Remarks on the stability of Cartesian PMLs in corners

This work is a contribution to the understanding of the question of stability of Perfectly Matched Layers (PMLs) in corners, at continuous and discrete levels. First, stability results are presented for the Cartesian PMLs associated to a general first-order hyperbolic system. Then, in the context of the pressure-velocity formulation of the acoustic wave propagation, an unsplit PML formulation is discretized with spectral mixed finite elements in space and finite differences in time. It is shown, through the stability analysis of two different schemes, how a bad choice of the time discretization can deteriorate the CFL stability condition. Some numerical results are finally presented to illustrate these stability results

Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability

Extended VersionInternational audienceIn this work we consider a problem of modelling of 2D anisotropic dispersive wave propagation in unbounded domains with the help of perfectly matched layers (PML). We study the Maxwell equations in passive media with a frequency-dependent diagonal tensor of dielectric permittivity and magnetic permeability. An application of the traditional PMLs to this kind of problems often results in instabilities. We provide a recipe for the construction of new, stable PMLs. For a particular case of non-dissipative materials, we show that a known necessary stability condition of the perfectly matched layers is also sufficient. We illustrate our statements with theoretical and numerical arguments

The fictitious domain method and applications in wave propagation

International audienceThis paper deals with the convergence analysis of the fictitious domain method used for taking into account the Neumann boundary condition on the surface of a crack (or more generally an object) in the context of acoustic and elastic wave propagation. For both types of waves we consider the first order in time formulation of the problem known as mixed velocity-pressure formulation for acoustics and velocity-stress formulation for elastodynamics. The convergence analysis for the discrete problem depends on the mixed finite elements used. We consider here two families of mixed finite elements that are compatible with mass lumping. When using the first one which is less expensive and corresponds to the choice made in a previous paper, it is shown that the fictitious domain method does not always converge. For the second one a theoretical convergence analysis was carried out in [7] for the acoustic case. Here we present numerical results that illustrate the convergence of the method both for acoustic and elastic waves

Stability and Convergence Analysis of Time-domain Perfectly Matched Layers for The Wave Equation in Waveguides

International audienceThis work is dedicated to the proof of stability and convergence of the Bérenger's perfectly matched layers in the waveguides for an arbitrary L ∞ damping function. The proof relies on the Laplace domain techniques and an explicit representation of the solution to the PML problem in the waveguide. A bound for the PML error that depends on the absorption parameter and the length of the PML is presented. Numerical experiments confirm the theoretical findings

Hyperboloidal layers for hyperbolic equations on unbounded domains

We show how to solve hyperbolic equations numerically on unbounded domains by compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with spatial compactification. We construct a new layer method based on this idea, called the hyperboloidal layer. The method is demonstrated on numerical tests including the one dimensional Maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques.Comment: 23 pages, 23 figure

An Analysis of New Mixed Finite Elements for the Approximation of Wave Propagation Problems

International audienceWe construct and analyze a new family of rectangular (two-dimensional) or cubic (three-dimensional) mixed finite elements for the approximation of the acoustic wave equations. The main advantage of this element is that it permits us to obtain through mass lumping an explicit scheme even in an anisotropic medium. Nonclassical error estimates are given for this new element

Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability

Extended VersionInternational audienceIn this work we consider a problem of modelling of 2D anisotropic dispersive wave propagation in unbounded domains with the help of perfectly matched layers (PML). We study the Maxwell equations in passive media with a frequency-dependent diagonal tensor of dielectric permittivity and magnetic permeability. An application of the traditional PMLs to this kind of problems often results in instabilities. We provide a recipe for the construction of new, stable PMLs. For a particular case of non-dissipative materials, we show that a known necessary stability condition of the perfectly matched layers is also sufficient. We illustrate our statements with theoretical and numerical arguments

Some New Mixed Finite Elements in View of the Numerical Solution of Time Dependent Wave Propagation Problems

We present the construction and the analysis of a new family of quadrangular (2D) or cubic (3D) mixed finite elements, leading to an explicit scheme (mass lumping) for the approximation of the acoustic or elastic wave equations, including the case of an anisotropic medium. Non classical error estimates are given for this new element

Would You Choose to be Happy? Tradeoffs Between Happiness and the Other Dimensions of Life in a Large Population Survey

A large literature documents the correlates and causes of subjective well-being, or happiness. But few studies have investigated whether people choose happiness. Is happiness all that people want from life, or are they willing to sacrifice it for other attributes, such as income and health? Tackling this question has largely been the preserve of philosophers. In this article, we find out just how much happiness matters to ordinary citizens. Our sample consists of nearly 13,000 members of the UK and US general populations. We ask them to choose between, and make judgments over, lives that are high (or low) in different types of happiness and low (or high) in income, physical health, family, career success, or education. We find that people by and large choose the life that is highest in happiness but health is by far the most important other concern, with considerable numbers of people choosing to be healthy rather than happy. We discuss some possible reasons for this preference

On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials

This work deals with Perfectly Matched Layers (PMLs) in the context of dispersive media, and in particular for Negative Index Metamaterials (NIMs). We first present some properties of dispersive isotropic Maxwell equations that include NIMs. We then demonstrate numerically the inherent instabilities of the classical PMLs applied to NIMs. We propose and analyse the stability of very general PMLs for a large class of dispersive systems using a new change of variable. We give necessary criteria for the stability of such models. For dispersive isotropic Maxwell equations, this analysis is completed by giving necessary and sufficient conditions of stability. Finally, we propose new PMLs that satisfy these criteria and demonstrate numerically their efficiency