Convergence of infinite element methods for scalar waveguide problems

We consider the numerical solution of scalar wave equations in domains which are the union of a bounded domain and a finite number of infinite cylindrical waveguides. The aim of this paper is to provide a new convergence analysis of both the Perfectly Matched Layer (PML) method and the Hardy space infinite element method in a unified framework. We treat both diffraction and resonance problems. The theoretical error bounds are compared with errors in numerical experiments

Fano resonances in acoustics

Contrary to completely open systems laterally confined domains can sustain localized, truly trapped modes with nominally zero radiation loss. These discrete resonant modes cannot be excited linearly by the continuous propagating duct modes due to symmetry constraints. If the symmetry of the geometry is broken the trapped modes become highly localized quasi-trapped modes which can interfere with the propagating duct modes. The resulting narrowband Fano resonances with resonance and antiresonance features are a generic phenomenon in all scattering problems with multiple resonant pathways. The present paper deals with the classical scattering of acoustic waves by various obstacles such as hard-walled single and multiple circular cylinders or rectangular and wedge-like screens in a two-dimensional duct without mean flow. The transmission and reflection coefficients as well as the (complex) resonances are computed numerically by means of the finite element method in conjunction with two different types of absorbing boundary conditions, namely the complex scaling method and the Hardy space method. The results exhibit the typical asymmetric Fano line shapes near the trapped-mode resonances if the symmetry of the geometry is broken

Trapped modes and Fano resonances in two-dimensional acoustical duct-cavity systems

Revisiting the classical acoustics problem of rectangular side-branch cavities in a two-dimensional duct of infinite length we use the finite element method to numerically compute the acoustic resonances as well as the sound transmission and reflection for an incoming fundamental duct mode. To satisfy the requirement of outgoing waves in the far field we use two different forms of absorbing boundary conditions, namely the complex scaling method and the Hardy space method. In general the resonances are damped due to radiation losses, but there also exist various types of localized trapped modes with nominally zero radiation loss. The most common type of trapped mode is antisymmetric about the duct axis and becomes quasi-trapped with very low damping if the symmetry about the duct axis is broken. In this case a Fano resonance results with resonance and antiresonance features and drastic changes in the sound transmission and reflection coefficient. Two other types of trapped modes, termed embedded trapped modes, result from the interaction of neighbouring modes or Fabry–P´erot interference in multi-cavity systems. These embedded trapped modes occur only for very particular geometry parameters and frequencies and become highly localized quasi-trapped modes as soon as the geometry is perturbed. We show that all three types of trapped modes are possible in duct-cavity systems and that embedded trapped modes continue to exist when a cavity is moved off centre. If several cavities interact the single-cavity trapped mode splits into several trapped supermodes which might be useful for the design of low-frequency acoustic filters

Resonance problems in acoustic waveguides

Acoustic resonance problems in closed systems can be solved numerically with finite element methods. The resulting real resonances become complex, if the system changes to an open system with an unbounded domain. While for a completely open system only damped resonances exist, laterally confined domains can sustain localized, truly trapped modes with nominally zero radiation loss. These discrete resonant modes cannot be excited linearly by continuous propagating duct modes due to symmetry constraints. If the symmetry of the geometry is broken the trapped modes become highly localized quasi-trapped modes which can interfere with the propagating duct modes. Since standard finite element methods are restricted to bounded domains, in a numerical treatment of such problems so called transparent or non-reflecting boundary conditions are needed. In this paper two different types of transparent boundary conditions namely the complex scaling method and the Hardy space infinite element method, are used to compute resonances as well as transmission and reflection coefficients for symmetric obstacles in a two dimensional waveguide