Detailed analysis of the effects of stencil spatial variations with arbitrary high-order finite-difference Maxwell solver

Due to discretization effects and truncation to finite domains, many electromagnetic simulations present non-physical modifications of Maxwell's equations in space that may generate spurious signals affecting the overall accuracy of the result. Such modifications for instance occur when Perfectly Matched Layers (PMLs) are used at simulation domain boundaries to simulate open media. Another example is the use of arbitrary order Maxwell solver with domain decomposition technique that may under some condition involve stencil truncations at subdomain boundaries, resulting in small spurious errors that do eventually build up. In each case, a careful evaluation of the characteristics and magnitude of the errors resulting from these approximations, and their impact at any frequency and angle, requires detailed analytical and numerical studies. To this end, we present a general analytical approach that enables the evaluation of numerical discretization errors of fully three-dimensional arbitrary order finite-difference Maxwell solver, with arbitrary modification of the local stencil in the simulation domain. The analytical model is validated against simulations of domain decomposition technique and PMLs, when these are used with very high-order Maxwell solver, as well as in the infinite order limit of pseudo-spectral solvers. Results confirm that the new analytical approach enables exact predictions in each case. It also confirms that the domain decomposition technique can be used with very high-order Maxwell solver and a reasonably low number of guard cells with negligible effects on the whole accuracy of the simulation.Comment: 33 pages, 14 figure

Scattering from finite structures : an extended Fourier modal method

The Fourier modal method (FMM) is widely used in the diffractive optics community as an effcient tool for simulating scattering from infinitely periodic gratings. In reality the gratings are finite in size, and in applications such as lithography, it is desirable to make them as small as possible. At a certain point the assumption on infinite periodicity loses its validity. This thesis addresses the issues of extending the FMM to finite structures and consequently improving the stability and efficiency of the newly developed method. The aperiodic Fourier modal method in contrast-field formulation (AFMM-CFF) is developed by placing perfectly matched layers at the lateral sides of the computational domain and reformulating the governing equations in terms of a contrast field which does not contain the incoming field. Due to the reformulation, the homogeneous system of second-order ordinary differential equations from the original FMM becomes non-homogeneous. Its solution is derived analytically and used in the established FMM framework. The technique is first demonstrated on a simple problem of planar scattering of TE-polarized light by a single rectangular line. Later the method is generalized to arbitrary shapes of scatterers and conical incidence. The contrast-field formulation of the equations modifies the structure of the resulting linear systems and makes the direct application of available stable recursion algorithms impossible. We adapt the well-known S-matrix algorithm for use with the AFMM-CFF. To this end stable recursive relations are derived for the new type of linear systems. The stability of the algorithm is confirmed by numerical results. The effciency of the AFMM-CFF is improved by exchanging the discretization directions.Classically, spectral discretization is used in the finite periodic direction and spatial dis- cretization in the normal direction. In the light of the fact that the structures of interest have a large width-to-height ratio and that the two discretization techniques have dif- ferent computational complexities, we propose exchanging the discretization directions. This step requires a projection of the background field on the new basis introduced by the alternative discretization. For scatterers with locally repeating patterns, such as finite gratings, exchanging the discretization directions facilitates the reuse of results of previous computations, thus making the method even more efficient. As shown by numerical experiments a considerable reduction of the computational costs can be achieved

Modelling Seismic Wave Propagation for Geophysical Imaging

International audienceThe Earth is an heterogeneous complex media from the mineral composition scale (10−6m) to the global scale ( 106m). The reconstruction of its structure is a quite challenging problem because sampling methodologies are mainly indirect as potential methods (Günther et al., 2006; Rücker et al., 2006), diffusive methods (Cognon, 1971; Druskin & Knizhnerman, 1988; Goldman & Stover, 1983; Hohmann, 1988; Kuo & Cho, 1980; Oristaglio & Hohmann, 1984) or propagation methods (Alterman & Karal, 1968; Bolt & Smith, 1976; Dablain, 1986; Kelly et al., 1976; Levander, 1988; Marfurt, 1984; Virieux, 1986). Seismic waves belong to the last category. We shall concentrate in this chapter on the forward problem which will be at the heart of any inverse problem for imaging the Earth. The forward problem is dedicated to the estimation of seismic wavefields when one knows the medium properties while the inverse problem is devoted to the estimation of medium properties from recorded seismic wavefields

Parsimonious finite-volume frequency-domain method for 2D P-SV-wave modeling

International audienceA new numerical technique for solving the 2D elastodynamic equations based on a finite volume approach is proposed. The associated discretization is through triangles. Only fluxes of required quantities are shared between cells, relaxing meshing conditions compared to finite element methods. The free surface is described along the edges of the triangles which may have different slopes. By applying a parsimonious strategy, stress components are eliminated from the discrete equations and only velocities are left as unknowns in triangles, minimizing the core memory requirement of the simulation. Efficient PML absorbing conditions have been designed for damping waves around the grid. Since the technique is devoted to full waveform inversion, we implemented the method in the frequency domain using a direct solver, an efficient strategy for multiple-source simulations. Standard dispersion analysis in infinite homogeneous media shows that numerical dispersion is similar to those of O(¢x2) staggeredgrid finite-difference formulations when considering structured triangular meshes. The method is validated against analytical solutions of several canonical problems and with numerical solutions computed with a well-established finite-difference time-domain method in heterogeneous media. In presence of a free surface, the finite-volume method requires ten triangles per wavelength for a flat topography and fifteen triangles per wavelength for more complex shapes, well below criteria required by the staircase approximation of finite-difference methods. Comparison between the frequency-domain finite-volume and the O(¢x2) rotated finite-difference methods also shows that the former is faster and less-memory demanding for a given accuracy level. We developed an efficient method for 2-D P-SV-wave modeling on structured triangular meshes as a tool for frequency-domain full-waveform inversion. Further work is required to assess the method on unstructured meshes

Simulation of sound propagation over porous barriers of arbitrary shapes

A time-domain solver using an immersed boundary method is investigated for simulating sound propagation over porous and rigid barriers of arbitrary shapes. In this study, acoustic propagation in the air from an impulse source over the ground is considered as a model problem. The linearized Euler equations are solved for sound propagation in the air and the Zwikker-Kosten equations for propagation in barriers as well as in the ground. In comparison to the analytical solutions, the numerical scheme is validated for the cases of a single rigid barrier with different shapes and for two rigid triangular barriers. Sound propagations around barriers with different porous materials are then simulated and discussed. The results show that the simulation is able to capture the sound propagation behaviors accurately around both rigid and porous barriers