Full-tensor Anisotropy Research Articles

Time-domain modeling of dispersive and lossy liquid-crystals for terahertz applications

A numerical framework based on the finite-difference time-domain method is proposed for the rigorous study of electro-optically tunable terahertz devices based on liquid crystals. The formulation accounts for both the liquid-crystal full-tensor anisotropy and the dispersion of its complex refractive indices, which is described via modified Lorentzian terms. Experimentally characterized liquid-crystalline materials in the terahertz spectrum are fitted and modeled in benchmark examples, directly compared with reference analytical or semi-analytical solutions. In addition, the efficiency of broadband time-domain modeling of the proposed technique is also demonstrated by accurately reproducing time-domain spectroscopy measurements.

A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids

A new family of cell-centered finite-volume schemes is presented for solving the general full-tensor pressure equation of subsurface flow in porous media on arbitary unstructured triangulations. The new schemes are flux continuous and have full pressure support (FPS) over each subcell with continuous pressure imposed across each control-volume sub-interface, in contrast to earlier formulations. The earlier methods are point-wise continuous in pressure and flux with triangle-pressure-support (TPS) which leads to a more limited quadrature range. An M-matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle. Conditions for the schemes to be positive definite are also derived. A range of computational examples are presented for unstructured triangular grids, including highly irregular grids, and the new FPS schemes are compared against the earlier pointwise continuous TPS formulations. The earlier pointwise TPS methods can induce strong spurious oscillations for problems involving strong full-tensor anisotropy where the M-matrix conditions are violated, and can lead to decoupled solutions in such cases. Unstructured cell-centered decoupling is investigated. In contrast to TPS, the new FPS formulation leads to well resolved solutions that are essentially free of spurious oscillations. A substantial degree of improved convergence behavior, for both pressure and velocity, is also observed in all convergence tests. This is particularly important for problems involving high anisotropy ratios. Also the new formulation proves to be highly beneficial for an upscaling example, where enhancement of convergence is highly significant for certain quadrature points, clearly demonstrating further advantages of the new formulation.

Nonlinear two-point flux approximation for modeling full-tensor effects in subsurface flow simulations

Subsurface flow models can exhibit strong full-tensor anisotropy due to either permeability or grid nonorthogonality effects. Upscaling procedures, for example, generate full-tensor effects on the coarse scale even for cases in which the underlying fine-scale permeability is isotropic. A multipoint flux approximation (MPFA) is often needed to accurately simulate flow for such systems. In this paper, we present and apply a different approach, nonlinear two-point flux approximation (NTPFA), for modeling systems with full-tensor effects. In NTPFA, transmissibility (which provides interblock connections) is determined from reference global flux and pressure fields for a specific flow problem. These fields can be generated using either fully resolved or approximate global simulations. The use of fully resolved simulations leads to an NTPFA method that corresponds to global upscaling procedures, while the use of approximate simulations gives a method corresponding to recently developed local–global techniques. For both approaches, NTPFA algorithms applicable to both single-scale full-tensor permeability systems and two-scale systems are described. A unified framework is introduced, which enables single-scale and two-scale problems to be viewed in a consistent manner. Extensive numerical results demonstrate that the global and local–global NTPFA techniques provide accurate flow predictions over wide parameter ranges for both single-scale and two-scale systems, though the global procedure is more accurate overall. The applicability of NTPFA to the simulation of two-phase flow in upscaled models is also demonstrated.

Accurate local upscaling with variable compact multipoint transmissibility calculations

We propose a new single-phase local upscaling method that uses spatially varying multipoint transmissibility calculations. The method is demonstrated on two-dimensional Cartesian and adaptive Cartesian grids. For each cell face in the coarse upscaled grid, we create a local fine grid region surrounding the face on which we solve two generic local flow problems. The multipoint stencils used to calculate the fluxes across coarse grid cell faces involve the six neighboring pressure values. They are required to honor the two generic flow problems. The remaining degrees of freedom are used to maximize compactness and to ensure that the flux approximation is as close as possible to being two-point. The resulting multipoint flux approximations are spatially varying (a subset of the six neighbors is adaptively chosen) and reduce to two-point expressions in cases without full-tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a local–global upscaling method.

Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods

Discretization methods are proposed for control-volume formulations on polygonal and triangular grid cells in two space dimensions. The methods are applicable for any system of conservation laws where the flow density is defined by a gradient law, like Darcy's law for porous-media flow. A strong feature of the methods is the ability to handle media inhomogeneities in combination with full-tensor anisotropy. This paper gives a derivation of the methods, and the relation to previously published methods is also discussed. A further discussion of the methods, including numerical examples, is given in the companion paper, Part II [SIAM J. Sci. Comput., pp. 1717--1736].

Discretization on Non-Orthogonal, Quadrilateral Grids for Inhomogeneous, Anisotropic Media

Two classes of discretization methods are proposed for control-volume formulations on quadrilateral grids in two space dimensions. Curvilinear grids are considered as a special case. The methods are applicable for any system of conservation laws where the flux is defined by a gradient law, like Darcy's law for porous-media flow. A strong feature of the methods is the ability to handle media inhomogeneities in combination with full-tensor anisotropy and/or non-orthogonality of the grid. Further properties of the methods will be discussed and examples of their use will be presented.