Numerical Methods for Scientists and Engineers

State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite-element methods are described. Stabilized, multiscale, and other wave-based discretization methods developed to reduce this error are reviewed. A review of finite-element methods for acoustic inverse problems and shape optimization is also given.

A review of finite-element methods for time-harmonic acoustics

The generalized finite element method for Helmholtz equation: Theory, computation, and open problems

This paper describes a finite element model for the solution of Helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element. The approach is based on partition of unity isoparametric elements. At each finite element node the potential is expanded in a discrete series of planar waves, each propagating at a specified angle. These angles can be uniformly distributed or may be carefully chosen. They can also be the same for all nodes of the studied mesh or may vary from one node to another. The implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres. The wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced. Issues related to the integration and the conditioning are also discussed.

Modelling of short wave diffraction problems using approximating systems of plane waves

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.

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

Computational aero-acoustics for fan duct propagation and radiation. Current status and application to turbofan liner optimisation

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.

https://hal.archives-ouvertes.fr/hal-01004921/document

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

Air travel in the UK has increased five-fold over the last 30 years. Half the population now flies at least once a year and freight traffic at UK airports has doubled since 1990. Similar statistics hold for other developed countries. It is likely that this growth will be eclipsed in the years to come by the expansion in civil aviation within emerging giants such as India and China. Civil aviation links distant communities for social and business purposes and gives affordable access to foreign travel for leisure purposes. It is unlikely however that the global community can continue to enjoy the benefits of air travel through increased capacity regardless of environmental cost. To tackle this problem, a balanced approach is essential which recognises the importance of air travel, but which also deals with environmental issues such as noise and emissions. Predicting the effects of noise and emissions is a first essential step towards such a solution. This article focuses on the prediction of aircraft noise and on developing noise reduction technologies. The aircraft noise issue is that of reducing the environmental impact of exterior noise at take-off and approach. This constitutes a major nuisance to residents near airports. It has resulted in stringent local airport regulations, such as the ‘quota count’ system at Heathrow, which are now more restrictive in defining acceptable limits for aircraft noise than are the international noise regulations promulgated by ICAO which first brought aircraft noise within a regulatory framework in the sixties and seventies. In this article, the major sources of noise in a modern aircraft are discussed, and current techniques for predicting aircraft noise are reviewed. These include the effects of source definition and acoustic treatment. Recent advances in noise reduction technology will be presented and their performance assessed in the light of noise targets set by the Advisory Council for Aeronautics Research in Europe (ACARE) in their ‘vision 2010’ and ‘vision 2020’ plans. Finally some specific noise reduction technologies will be described which have been developed and assessed in recent EC projects.

Predicting and reducing aircraft noise

This paper presents a prediction method for fan tone noise propagation and radiation, and a study on the modal power distribution required to accurately model fan broadband noise. The results from the predictions are validated against rig measurements. In the prediction method for fan tone noise propagation and radiation, the source can be described as multimodal at subsonic tip speeds, and composed of a multimodal content and a high pressure amplitude rotor-locked content at supersonic tip speeds. In-duct sound pressure level (SPL) measurements from mode detection have been used to calibrate the predictions

Rie Sugimoto

Papers

Computational aero-acoustics for fan duct propagation and radiation. Current status and application to turbofan liner optimisation

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

Predicting and reducing aircraft noise

Prediction of Forward Fan Noise Propagation and Radiation from Intakes

A review of CAA for fan duct propagation and radiation, with application to liner optimisation