J. Trevelyan

Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering.

Classical finite–element and boundary–element formulations for the Helmholtz equation are presented, and their limitations with respect to the number of variables needed to model a wavelength are explained. A new type of approximation for the potential is described in which the usual finite–element and boundary–element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions evenly distributed on the unit sphere. Compared with standard piecewise polynomial approximation, the plane–wave basis is shown to give considerable reduction in computational complexity. In practical terms, it is concluded that the frequency for which accurate results can be obtained, using these new techniques, can be up to 60 times higher than that of the conventional finite–element method, and 10 to 15 times higher than that of the conventional boundary–element method.

Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications

The classical boundary element formulation for the Helmholtz equation is rehearsed, and its limitations with respect to the number of variables needed to model a wavelength are explained. A new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions. This is termed the plane wave basis boundary element method. The modifications needed to the classical procedures, in terms of integration of the element matrices, and location of collocation points are described. The well-known Singular Value Decomposition solution technique, which is adopted here for the solution of the system matrix equation in its complex form, is briefly outlined. The conditioning of the system matrix is analysed for a simple radiation problem. The corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions. The CHIEF method is adopted to enhance the quality of the solution, particularly in the vicinity of irregular frequencies. The plane wave basis boundary element method is then applied to two problems: scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem. In both cases results are compared with analytical solutions. The results clearly demonstrate that the new method is considerably more efficient than the classical approach. For a given number of degrees of freedom, the frequency for which accurate results can be obtained, using the new technique, can be up to three or four times higher than that of the classical method. This makes the method a powerful new addition to our tools for tackling high-frequency radiation and scattering problems.

Wave boundary elements: a theoretical overview presenting applications in scattering of short waves

It is well known that the use of conventional discrete numerical methods of analysis (FEM and BEM) in the solution of Helmholtz and elastodynamic wave problems is limited by an upper bound on frequency. The current work addresses this problem by incorporating the underlying wave behaviour of the solution into the formulation of a boundary element, using ideas arising from the Partition of Unity finite element methods. The resulting ‘wave boundary elements’ have been found to provide highly accurate solutions (10 digit accuracy in comparison with analytical solutions is not uncommon). Moreover, excellent results are presented for models in which each element may span many full wavelengths. It has been found that the wave boundary elements have a requirement to use only around 2.5 degrees of freedom per wavelength, instead of the 8–10 degrees of freedom per wavelength required by conventional direct collocation elements, extending the supported frequency range for any given computational resources by a factor of three for 2D problems, or by a factor of 10–15 for 3D problems. This is expected to have a significant impact on the range of simulations available to engineers working in acoustic simulation. This paper presents an outline of the formulation, a description of the most important considerations for numerical implementation, and a range of application examples.

A numerical integration scheme for special finite elements for the Helmholtz equation

The theory for integrating the element matrices for rectangular, triangular and quadrilateral finite elements for the solution of the Helmholtz equation for very short waves is presented. A numerical integration scheme is developed. Samples of Maple and Fortran code for the evaluation of integration abscissae and weights are made available. The results are compared with those obtained using large numbers of Gauss-Legendre integration points for a range of testing wave problems. The results demonstrate that the method gives correct results, which gives confidence in the procedures, and show that large savings in computation time can be achieved.

Use of wave boundary elements for acoustic computations.

Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this work, a new type of interpolation for the acoustic field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation has been found in the current work to be highly successful. Results are shown for a variety of classical scattering problems, and also for scattering from nonconvex obstacles. Notable results include a conclusion that, using this new formulation, only approximately 2.5 degrees of freedom per wavelength are required. Compared with the 8 to 10 degrees of freedom normally required for conventional boundary (and finite) elements, this shows the marked improvement in storage requirement. Moreover, the new formulation is shown to be extremely accurate. It is estimated that for 2D Helmholtz problems, and for a given computational resource, the frequency range allowed by this method is extended by a factor of three over conventional direct collocation Boundary Element Method. Recent successful developments of the current method for plane elastodynamics problems are also briefly outlined.

On wave boundary elements for radiation and scattering problems with piecewise constant impedance

Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this paper, a new type of interpolation for the wave field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation is found in this paper to be highly successful. Results are shown for a variety of scattering/radiating problems from convex and nonconvex obstacles on which are prescribed piecewise constant Robin conditions. Notable results include a conclusion that, using this new formulation, only approximately three degrees of freedom per wavelength are required.

Numerical evaluation of the two-dimensional partition of unity boundary integrals for Helmholtz problems

There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.

Characteristics of group velocities of backward waves in a hollow cylinder

It is known that modes in axially uniform waveguides exhibit backward-propagation characteristics for which group and phase velocities have opposite signs. For elastic plates, group velocities of backward Lamb waves depend only on Poissons ratio. This paper explores ways to achieve a large group velocity of a backward mode in hollow cylinders by changing the outer to inner radius ratio, in order that such a mode with strong backward-propagation characteristics may be used in acoustic logging tools. Dispersion spectra of guided waves in hollow cylinders of varying radii are numerically simulated to explore the existence of backward modes and to choose the clearly visible backward modes with high group velocities. Analyses of group velocity characteristics show that only a small number of low order backward modes are suitable for practical use, and the radius ratio to reach the highest group velocity corresponds to the accidental degeneracy of neighboring pure transverse and compressional modes at the wavenumber k = 0. It is also shown that large group velocities of backward waves are achievable in hollow cylinders made of commonly encountered materials, which may bring cost benefits when using acoustic devices which take advantage of backward-propagation effects.

Bearing currents in wind turbine generators

Increasing the availability of multi-megawatt wind turbines (WT) is necessary if the cost of energy generated by wind is to be reduced. It has been found that WT generator bearings have a surprisingly high failure rate, with failures happening too early to be due to classical rolling contact fatigue. One potentially important root cause of bearings failures, bearing currents, has been investigated in this paper. The use of pulse-width modulated power electronic converters in variable speed WTs results in the presence of a common-mode voltage which may drive stray currents through a parasitic circuit in the generator structure. In this paper it is shown that if appropriate mitigation strategies are not employed, the bearing lubricant may experience electric stress in excess of its dielectric strength resulting in electrostatic discharge machining (EDM) of the bearing. Moreover it is shown that rotor-fed machines are more susceptible than stator-fed machines due to the presence of a larger coupling capacita...

Time-independent hybrid enrichment for finite element solution of transient conduction-radiation in diffusive grey media

We investigate the effectiveness of the partition-of-unity finite element method for transient conduction-radiation problems in diffusive grey media. The governing equations consist of a semi-linear transient heat equation for the temperature field and a stationary diffusion approximation to the radiation in grey media. The coupled equations are integrated in time using a semi-implicit method in the finite element framework. We show that for the considered problems, a combination of hyperbolic and exponential enrichment functions based on an approximation of the boundary layer leads to improved accuracy compared to the conventional finite element method. It is illustrated that this approach can be more efficient than using h adaptivity to increase the accuracy of the finite element method near the boundary walls. The performance of the proposed partition-of-unity method is analyzed on several test examples for transient conduction-radiation problems in two space dimensions.