Uniform Grid Spacing Research Articles

A 3D frequency-domain electromagnetic solver employing a high order compact finite-difference scheme

Three-dimensional (3D) electromagnetic (EM) modeling is an important tool for geophysical applications. In EM induction methods the modified Helmholtz equation is often used to describe scattered or residual electric fields in three dimensions. Throughout this paper, a high order compact finite difference scheme for the solution of that equation for vertical magnetic dipole source (VMD) is presented. The approximation of the residual electric field intensity using a fourth order compact finite difference (FD) discretizer is achieved with the solution of a block linear system, the coefficient matrices are large and sparse with a particular structure, implying the application of a matrix-free Krylov subspace method for an efficient numerical solution. The proposed solver is being examined using a number of test problems in uniform and non-uniform grid spacing where numerical and analytical solutions for the homogeneous half-space cases are being compared.

Direct simulation of stably stratified wall-bounded turbulence using the lattice Boltzmann method

The lattice Boltzmann method (LBM) is employed to simulate stratified plane Couette (SPC) flows in their statistically stationary turbulent state. The aim is to assess the suitability of the LBM for direct simulation of wall-bounded, sheared turbulence under the influence of stable stratification. The SPC flow is generated by two parallel plates moving in opposite directions with velocities ± U w, and the buoyancy is fixed at ± b w at the upper and lower plates, respectively. The Reynolds number Re = U w h / ν, where h is the half-gap height, and ν is the kinematic viscosity, varies from 1000 to 3000. The Richardson number Ri = b w h / U w 2 is set to 0 or 0.01. The LBM results are compared to direct numerical simulations using the conventional pseudo-spectral method, and good agreement is found in various turbulence statistics, such as mean and fluctuation velocity and buoyancy, Reynolds stress, turbulent heat flux, dissipation rate, wall fluxes of momentum and heat, and longitudinal and transverse turbulence spectra. The results from grid-sensitivity tests indicate that the uniform isotropic grid spacing Δ x in LBM needs to be no greater than approximately the near-wall viscous length scale δ ν to achieve adequate resolution of stratified wall-bounded turbulence.

Assessment of high order regularized hybrid lattice Boltzmann scheme for turbulent thermal convection

In this paper, fluid behavior and the second order statistics of developed turbulent natural convection is numerically predicted by the hybrid lattice Boltzmann method. In order to simulate the high Rayleigh number flow, the high order regularized lattice Boltzmann scheme combined with the finite difference solution of energy equation is used. An in-house code was carefully validated against the reference data computed by different numerical macroscopic and mesoscopic techniques. Simulations were performed for cavity aspect ratio 1 ≤ Ar ≤ 4 and Rayleigh number 1010 ≤ Ra ≤ 1011. It was found that two-dimensional (2D) models overpredicted turbulent statistics in comparison with three-dimensional (3D) models. Moreover, less stratified thermal fields are observed under 2D simulations. However, the mean Nusselt numbers are well agreed both for 2D and 3D models. The proposed hybrid model with uniform grid spacing reproduced integral heat transfer characteristics with acceptable accuracy for engineering applications. In particular, the relative error in the mean Nusselt number is within 5% when comparing the results with conventional fully resolved Navier-Stokes formulation.

A New Surface Node Method to Accurately Model the Mechanical Behavior of the Boundary in 3D State-Based Peridynamics

Peridynamics is a non-local continuum theory capable of modeling crack initiation and propagation in solid bodies. However, the layer near the boundary of the body exhibits a stiffness fluctuation due to the so-called surface effect and the inaccurate way of imposing the boundary conditions. Moreover, in numerical models discretized using the meshfree method with uniform grid spacing, there are no nodes on the external surface of the body where the boundary conditions should be applied. Inspired by the method of the fictitious nodes with the Taylor-based extrapolation, we propose an innovative method that introduces a new type of nodes lying on the external surface of the body, i.e., the surface nodes. These nodes represent the interactions between the nodes within the body and the fictitious nodes surrounding the body, and they are used to mitigate the surface effect and properly impose the boundary conditions via the concept of force flux. Moreover, a procedure to compute the analytical solution of peridynamic problems is developed: a manufactured displacement field is prescribed and the volume and surface forces, to obtain that displacement field, are computed. The benefits of the surface node method are shown by means of several 2D and 3D quasi-static examples by comparing the numerical results with other methods with or without boundary corrections.

On the application and grid-size sensitivity of the urban dispersion model CAIRDIO v2.0 under real city weather conditions

Abstract. There is a gap between the need for city-wide air-quality simulations considering the intra-urban variability and mircoscale dispersion features and the computational capacities that conventional urban microscale models require. This gap can be bridged by targeting model applications on the gray zone situated between the mesoscale and large-eddy scale. The urban dispersion model CAIRDIO is a new contribution to the class of computational-fluid dynamics models operating in this scale range. It uses a diffuse-obstacle boundary method to represent buildings as physical obstacles at gray-zone resolutions in the order of tens of meters. The main objective of this approach is to find an acceptable compromise between computationally inexpensive grid sizes for spatially comprehensive applications and the required accuracy in the description of building and boundary-layer effects. In this paper, CAIRDIO is applied on the simulation of black carbon and particulate matter dispersion for an entire mid-size city using a uniform horizontal grid spacing of 40 m. For model evaluation, measurements from five operational air monitoring stations representative for the urban background and high-traffic roads are used. The comparison also includes the mesoscale host simulation, which provides the boundary conditions. The measurements show a dominant influence of the mixing layer evolution at background sites, and therefore both the mesoscale and large-eddy simulation (LES) results are in good agreement with the observed air pollution levels. In contrast, at the high-traffic sites the proximity to emissions and the interactions with the building environment lead to a significantly amplified diurnal variability in pollutant concentrations. These urban road conditions can only be reasonably well represented by CAIRDIO while the meosocale simulation indiscriminately reproduces a typical urban-background profile, resulting in a large positive model bias. Remaining model discrepancies are further addressed by a grid-spacing sensitivity study using offline-nested refined domains. The results show that modeled peak concentrations within street canyons can be further improved by decreasing the horizontal grid spacing down to 10 m, but not beyond. Obviously, the default grid spacing of 40 m is too coarse to represent the specific environment within narrow street canyons. The accuracy gains from the grid refinements are still only modest compared to the remaining model error, which to a large extent can be attributed to uncertainties in the emissions. Finally, the study shows that the proposed gray-scale modeling is a promising downscaling approach for urban air-quality applications. The results, however, also show that aspects other than the actual resolution of flow patterns and numerical effects can determine the simulations at the urban microscale.

Seismotectonic evidence for subduction beneath the Eastern Greater Caucasus

SUMMARY We generated high-resolution 3-D seismic velocity models as well as a relocated earthquake catalogue across the eastern Greater Caucasus and Kura basins. This work was done using data from the recently upgraded Republic Seismological Survey Center's (RSSC) seismic network. We generated our tomographic images of crustal velocity structure in Azerbaijan using double-difference inversions (i.e. tomoDD and hypoDD). Earthquake catalogues from the RSSC between 2011 and 2016 were used; these catalogues include absolute arrival times of 103 288 P- and 120 952 S-wave traveltime picks for 7574 events recorded at 35 stations in Azerbaijan. Beginning with a layered, 1-D velocity model that was estimated using VELEST, we inverted simultaneously for relative location, Vp and Vs on a 3-D grid with dimensions 670 × 445 × 45 km, with a uniform grid spacing of 55 × 55 × 5 km for all of Azerbaijan. We observe that the relocated hypocentres cluster into two depth ranges, at the surface and at depth, that appear to correspond to major fault zones and the top of a subducting plate. Additionally, we note intermediate depth seismicity (∼50–60 km) beneath the Kura Basin, and a northward deepening of earthquake depths. Seismic velocities vary significantly throughout the study region; we observe very slow velocities throughout the Kura Basin between 5 and 15 km, and elevated velocities at 20–35 km. The wholesale velocity structure and seismic structure of Kura Basin strongly mirrors that of the Caspian Sea, which suggests that the geodynamics of the Caspian continue westwards into Azerbaijan. The key results of this study suggest that the northward subduction observed in the Caspian Sea continues beneath the Eastern Greater Caucasus, as well as provides evidence for active faulting along the southern margin of the mountain range.

Optimal Design of Unequal Conductor Spacing In Grounding Grid by Modern Computational Intelligent Techniques

With uniform grid spacing current distribution among grid conductors is non-uniform; conductors towards the periphery carry more current and conductors towards the centre carry less current. Non- uniform grid spacing increases leakage current in the centre conductors while decreases in the fringe conductors bringing about uniform distribution of leakage current. This paper presents optimal design of unequal grid spacing in grounding grid design by two modern computational intelligent techniques: Particle Swarm Optimization technique and Bat Algorithm technique. Compression ratio is the method adapted in creating unequal grid conductor spacing, in which compression ratio C is varied from 0.1 to 1 to result in numerous unequal grid configurations. Modern intelligent techniques such as PSO and BAT algorithm are implemented to find the optimal value of compression ratio C so as to achieve one best unequal grid configuration among numerous feasible unequal grid configurations. Mat Lab software is developed for both PSO and BAT algorithm which are extremely adaptable to examine the unequal grid spacing. Optimal value of compression ratio C is obtained for different number of grid conductors along the x-axis and y-axis and unequal grid spacing is analysed through distinct graphs.

An efficient coding theory for a dynamic trajectory predicts non-uniform allocation of entorhinal grid cells to modules.

Grid cells in the entorhinal cortex encode the position of an animal in its environment with spatially periodic tuning curves with different periodicities. Recent experiments established that these cells are functionally organized in discrete modules with uniform grid spacing. Here we develop a theory for efficient coding of position, which takes into account the temporal statistics of the animal’s motion. The theory predicts a sharp decrease of module population sizes with grid spacing, in agreement with the trend seen in the experimental data. We identify a simple scheme for readout of the grid cell code by neural circuitry, that can match in accuracy the optimal Bayesian decoder. This readout scheme requires persistence over different timescales, depending on the grid cell module. Thus, we propose that the brain may employ an efficient representation of position which takes advantage of the spatiotemporal statistics of the encoded variable, in similarity to the principles that govern early sensory processing.

Seismic tomography of compressional wave attenuation structure for Kı̄lauea Volcano, Hawai‘i

AbstractWe present a frequency‐independent three‐dimensional (3‐D) compressional wave attenuation model (indicated by the reciprocal of quality factor Qp) for Kı̄lauea Volcano in Hawai‘i. We apply the simul2000 tomographic algorithm to the attenuation operator t* values for the inversion of Qp perturbations through a recent 3‐D seismic velocity model and earthquake location catalog. The t* values are measured from amplitude spectra of 26708 P wave arrivals of 1036 events recorded by 61 seismic stations at the Hawaiian Volcanology Observatory. The 3‐D Qp model has a uniform horizontal grid spacing of 3 km, and the vertical node intervals range between 2 and 10 km down to 35 km depth. In general, the resolved Qp values increase with depth, and there is a correlation between seismic activity and low‐Qp values. The area beneath the summit caldera is dominated by low‐Qp anomalies throughout the entire resolved depth range. The Southwest Rift Zone and the East Rift Zone exhibit very high Qp values at about 9 km depth, whereas the shallow depths are characterized with low‐Qp anomalies comparable with those in the summit area. The seismic zones and fault systems generally display relatively high Qp values relative to the summit. The newly developed Qp model provides an important complement to the existing velocity models for exploring the magmatic system and evaluating and interpreting intrinsic physical properties of the rocks in the study area.

Three-Dimensional Compressional Attenuation Model (QP) for the Salton Trough, Southern California

Abstract I present a frequency‐independent 3D seismic attenuation model (indicated by ) for the crust of the Salton trough and the adjacent regions in southeastern southern California. The simul2000 tomographic algorithm was used to invert the frequency‐independent attenuation operator t * values measured from amplitude spectra of 23,378 P ‐wave arrivals of 1203 events through a recently developed 3D velocity model. The Q P model has a uniform horizontal grid spacing of 5 km, and the vertical‐node intervals range between 2 and 5 km down to 27 km depth. In general, the Q P values increase with depth and agree with the surface geology in the shallow depth layers. Low Q P values are observed in the Imperial Valley, California, which are consistent with the sedimentary deposits and may also reflect the presence of pore fluid in the active fault zones, whereas greatly elevated Q P values are shown in the surrounding crystalline ranges. The new Q P model provides an important complement to the existing velocity models for interpreting structural heterogeneity and fluid saturation of rocks in the study area. Online Material: Figures of checkerboard resolution tests, Q P versus V P , and Q P versus heat flow.

High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates

High-order reconstruction schemes for the solution of hyperbolic conservation laws in orthogonal curvilinear coordinates are revised in the finite volume approach. The formulation employs a piecewise polynomial approximation to the zone-average values to reconstruct left and right interface states from within a computational zone to arbitrary order of accuracy by inverting a Vandermonde-like linear system of equations with spatially varying coefficients. The approach is general and can be used on uniform and non-uniform meshes although explicit expressions are derived for polynomials from second to fifth degree in cylindrical and spherical geometries with uniform grid spacing. It is shown that, in regions of large curvature, the resulting expressions differ considerably from their Cartesian counterparts and that the lack of such corrections can severely degrade the accuracy of the solution close to the coordinate origin. Limiting techniques and monotonicity constraints are revised for conventional reconstruction schemes, namely, the piecewise linear method (PLM), third-order weighted essentially non-oscillatory (WENO) scheme and the piecewise parabolic method (PPM).The performance of the improved reconstruction schemes is investigated in a number of selected numerical benchmarks involving the solution of both scalar and systems of nonlinear equations (such as the equations of gas dynamics and magnetohydrodynamics) in cylindrical and spherical geometries in one and two dimensions. Results confirm that the proposed approach yields considerably smaller errors, higher convergence rates and it avoid spurious numerical effects at a symmetry axis.

Three‐dimensional compressional wave attenuation tomography for the crust and uppermost mantle of Northern and central California

AbstractI present a frequency‐independent three‐dimensional (3‐D) compressional wave attenuation model (indicated by quality factor Qp) for the crust and uppermost mantle of Northern and central California. The tomographic inversion used t∗ values measured from amplitude spectra of 80,988 P wave arrivals of 3247 events recorded by 463 network stations through a 3‐D seismic velocity model. The model has a uniform horizontal grid spacing of 15 km, and the vertical node intervals range between 2 and 10 km down to 45 km depth. In general, the resulting Qp values increase with depth and agree with the surface geology at shallow depth layers. The most significant features observed in the Qp model are the high Qp values in the Sierra Nevada mountains and low Qp anomalies in the western fault zones. Low Qp values are also imaged in Owens Valley and Long Valley at shallow depths and the Cape Mendocino region in the lower crust (∼25 km depth). An overall contrast of Qp values across the fault is observed in the creeping, Parkfield and Cholame‐Carrizo sections of the San Andreas Fault. The new 3‐D Qp model provides an important complement to the existing regional‐scale velocity models for interpreting structural heterogeneity and fluid saturation of rocks in the study area.

Application of an Iterative High Order Difference Scheme Along With an Explicit System Solver for Solution of Stream Function-Vorticity Form of Navier–Stokes Equations

This paper describes the general convection-diffusion equation in 2D domain based on a particular fourth order finite difference method. The current fourth-order compact formulation is implemented for the first time, which offers a semi-explicit method of solution for the resulting equations. A nine point finite difference scheme with uniform grid spacing is also put into action for discretization purpose. The proposed numerical model is based on the Navier–Stokes equations in a stream function-vorticity formulation. The fast convergence characteristic can be mentioned as an advantage of this scheme. It combines the enhanced Fournié's fourth order scheme and the expanded fourth order boundary conditions, while offering a semi-explicit formulation. To accomplish this, some coefficients which do not influence the solutions are also omitted from Fournié's formulation. Consequently, very accurate results can be acquired with a relatively coarse mesh in a short time. The robustness and accuracy of the proposed scheme is proved using the benchmark problems of flow in a driven square cavity at medium and relatively high Reynolds numbers, flow over a backward-facing step, and flow in an L-shaped cavity.

Numerical Simulations of Flow around Square Cylinder Using an Iterative High Order Difference Scheme

A time dependent two-dimensional numerical simulation of flow around a rectangular cylinder is conducted using a particular high order compact finite difference method. The current forth-order compact scheme is implemented for the simulation of flow around bluff bodies for the first time. This formulation offers a semiexplicit method for the solution of Navier-Stokes equations in stream function-vorticity form and based on a nine point difference scheme with uniform grid spacing. The main advantage of the suggested method is its fast convergence and high accuracy. The proposed numerical method combines the enhanced Fourni’e’s forth order scheme with a semi-explicit formulation. By this combination, very accurate results can be obtained with a relatively coarse mesh in a short time. Uniform grids are applied with this high performance algorithm, and the accuracy of the scheme is proved using the benchmark problem of incompressible laminar flow around a rectangular cylinder at medium and high Reynolds numbers. Current results have good agreement with other numerical and experimental values. Generally, it is shown that, using the high order finite difference method demonstrates fairly competitive results for the current problem.

Numerical solution of the advection-diffusion equation using Laplace transform finite analytical method

In this study, a numerical method has been investigated and developed to solve the one-dimensional advection-diffusion equation to predict the quality of water in rivers. In this method, time variable is eliminated first by the Laplace transformation, and then a finite analytical method is applied in space. Both are based on a local element of the discretized domain in a finite-volume method. Since the Laplace transformation has been used for temporal approximation, an efficient and accurate inverse Laplace transform method of De Hoog 1982 [An improved method for numerical inversion of Laplace transform. SIAM, Journal of Scientific and Statistical Computing, 3 (3), 357–366] is employed to obtain the solution in real time. The proposed method is compared against analytical solutions and two finite-difference methods. The present computations and comparisons show that the proposed method is superior to the finite-difference methods. The results of the proposed method also agree with analytical solutions without numerical oscillation or diffusion. The present method is applied to steady and unsteady flows and it also provides flexibility for uniform and non-uniform grid spacing and for a wide range of Péclet numbers. It takes less computational effort than finite-difference methods.

Relativistic magnetohydrodynamics in dynamical spacetimes: A new adaptive mesh refinement implementation

We have written and tested a new general relativistic magnetohydrodynamics (GRMHD) code, capable of evolving MHD fluids in dynamical spacetimes with adaptive-mesh refinement (AMR). Our code solves the Einstein-Maxwell-MHD system of coupled equations in full 3+1 dimensions, evolving the metric via the Baumgarte-Shapiro Shibata-Nakamura (BSSN) formalism and the MHD and magnetic induction equations via a conservative, high-resolution shock-capturing scheme. The induction equations are recast as an evolution equation for the magnetic vector potential, which exists on a grid that is staggered with respect to the hydrodynamic and metric variables. The divergenceless constraint div(B)=0 is enforced by the curl of the vector potential. Our MHD scheme is fully compatible with AMR, so that fluids at AMR refinement boundaries maintain div(B)=0. In simulations with uniform grid spacing, our MHD scheme is numerically equivalent to a commonly used, staggered-mesh constrained-transport scheme. We present code validation test results, both in Minkowski and curved spacetimes. They include magnetized shocks, nonlinear Alfv\'en waves, cylindrical explosions, cylindrical rotating disks, magnetized Bondi tests, and the collapse of a magnetized rotating star. Some of the more stringent tests involve black holes. We find good agreement between analytic and numerical solutions in these tests, and achieve convergence at the expected order.

High-precision potential-field and gradient-component transformations and derivative computations using cubic B-splines

Potential-field and gradient-component transformations and derivative computations are necessary for many techniques of data enhancement, direct interpretation, and inversion. We advance new unified formulas for fast interpolation, differentiation, and integration and propose flexible high-precision algorithms to perform 3D and 2D potential-field- and gradient component transformations and derivative computations in the space domain using cubic B-splines. The spline-based algorithms are applicable to uniform or nonuniform rectangular grids for the 3D case and to regular or irregular grids for the 2D case. The fast Fourier transform (FFT) techniques require uniform grid spacing. The spline-based horizontal-derivative computations can be done at any point in the computational domain, whereas the FFT methods use only grid points. Comparisons between spline and FFT techniques through two gravity-gradient examples and one magnetic example show that results computed with the spline technique agree better with the exact theoretical data than do results computed with the FFT technique.

Shallow water model on cubed-sphere by multi-moment finite volume method

A global numerical model for shallow water flows on the cubed-sphere grid is proposed in this paper. The model is constructed by using the constrained interpolation profile/multi-moment finite volume method (CIP/MM FVM). Two kinds of moments, i.e. the point value (PV) and the volume-integrated average (VIA) are defined and independently updated in the present model by different numerical formulations. The Lax–Friedrichs upwind splitting is used to update the PV moment in terms of a derivative Riemann problem, and a finite volume formulation derived by integrating the governing equations over each mesh element is used to predict the VIA moment. The cubed-sphere grid is applied to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry. Highly localized reconstruction in CIP/MM FVM is well suited for the cubed-sphere grid, especially in dealing with the discontinuity in the coordinates between different patches. The mass conservation is completely achieved over the whole globe. The numerical model has been verified by Williamson’s standard test set for shallow water equation model on sphere. The results reveal that the present model is competitive to most existing ones.

The groundwater modeling tool for GRASS (GMTG): Open source groundwater flow modeling

Geographic Information Systems (GIS) are used to store, manipulate and visualize both spatial and non-spatial data. Because of their data manipulating capabilities, GIS have been linked to different simulation models in different research areas and are commonly used for both surface and ground water modeling. Unfortunately this has been done mainly with proprietary GIS which are expensive and which do not provide access to their source code thus making them hard to customize. In order to overcome these problems, a module was created in the Open Source Geographic Resources Analysis Support System (GRASS) GIS to integrate it with the finite difference groundwater flow model MODFLOW, to take full advantage of the GIS capabilities. The results obtained with this module, when compared to those obtained with an existing MODFLOW pre and post-processor show that it can be used to develop groundwater flow models using uniform grid spacing on the horizontal plane. This module provides a tool for groundwater flow modeling to those users who cannot afford the commercially available processors and/or to those who wish to develop their models within a GIS.

High order marching schemes for the wave equation in complex geometry

We present a new class of explicit marching schemes for the wave equation in complex geometry. They rely on a simple embedding of the domain in a uniform Cartesian grid, which allows for efficient and automatic implementation but creates irregular cells near the boundary. While existing explicit finite difference schemes are generally restricted in the size of the time step that can be taken by the dimensions of the smallest cell, the schemes described here are capable of taking time steps dictated by the uniform grid spacing. This should be of significant benefit in a wide variety of simulation efforts.