Rigorous analysis of internal resonances in 3-D hybrid FE-BIE formulations by means of the Poincaré-Steklov operator

3-D hybrid finite-element (FE) boundary integral equation (BIE) formulations are widely used because of their ability to simulate large inhomogeneous structures in both open and bounded simulation domains by applying each method where it is the most efficient. However, some formulations suffer from breakdown frequencies at which the solution is not uniquely defined and errors are introduced due to internal resonances. In this paper, we investigate the occurrence of spurious solutions resulting from these resonances by using the concept of the Poincare-Steklov or Dirichlet-to-Neumann operator, which provides a relation between the tangential electric field and the electric current on the boundary of a domain. By identifying this operator in both the FE and BIE method, several new properties of internal resonances in 3-D hybrid FE-BIE formulations are easily derived. Several conformal and nonconformal formulations are studied and the theory is then applied to a scattering problem

JCMmode: An Adaptive Finite Element Solver for the Computation of Leaky Modes

We present our simulation tool JCMmode for calculating propagating modes of an optical waveguide. As ansatz functions we use higher order, vectorial elements (Nedelec elements, edge elements). Further we construct transparent boundary conditions to deal with leaky modes even for problems with inhomogeneous exterior domains as for integrated hollow core Arrow waveguides. We have implemented an error estimator which steers the adaptive mesh refinement. This allows the precise computation of singularities near the metal's corner of a Plasmon-Polariton waveguide even for irregular shaped metal films on a standard personal computer.Comment: 11 page

Exponential Krylov time integration for modeling multi-frequency optical response with monochromatic sources

Light incident on a layer of scattering material such as a piece of sugar or white paper forms a characteristic speckle pattern in transmission and reflection. The information hidden in the correlations of the speckle pattern with varying frequency, polarization and angle of the incident light can be exploited for applications such as biomedical imaging and high-resolution microscopy. Conventional computational models for multi-frequency optical response involve multiple solution runs of Maxwell's equations with monochromatic sources. Exponential Krylov subspace time solvers are promising candidates for improving efficiency of such models, as single monochromatic solution can be reused for the other frequencies without performing full time-domain computations at each frequency. However, we show that the straightforward implementation appears to have serious limitations. We further propose alternative ways for efficient solution through Krylov subspace methods. Our methods are based on two different splittings of the unknown solution into different parts, each of which can be computed efficiently. Experiments demonstrate a significant gain in computation time with respect to the standard solvers.Comment: 22 pages, 4 figure

Boosting the Maxwell double layer potential using a right spin factor

We construct new spin singular integral equations for solving scattering problems for Maxwell's equations, both against perfect conductors and in media with piecewise constant permittivity, permeability and conductivity, improving and extending earlier formulations by the author. These differ in a fundamental way from classical integral equations, which use double layer potential operators, and have the advantage of having a better condition number, in particular in Fredholm sense and on Lipschitz regular interfaces, and do not suffer from spurious resonances. The construction of the integral equations builds on the observation that the double layer potential factorises into a boundary value problem and an ansatz. We modify the ansatz, inspired by a non-selfadjoint local elliptic boundary condition for Dirac equations

Comparison of outflow boundary conditions for subsonic aeroacoustic simulations

Aeroacoustics simulations require much more precise boundary conditions than classical aerodynamics. Two classes of non-reflecting boundary conditions for aeroacoustics are compared in the present work: characteristic analysis based methods and Tam and Dong approach. In characteristic methods, waves are identified and manipulated at the boundaries while Tam and Dong use modified linearized Euler equations in a buffer zone near outlets to mimic a non-reflecting boundary. The principles of both approaches are recalled and recent characteristic methods incorporating the treatment of transverse terms are discussed. Three characteristic techniques (the original NSCBC formulation of Poinsot and Lele and two versions of the modified method of Yoo and Im) are compared to the Tam and Dong method for four typical aeroacoustics problems: vortex convection on a uniform flow, vortex convection on a shear flow, acoustic propagation from a monopole and from a dipole. Results demonstrate that the Tam and Dong method generally provides the best results and is a serious alternative solution to characteristic methods even though its implementation might require more care than usual NSCBC approaches

Computation of electromagnetic fields inside three dimensional inhomogeneous dielectrics using a buffered block forward backward algorithm

The paper is concerned with the electromagnetic scattering from a three-dimensional inhomogeneous dielectric object. In particular, the paper compares the use of a buffered block forward backward (BBFB) algorithm to the use of the commonly employed weak form of the CG-FFT method for the numerical solution of the resultant electric field integral equation (EFIE). The BBFB method is based on the spatial segmentation of the dielectric into smaller pieces. Results are shown which illustrate the convergence of the algorithm and its superior performance to the CG-FFT

High-order, Dispersionless "Fast-Hybrid" Wave Equation Solver. Part I: O(1)\mathcal{O}(1) Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.Comment: 33 pages, 8 figures, revised and extended manuscript (and now including direct comparisons to existing CQ and TDIE solver implementations) (Part I of II

Modeling seismic wave propagation and amplification in 1D/2D/3D linear and nonlinear unbounded media

To analyze seismic wave propagation in geological structures, it is possible to consider various numerical approaches: the finite difference method, the spectral element method, the boundary element method, the finite element method, the finite volume method, etc. All these methods have various advantages and drawbacks. The amplification of seismic waves in surface soil layers is mainly due to the velocity contrast between these layers and, possibly, to topographic effects around crests and hills. The influence of the geometry of alluvial basins on the amplification process is also know to be large. Nevertheless, strong heterogeneities and complex geometries are not easy to take into account with all numerical methods. 2D/3D models are needed in many situations and the efficiency/accuracy of the numerical methods in such cases is in question. Furthermore, the radiation conditions at infinity are not easy to handle with finite differences or finite/spectral elements whereas it is explicitely accounted in the Boundary Element Method. Various absorbing layer methods (e.g. F-PML, M-PML) were recently proposed to attenuate the spurious wave reflections especially in some difficult cases such as shallow numerical models or grazing incidences. Finally, strong earthquakes involve nonlinear effects in surficial soil layers. To model strong ground motion, it is thus necessary to consider the nonlinear dynamic behaviour of soils and simultaneously investigate seismic wave propagation in complex 2D/3D geological structures! Recent advances in numerical formulations and constitutive models in such complex situations are presented and discussed in this paper. A crucial issue is the availability of the field/laboratory data to feed and validate such models.Comment: of International Journal Geomechanics (2010) 1-1

Hermite regularization of the Lattice Boltzmann Method for open source computational aeroacoustics

The lattice Boltzmann method (LBM) is emerging as a powerful engineering tool for aeroacoustic computations. However, the LBM has been shown to present accuracy and stability issues in the medium-low Mach number range, that is of interest for aeroacoustic applications. Several solutions have been proposed but often are too computationally expensive, do not retain the simplicity and the advantages typical of the LBM, or are not described well enough to be usable by the community due to proprietary software policies. We propose to use an original regularized collision operator, based on the expansion in Hermite polynomials, that greatly improves the accuracy and stability of the LBM without altering significantly its algorithm. The regularized LBM can be easily coupled with both non-reflective boundary conditions and a multi-level grid strategy, essential ingredients for aeroacoustic simulations. Excellent agreement was found between our approach and both experimental and numerical data on two different benchmarks: the laminar, unsteady flow past a 2D cylinder and the 3D turbulent jet. Finally, most of the aeroacoustic computations with LBM have been done with commercial softwares, while here the entire theoretical framework is implemented on top of an open source library (Palabos).Comment: 34 pages, 12 figures, The Journal of the Acoustical Society of America (in press