate the dispersion effect and provide exponential convergence rate.

In this thesis we mainly focus on the acoustic part of the acoustic-fluid inter- action problem for both monochromatic and polychromatic waves. First of all, a complete list of errors involved in numerical calculation are considered, and stud- ied in edge elements. The stringent definition of qualified error estimator is then discussed. Secondly, the FEM solver for elliptic PDEs within the mesh oriented finite element method code (MoFEM) is developed, examples of thermal conduc- tive problems are solved. Finally, numerical results are compared with commercial software (presented in Appendix).

In what follows, we build up the Helmholtz operator based on previous developed programming strategies (associated with elliptic type PDE), and then implement the relative L2 and H1 error estimation norms together with five benchmark exam-

ples. Initially, the problem of wave-guide and a plane wave impinging on the sound hard surface of a cylinder are considered, where solution convergence is studied for both geometry and multiple physical fields. Moreover, the numerical efficiency and accuracy of the hp-adaptivity in the context of the Helmholtz equation are investi- gated. The application of hierarchical finite element approximation together with generalized Duffy transformation drastically improves the computational efficiency and accuracy of the acoustic solver, while removing the singularity and uncertainty from numerical integration techniques [10, 137]. Subsequently, the problem of SAW actuation of a fluid droplet is considered. Using hierarchical finite element approx- imation basis (Lobatto and Legendre) on unstructured meshes [10], both acoustic potential (pressure) field and geometry are independently approximated with arbi- trary and heterogeneous polynomial order.

In detailed exposition, the Rayleigh wave propagating on a SAW device medium can be regarded as a plane wave problem [291], which can be resolved with time- harmonic damped wave equation. Furthermore, a closed form leaky SAW is derived based on results and information from [270] and [320]. In our procedure, arbitrary signals can be applied to the medium. Analysis can be done in either frequency or time domain. Moreover, Reynolds’ stress is retrieved from the acoustic velocity by taking unsteady part of the gradient of the acoustic potential multiplied by the den- sity. In the sequel, the problem was expanded to polychromatic waves via Fourier transformation in both spatial domain and frequency domain with respect to the input signal. The acoustic pressure distribution inside a droplet is observed by ma- nipulating the physical properties of the modelling, in order to further understand the underlying physics of the application.