An accurate and efficient multiscale finite-difference frequency-domain method for the scalar Helmholtz equation

Frequency-domain numerical modeling of the seismic wave equation can readily describe frequency-dependent seismic wave behaviors, yet is computationally challenging to perform in finely discretized or large-scale geological models. Conventional finite-difference frequency-domain (FDFD) methods for solving the Helmholtz equation usually lead to large linear systems that are difficult to solve with a direct or iterative solver. Parallel strategies and hybrid solvers can partially alleviate the computational burden by improving the performance of the linear system solver. We develop a novel multiscale FDFD method to eventually construct a dimension-reduced linear system from the scalar Helmholtz equation based on the general framework of heterogeneous multiscale method (HMM). The methodology associated with multiscale basis functions in the multiscale finite-element method (MsFEM) is applied to the local microscale problems of this multiscale FDFD method. Solved from frequency- and medium-dependent local Helmholtz problems, these multiscale basis functions capture fine-scale medium heterogeneities and are finally incorporated into the dimension-reduced linear system by a coupling of scalar Helmholtz problem solutions at two scales. We use several highly heterogeneous models to verify the performance in terms of the accuracy, efficiency, and memory cost of our multiscale method. The results show that our new method can solve the scalar Helmholtz equation in complicated models with high accuracy and quite low time and memory costs compared with the conventional FDFD methods.

Adaptive 27-point finite difference frequency domain method for wave simulation of 3D acoustic wave equation

The finite-difference frequency-domain (FDFD) method has important applications in the wave simulation of various wave equations. To promote the accuracy and efficiency for wave simulation with the FDFD method, we have developed a new 27-point FDFD stencil for 3D acoustic wave equation. In the developed stencil, the FDFD coefficients not only depend on the ratios of cell sizes in the x-, y-, and z-directions, but we also depend on the spatial sampling density (SD) in terms of the number of wavelengths per grid. The corresponding FDFD coefficients can be determined efficiently by making use of the plane-wave expression and the lookup table technique. We also develop a new way for designing an adaptive FDFD stencil by directly adding some correction terms to the conventional second-order FDFD stencil, which is simpler to use and easier to generalize. Corresponding dispersion analysis indicates that, compared to the optimal 27-point stencil derived from the average-derivative method (ADM), the developed adaptive 27-point stencil can reduce the required SD from approximately 4 to 2.2 points per wavelength (PPW) for a cubic mesh and to 2.7 PPW for a general cuboid mesh. Numerical examples of a 3D homogeneous model and SEG/EAGE salt-dome model indicate that the developed stencil is more accurate than the ADM 27-point stencil for cubic and general cuboid meshes, while requiring similar CPU time and computational memory as the ADM 27-point stencil for direct and iterative solvers.

Trapezoid-grid finite difference frequency domain method for seismic wave simulation

The large computational cost and memory requirement for the finite difference frequency domain (FDFD) method limit its applications in seismic wave simulation, especially for large models. For conventional FDFD methods, the discretisation based on minimum model velocity leads to oversampling in high-velocity regions. To reduce the oversampling of the conventional FDFD method, we propose a trapezoid-grid FDFD scheme to improve the efficiency of wave modeling. To alleviate the difficulty of processing irregular grids, we transform trapezoid grids in the Cartesian coordinate system to square grids in the trapezoid coordinate system. The regular grid sizes in the trapezoid coordinate system correspond to physical grid sizes increasing with depth, which is consistent with the increasing trend of seismic velocity. We derive the trapezoid coordinate system Helmholtz equation and the corresponding absorbing boundary condition, then get the FDFD stencil by combining the central difference method and the average-derivative method (ADM). Dispersion analysis indicates that our method can satisfy the requirement of maximum phase velocity error less than $1\%$ with appropriate parameters. Numerical tests on the Marmousi model show that, compared with the regular-grid ADM 9-point FDFD scheme, our method can achieve about $80\%$ computation efficiency improvement and $80\%$ memory reduction for comparable accuracy.

Metasurface Design Using Electromagnetic Inversion

<p>This paper summarizes and synthetically evaluates a method proposed for metasurface design. The method takes as input a set of desired far-field (FF) performance specifications and produces an effective susceptibility distribution that, when illuminated with a known incident field, produces a FF radiation pattern exhibiting the desired specifications. To this end, electromagnetic inversion (inverse source) is used to solve for the desired tangential fields on the output side of the metasurface. A finite-difference frequency-domain solver, recently developed for simulation of metasurfaces, is used to synthetically evaluate the proposed method using a two-dimensional (2D) example. It should be noted that microscopic metasurface design (i.e., the design of the physical metasurface implementation) is beyond the scope of this paper.</p>