to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

流体力学用語集 非線形音響学(Nonlinear acoustics)

A discrete technique of the Schwarz alternating method is presented in this last chapter, to combine the Ritz-Galerkin and finite element methods. This technique is well suited for solving singularity problems in parallel, and requires a little more computation for large overlap of subdomains. The convergence rate of the iterative procedure, which depends upon overlap of subdomains, will be studied. Also a balance strategy will be proposed to couple the iteration number with the element size used in the FEM. For the crack-infinity problem of singularity the total CPU time by the technique in this chapter is much less than that by the nonconforming combination in Chapter 12.

Schwarz Alternating Method

Integral transforms (Laplace, Fourier and Mellin) are introduced with their properties, the relationship between these transforms was discussed and some important applications in physics and engineering were given. ااااااا دقل مت ضارعتسإ ةساردو ل ةيلماكتلا تليوحتلا لك ، سلبل تلوحت نم روف ي ر نيليمو عم ةشقانم كلذكو ،اهنم لك صاوخ و صئاصخ ةقلعلا ةشقانم مت هذه نيب طبرلاو و ،تليوحتلا مت ميدقت تاقيبطتلا ضعب تليوحتلا هذهل ةمهملا يف تلاجم ءايزيفلا ةسدنهلاو.

Integral transforms and their applications

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega>0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Despres, SIAM J. Numer. Anal., 35 (1998), pp. 255-299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of $p$-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121-136], and plane wave approximation theory.

/pdf/plane-wave-discontinuous-galerkin-methods-for-the-2d-tf8l9vefgi.pdf

Plane Wave Discontinuous Galerkin Methods for the 2D Helmholtz Equation: Analysis of the $p$-Version

Several special finite element methods have been proposed to solve Helmholtz problems in the mid-frequency regime, such as the Partition of Unity Method, the Ultra Weak Variational Formulation and the Discontinuous Enrichment Method. The first main purpose of this paper is to present a discontinuous Galerkin method with plane waves (which is a variant of the Discontinuous Enrichment Method) to solve the displacement-based acoustic equation. The use of the displacement variable is often necessary in the context of fluid–structure interactions. A well-known issue with this model is the presence of spurious vortical modes when one uses standard finite elements such as Lagrange elements. This problem, also known as the locking phenomenon, is observed with several other vector based equations such as incompressible elasticity and electromagnetism. So this paper also aims at assessing if the special finite element methods suffer from the locking phenomenon in the context of the displacement acoustic equation. The discontinuous Galerkin method presented in this paper is shown to be very accurate and stable, i.e. no spurious modes are observed. The optimal choice of the various parameters are discussed with regards to numerical accuracy and conditioning. Some interesting properties of the mixed displacement–pressure formulation are also presented. Furthermore, the use of the Partition of Unity Method is also presented, but it is found that spurious vortical modes may appear with this method.

Discontinuous Galerkin methods with plane waves for the displacement‐based acoustic equation

This paper discusses the use of mode-matching techniques to model sound propagation in lined ducts with flow. With no flow mode-matching methods are relatively well established and the continuity of acoustic pressure and axial velocity is generally applied. But with a base flow the behaviour of the solution at the transition between, say, a hard wall and a lined wall is not well understood. For this reason different assumptions can be made on the behaviour of the acoustic field and these lead to different models. This paper presents a comparison of different models and illustrates that significant differences can be observed. A modified mode-matching scheme based on conservation of mass and momentum is proposed. The conservation of sound power is investigated for different matching conditions. Finally, the link between the solutions obtained with mode-matching methods and finite element methods is discussed, and the present analysis can shed some light on the underlying assumptions used in finite element models.

Mode-Matching Techniques for Sound Propagation in Lined Ducts with Flow

https://hal.archives-ouvertes.fr/hal-01064463/document

A numerical method for vibro-acoustic problems with sheared mean flows

implementations of the method are discussed. Simulation results are presented for an isolated at plate interacting with inhomogeneous non-stationary turbulence. The sensitivity of noise levels to the upstream turbulence is investigated by considering trains of Gaussian wakes with dierent widths and separations. Far-eld noise levels are validated against Amiet’s analytical solution modied to include the eects of the wakes.

Broadband Fan Interaction Noise using Synthetic Inhomogeneous Non-stationary Turbulence

https://eprints.soton.ac.uk/365245/1/brambley14.pdf

Gwenael Gabard

Papers

Discontinuous Galerkin methods with plane waves for the displacement‐based acoustic equation

Mode-Matching Techniques for Sound Propagation in Lined Ducts with Flow

A numerical method for vibro-acoustic problems with sheared mean flows

Broadband Fan Interaction Noise using Synthetic Inhomogeneous Non-stationary Turbulence

Reflection of an acoustic line source by an impedance surface with uniform flow