Numerical Spectral Approach Research Articles

Numerical spectral approach for studying activation energy behavior in viscoelastic fluid flow through non-Darcian medium

The work examines the effects of MHD incompressible viscoelastic fluid under nonlinear thermal radiation while flowing in a non-Darcy porous medium. The utilization of the Darcy-Brinkmann-Forchheimer model has thus been implemented in the formulation of the momentum equation. The study posits the existence of a chemical phenomenon which involves elements A and B as two chemical species. The diffusion coefficient of chemical species exhibits inequality when considered for viscoelastic fluid. Numerical solutions are generated using SRM, spectral relaxation technique for governing flow problems. The current algorithm is formulated and implemented using the computational software MATLAB. The results exhibit a considerable degree of agreement with formerly established numerical outcomes within the framework of a specific scenario. The observations from this study provide evidence that chemical species B operates as a catalyst in a subsurface environment that is oriented horizontally. As a result, the connection between the homogeneous-heterogeneous scheme can be observed in the context of isothermal cubic autocatalytic occurrences and processes of the first order, subsequently. The utilization of the heterogeneous reaction parameter offers notable advantages in reducing the concentration of the bulk-fluid while simultaneously enhancing the concentration of the catalyst situated over the surface. The current findings contribute to the exploration of the computational study of diffusivities in MHD viscoelastic fluid flow. Specifically, this research inquires the effects of nonlinear thermal radiation and homogeneous-heterogeneous reactions.

Field Dislocation Mechanics for heterogeneous elastic materials: A numerical spectral approach

Spectral methods using Fast Fourier Transform (FFT) algorithms have recently seen a surge in interest in the mechanics of materials community. The present contribution addresses the critical question of determining accurate local mechanical fields using FFT methods without artificial fluctuations arising from materials and defects induced discontinuities. Precisely, the present work introduces a numerical approach based on intrinsic discrete Fourier transforms for the simultaneous treatment of material discontinuities arising from the presence of dislocations and from elastic stiffness heterogeneities. To this end, the elasto-static equations of the field dislocation mechanics theory for periodic heterogeneous materials are numerically solved with FFT in the case of dislocations in proximity of inclusions of varying stiffness. An optimal intrinsic discrete Fourier transform method is sought based on two distinct schemes. A centered finite difference scheme for differential rules are used for numerically solving the Poisson-type equation in the Fourier space, while centered finite differences on a rotated grid is chosen for the computation of the modified Fourier–Green’s operator associated with the Lippmann–Schwinger-type equation. By comparing different methods with analytical solutions for an edge dislocation in a composite material, it is found that the present spectral method is accurate, devoid of any numerical oscillation, and efficient even for an infinite phase elastic contrast like a hole embedded in a matrix containing a dislocation. The present FFT method is then used to simulate physical cases such as the elastic fields of dislocation dipoles located near the matrix/inclusion interface in a 2D composite material and the ones due to dislocation loop distributions surrounding cubic inclusions in 3D composite material. In these configurations, the spectral method allows investigating accurately the elastic interactions and image stresses due to dislocation fields in the presence of elastic inhomogeneities.

A numerical spectral approach to solve the dislocation density transport equation

A numerical spectral approach is developed to solve in a fast, stable and accurate fashion, the quasi-linear hyperbolic transport equation governing the spatio-temporal evolution of the dislocation density tensor in the mechanics of dislocation fields. The approach relies on using the Fast Fourier Transform algorithm. Low-pass spectral filters are employed to control both the high frequency Gibbs oscillations inherent to the Fourier method and the fast-growing numerical instabilities resulting from the hyperbolic nature of the transport equation. The numerical scheme is validated by comparison with an exact solution in the 1D case corresponding to dislocation dipole annihilation. The expansion and annihilation of dislocation loops in 2D and 3D settings are also produced and compared with finite element approximations. The spectral solutions are shown to be stable, more accurate for low Courant numbers and much less computation time-consuming than the finite element technique based on an explicit Galerkin-least squares scheme.

A numerical spectral approach for solving elasto-static field dislocation and g-disclination mechanics

A spectral approach is developed to solve the elasto-static equations of field dislocation and g-disclination mechanics in periodic media. Given the spatial distribution of Nye’s dislocation density and/or g-disclination density tensors in heterogeneous or homogenous linear elastic media, the incompatible and compatible elastic distortions are respectively obtained from the solutions of Poisson and Navier-type equations in the Fourier space. Intrinsic discrete Fourier transforms solved by the Fast Fourier Transform (FFT) method, which are consistent with the pixel grid for the calculation of first and second order spatial derivatives, are preferred and compared to the classical discrete approximation of continuous Fourier transforms when deriving elastic fields of defects. Numerical examples are provided for homogeneous linear elastic isotropic solids. For various defects, a regularized defect density in the core is considered and smooth elastic fields without Gibbs oscillations are obtained, when using intrinsic discrete Fourier transforms. The results include the elastic fields of single screw and edge dislocations, standard wedge disclinations and associated dipoles, as well as “twinning g-disclinations”. In order to validate the present spectral approach, comparisons are made with analytical solutions using the Riemann–Graves integral operator and with numerical simulations using the finite element approximation.

Effects of First‐Order Approximations on Head and Specific Discharge Covariances in High‐Contrast Log Conductivity

The hydraulic head and the specific discharge fluctuations depend nonlinearly on the hydraulic conductivity. However, the methods most commonly used in the stochastic analysis of groundwater flow are based upon the linearization of these relations. In this paper we apply a numerical spectral approach to investigate the range of validity of the small perturbation approximation for head and specific discharge moments in two‐dimensional finite domains. We find that the small perturbation approximation tends to underestimate the variance of large‐scale head and specific discharge fluctuations and error increases with increasing log‐conductivity variance and increasing domain size. The head fluctuations do not appear ergodic even when the small perturbation approximation predicts they will be ergodic. The specific discharge fluctuations, on the other hand, do appear ergodic. The small perturbation approximation performs well in estimating specific discharge variance in the longitudinal direction but significantly underestimates transverse specific discharge variance.

Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach. 2. Results : Dykaar, B B; Kitandis, P K Water Resour ResV28, N4, April 1991, P1167–1178

Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach. 2. Results : Dykaar, B B; Kitandis, P K Water Resour ResV28, N4, April 1991, P1167–1178

Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach. 1. Method : Dykaar, B B; Kitandis, P K Water Resour ResV28, N4, April 1992, P1155–1166

Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach. 1. Method : Dykaar, B B; Kitandis, P K Water Resour ResV28, N4, April 1992, P1155–1166

Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach: 2. Results

Using the numerical spectral method described in the companion paper (Dykaar and Kitanidis, this issue) the effective conductivity of a three‐dimensional, Isotropic, stationary, lognormally distributed hydraulic conductivity is computed. Six cases were investigated for variances in log conductivity ranging between 1 and 6. The results show that averaging volumes of at least 30 integral scales are required before the effective conductivity reaches its asymptotic value. The results support Matheron's (1967) hypothesis that Ke = KG exp (σY2/6), where Ke is the effective hydraulic conductivity, and KG and σY2 are the geometric mean and variance, respectively, of the hydraulic conductivity field. We also find that for a Gaussian covariance function, heterogeneities smaller than about 1.3 integral scales do not significantly contribute to the effective conductivity. In two dimensions, averaging volumes of more than 80 integral scales are required before the effective conductivity reaches the analytic infinite‐domain result of the geometric mean, while the effective conductivity is insensitive to heterogeneities less than 2 integral scales in size. The method is also applied to data from a shale and sandstone formation, and the results are compared with those from other methods.

Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach: 1. Method

A numerical spectral method is described for solving the equations governing the effective hydraulic conductivity of a stationary heterogeneous porous medium. The numerical method is efficient in the sense that the solution can be achieved in order Nt ln (Nt) operations and order Nt storage, where Nt is the total number of sampling points used in an n‐dimensional flow domain. The numerical method is extensively tested and shown to be quite robust. In the test cases, very accurate results are achieved even for an extremely heterogeneous porous medium. The developed numerical algorithm is quite general and can be used to compute effective conductivities for cases where analytic results are unavailable. In the companion paper (Dykaar and Kitanidis, this issue) the method is used to find the effective conductivity of (1) multidimensional isotropic, stationary, lognormally distributed conductivity and (2) core samples of a shale and sandstone formation.

A numerical spectral approach for the derivation of piezometric head covariance functions

Relating the variability of permeability to the variability of head is a central part of linear estimation techniques such as cokriging. Only a few analytic relationships between log permeability covariances and head covariances presently exist. This paper describes a general numerical procedure which computes head covariances (ordinary or generalized) and cross covariances for any proper log permeability covariance. The numerical spectral method, a discrete analog of Fourier‐Stieltjes analysis, employs the pertinent linearized (small‐perturbation approximation) equations describing the physics of flow. The domain is taken as finite, with boundary effects considered negligible. The numerical spectral method can reproduce all pertinent analytic results with excellent agreement. Furthermore, we demonstrate the method's generality by finding the covariance relations for a case where no analytical results presently exist.