Emmanuel Perrey-Debain

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 mode matching method for modeling dissipative silencers lined with poroelastic materials and containing mean flow

A mode matching method for predicting the transmission loss of a cylindrical shaped dissipative silencer partially filled with a poroelastic foam is developed. The model takes into account the solid phase elasticity of the sound-absorbing material, the mounting conditions of the foam, and the presence of a uniform mean flow in the central airway. The novelty of the proposed approach lies in the fact that guided modes of the silencer have a composite nature containing both compressional and shear waves as opposed to classical mode matching methods in which only acoustic pressure waves are present. Results presented demonstrate good agreement with finite element calculations provided a sufficient number of modes are retained. In practice, it is found that the time for computing the transmission loss over a large frequency range takes a few minutes on a personal computer. This makes the present method a reliable tool for tackling dissipative silencers lined with poroelastic materials.

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.

A displacement-pressure finite element formulation for analyzing the sound transmission in ducted shear flows with finite poroelastic lining

In the present work, the propagation of sound in a lined duct containing sheared mean flow is studied. Walls of the duct are acoustically treated with absorbent poroelastic foams. The propagation of elasto-acoustic waves in the liner is described by Biots model. In the fluid domain, the propagation of sound in a sheared mean flow is governed by the Galbruns equation. The problem is solved using a mixed displacement-pressure finite element formulation in both domains. A 3D implementation of the model has been performed and is illustrated on axisymmetric examples. Convergence and accuracy of the numerical model are shown for the particular case of the modal propagation in a infinite duct containing a uniform flow. Practical examples concerning the sound attenuation through dissipative silencers are discussed. In particular, effects of the refraction effects in the shear layer as well as the mounting conditions of the foam on the transmission loss are shown. The presence of a perforate screen at the air-porous interface is also considered and included in the model.

The Partition of Unity Finite Element Method for the simulation of waves in air and poroelastic media

Recently Chazot et al. [J. Sound Vib. 332, 1918-1929 (2013)] applied the Partition of Unity Finite Element Method for the analysis of interior sound fields with absorbing materials. The method was shown to allow a substantial reduction of the number of degrees of freedom compared to the standard Finite Element Method. The work is however restricted to a certain class of absorbing materials that react like an equivalent fluid. This paper presents an extension of the method to the numerical simulation of Biots waves in poroelastic materials. The technique relies mainly on expanding the elastic displacement as well as the fluid phase pressure using sets of plane waves which are solutions to the governing partial differential equations. To show the interest of the method for tackling problems of practical interests, poroelastic-acoustic coupling conditions as well as fixed or sliding edge conditions are presented and numerically tested. It is shown that the technique is a good candidate for solving noise control problems at medium and high frequency.

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.

Analysis of convergence and accuracy of the DRBEM for axisymmetric Helmholtz-type equation

This paper presents a study of the convergence properties of a new axisymmetric approximating function. This function, associated with a dual reciprocity boundary element model for axisymmetric Helmholtz-type equation, is independent of the wave number in order to avoid occasional singular matrix due to some particular wave numbers. Interpolation functions are derived from the approximating functions. Their properties are studied numerically and their local behaviour is illustrated. Numerical tests, carried out in the last section, show a reasonably good agreement with analytical solutions of two simple aeroacoustic problems. Some criteria concerning the number of nodes per wavelength needed to obtain satisfactory results are also presented.

Influence of solid phase elasticity in poroelastic liners submitted to grazing flows

In the present work, we study the sound propagation in a duct treated with a poroelastic liner exposed to a grazing flow. Acoustic propagation in the liner and in the fluid domain is respectively governed by Biots model and Galbruns equation. Here, the coupling between Galbruns and Biots equation is carried out with a mixed pressure‐displacement FE. On one hand, a mixed formulation is used in Galbruns equation to avoid numerical locking. And on the other hand, in poroelastic media, the description of both phases involves the displacement of the solid phase and the pressure in the fluid phase. In addition of using the complete Biots model, simplified models are also tested. A fluid equivalent model that does not take into account solid phase elasticity and a model that neglects only the shear stress are hence used. These two simplified models enable to evaluate the contributions of the compressional and shear waves in the solid phase. Finally, validity of each simplified model in the specific case of...