Staggered-grid Method Research Articles

A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

Computational Parallel on Simulation of Wave Attenuation by Mangrove Forest

Coastal ecosystems, specifically mangrove trees, safeguard coastal regions against natural disasters like erosion, floods, and tsunamis. Numerical simulations employing the Shallow Water Equation (SWE), encompassing mass and momentum conservation equations, are used to comprehend how mangroves attenuate wave energy. The SWE incorporates Manning's friction term, which is directly influenced by mangrove forests. However, the SWE's complexity and sensitivity to initial conditions hinder analytical solutions. Despite its increasing computational demands, we utilize the robust staggered grid method to address this challenge. Our study examines mangroves' wave-attenuating effects and introduces a parallel computational model using OpenMP to expedite computations. Findings reveal that mangroves can reduce wave amplitudes by up to 33% when employing a Manning's coefficient of 0.3 within confined basin simulations. Furthermore, our parallel computing experiments demonstrate substantial computation speed enhancements; the speedup improves up to a point, with a notable 7.26-fold acceleration observed when utilizing eight threads compared to a single line. Moreover, more than a 10-fold acceleration is observed when the number of threads is greater than 16. This underscores the significance of parallelization in exploring mangrove contributions to coastal protection.

Coupling staggered-grid and vertex-centered finite-volume methods for coupled porous-medium free-flow problems

A new discretization approach for coupled free and porous-medium flows is introduced, which uses a finite-volume staggered-grid method for the discretization of the Navier-Stokes equations in the free-flow subdomain, while a vertex-centered finite-volume method is employed in the porous medium. The latter allows for the use of unstructured grids to discretize the porous medium, and the presented method is capable of handling non-matching grids at the interface. Moreover, the basis functions of the vertex-centered finite-volume method, with their degrees of freedom located at the interface, allow for an accurate evaluation of the coupling terms and of additional nonlinear velocity-dependent terms in the porous medium. The available advantages of this coupling method are investigated in a series of tests: a convergence test for various grid types, an evaluation of the implementation of the coupling conditions, and an example using the velocity-dependent Forchheimer term in the porous-medium subdomain.

K-space dispersion error compensators for the fractional spatial derivatives based constant-Q viscoelastic wave equation modeling

Unlike the fractional temporal derivatives based constant-Q viscoelastic wave equation, the fractional spatial derivatives based one is preferred for the wavefield simulation due to its tractable memory consumption. This viscoelastic wave equation is often solved by the staggered-grid pseudospectral method, which has the spatial spectral accuracy but temporal low order accuracy. The k-space method is one of the potential remedies for the temporal discretization errors. However, it is difficult to fully remove the temporal dispersion errors by directly applying the elastic k-space error compensators to the viscoelastic wave equation, which contains additional temporal derivatives associated with the amplitude decay terms of the P- and S- waves. To tackle the issue, we have derived a new wavenumber-time domain viscoelastic k-space equation by the eigenvalue decomposition method and by the solution to the matrix differential equations. The error compensation terms inside our k-space equation can simultaneously correct the discretization errors of different types of temporal derivatives and are wave mode adaptive. The accuracy of the k-space wave equation is verified by the viscoelastic Green's function's solution. To further boost the simulation efficiency of the k-space method in heterogeneous media, we have proposed an improved k-space method, which decreases the number of the costly mixed-domain operators. The simulation results from the simple and complex models demonstrate that our two k-space methods are superior to the conventional staggered-grid pseudospectral method for the viscoelastic wave modeling.

Two exact first-order k-space formulations for low-rank viscoacoustic wave propagation on staggered grids

Wave propagation in the viscoacoustic media is physically dispersive and dissipated. Completely excluding the numerical dispersion error from the physical dispersion in the viscoacoustic wave simulation is indispensable to understanding the intrinsic property of the wave propagation in attenuated media for the petroleum exploration geophysics. In recent years, a viscoacoustic wave equation characterized by fractional Laplacian gains wide attention in geophysical community. However, the first-order form of the viscoacoustic wave equation, often solved by the conventional staggered-grid pseudospectral method, suffers from the numerical dispersion error in time due to the low-order finite-difference approximation. It is challenging to completely eliminate the error because the viscoacoustic wave equation contains two temporal derivatives, which stem from the time stepping and the amplitude attenuation terms, respectively. To tackle the issue, we derive two exact first-order k-space viscoacoustic formulations that can fully exclude the numerical error from the physical dispersion. For the homogeneous case, two formulations agree with the viscoacoustic analytical solution very well and have the same efficiency. For the heterogeneous case, our second k-space formulation is more efficient than the first one because the second formulation significantly reduces the number of the wavenumber-space mixed-domain operators, which are the expensive part of the viscoacoustic k-space simulation. Numerical cases validate that the two first-order k-space formulations are effective and efficient alternatives to the current staggered-grid pseudospectral formulation for the viscoacoustic wave simulation.

Application of phase field model coupled with convective effects in binary alloy directional solidification and roll casting processes

Based on the Kim-Kim-Suzuki (KKS) phase field model coupled with the thermodynamic parameters, the transformation process from columnar dendrites to equiaxed crystals during directional solidification of aluminium alloy was simulated, and the effects of phase field parameters on the growth morphology and dendrite segregation were discussed. Furthermore, considering the effect of the microcosmic flow field, the convection influence gradient term is introduced into KKS formula near the solid-liquid interface, and the phase field model considering flow field was applied to the inherent convective environment of the actual roll casting process, also the multiple dendrites growth behavior of magnesium alloy under the action of microscopic convection was further explored. When coupling calculation of microscopic velocity field and pressure field, the staggered grid method was used to deal with the complex interface. The combined solution of Marker in Cell (MAC) algorithm and phase field discrete calculation was realized. In order to further describe the influence of convection on the solidification process, the roll casting experiments are used to verify the impact growth of multiple dendrites under convection. The results show that the dendrites undergo solute remelting and the dendrites melt into equiaxed crystals, showing the phenomenon of Columnar to Equiaxed Transition (CET).

Coupling Staggered-Grid and Vertex-Centered Finite-Volume Methods for Coupled Porous-Medium Free-Flow Problems

Coupling Staggered-Grid and Vertex-Centered Finite-Volume Methods for Coupled Porous-Medium Free-Flow Problems

An Atomization Model of Air Spraying Using the Volume-of-Fluid Method and Large Eddy Simulation

When painting complex surfaces, such as large-curvature surfaces, poor coating quality is often obtained, which may be caused by lack of an appropriate atomization model, insufficient understanding of atomization mechanisms and laws, and improper painting parameters. This paper presents a numerical model of paint atomization of air spraying using the volume-of-fluid method and large eddy simulation. The interface capture and the turbulent flow were mainly considered in the model: the former was tracked by the volume-of-fluid method and the latter was predicted by the large eddy simulation. After the computational domain being meshed by the staggered-grid method, the governing equations were discretized by the finite volume method and were solved by the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) Consistent algorithm. The results of numerical simulations show that the characteristics of atomization flow field, such as velocity variation, pressure distribution, and paint volume fraction are in agreement with the regularities of atomization. Moreover, the primary and secondary atomization phenomena can be clearly observed: as soon as the paint issues from the nozzle, the paint flow begins to distort and the paint fragments continuously eject from the main paint flow and then these paint fragments distort and disintegrate into smaller elements. A comparison with the experimental data from the literature proves that the model of the whole atomization process of air spray is effective. The model is suitable for simulating the whole atomization process and easy to obtain initial conditions, which can be applied to set the appropriate painting parameters and study paint atomization mechanisms and laws in depth.

Numerical Simulation of Flood Routing through Channels with Area Variation using Staggered Grid Method

In the event that the discharge goes to enter the channel surpasses the channel ability, than would happen floods of genuine harm to the environment around the channel. Applying the dynamic wave model, the discharge amount could be predicted in the frequent flood region if the information of discharge upstream is found out. Many numerical schemes could be utilized to discover answers to the system of differential partial equations. We proposed the usage of the staggered method to find out the solution of the system. The numerical scheme shall be used to predict the top discharge in a frequent flood region and create a flooded pocket to diminish the top discharge in the region. Based on the outcome of the numerical simulation, we may claim that a stable scheme has been created to acquire the approach resolve of the equation partial system. The staggered scheme is capable of investigating the impact of the presence of the flood pocket especially lessening the top discharge.

Viscoelastic Wave Propagation Simulation Using New Spatial Variable-Order Fractional Laplacians

ABSTRACTTime-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.

Small-scale medium variations with high-order finite-difference and pseudospectral schemes

The accuracy of implementing interfaces with coarse-grid methods such as the pseudospectral method and high-order finite differences has been considered to be low. Our focus is on variations in interface locations and on inclusions that are significantly smaller than the grid step sizes. Classic implementations of these staggered-grid high-order methods are used. Band-limited versions of the Heaviside step function are used to manipulate the material-parameter grids. Interfaces can be implemented with accuracy that is one order of magnitude smaller than the step size. Small medium inclusions, diffractors, can, be three orders of magnitude smaller in area than the typical cell size and still be modeled with good accuracy. If the method used to implement an interface or to implement a small-scale inclusion is viewed as a filter, then this filter must be accurate up to the spatial Nyquist wavenumber of the simulation grid.

Simulation and characteristics analysis of a wavefield in a thermoelastic medium adopting the rotated staggered-grid pseudo-spectral method and L-S theory

Simulation and characteristics analysis of a wavefield in a thermoelastic medium adopting the rotated staggered-grid pseudo-spectral method and L-S theory

Numerical simulation of seismic wave triggered by low-frequency sound source with 3D staggered-grid difference method in shallow water

Ship seismic wave is often used for targeting ship location. This elastic wave that traveled in the seabed layer is triggered by low-frequency ship noise. At present, most of the theoretical studies of ship seismic wave were conducted in the frequency domain and the wave components such as P-wave, S-wave, and normal mode wave cannot be distinguished. In this study, the propagation characteristics were studied by numerical simulation in the time domain based on 3D staggered-grid finite difference method whose availability was firstly verified through a theoretical solution. The numerical simulation results show that seismic waves in the seabed layer were dominated by transmitted S-wave and interface wave in the near-field area, and transmitted P-wave is not obvious here, while seismic waves were dominated by normal mode and interface wave in the far-field area, and when the sound source is moved away from the seafloor, the amplitude of normal mode wave becomes larger, but the amplitude of interface wave becomes smaller.

Stability Analysis for Wave Simulation in 3D Poroelastic Media with the Staggered-Grid Method

Stability Analysis for Wave Simulation in 3D Poroelastic Media with the Staggered-Grid Method

A Staggered Grid Method for Solving Incompressible Flow on Unstructured Meshes

A finite volume method based unstructured grid is presented to solve the two dimensional viscous and incompressible flow. The method is based on the pressure-correction concept and solved by using a semi-staggered grid technique. The computational procedure can handle cells of arbitrary shapes, although solutions presented in this paper were only involved with triangular and quadrilateral cells. The pressure or pressure-correction value was stored on the vertex of cells. The mass conservation equation was discretized on the dual cells surrounding the vertex of primary cells, while the velocity components and other scale variables were saved on the central of primary cells. Since the semi-staggered arrangement can’t guarantee a strong coupling relationship between pressure and velocity, thus a weak coupling relationship leads to the oscillations for pressure and velocity. In order to eliminate such an oscillation, a special interpolation scheme was used to construct the pressure-correction equation. Computational results of several viscous flow problems show good agreement with the analytical or numerical results in previous literature. This semi-staggered grid method can be applied to arbitrary shape elements, while it has the most efficiency for triangular cells.

3D FDTD anisotropic and dispersive modeling for GPR using rotated staggered grid method

The finite-difference time-domain (FDTD) method is widely used in numerical modeling of ground penetrating radar (GPR). The well-known Yee's method is the earliest and most popular FDTD method. However, using Yee's method to solve an anisotropic case requires considerable interpolation of electric fields (assuming media to be non-magnetic). In this paper we present a different approach called rotated staggered grid (RSG) method to simulate electromagnetic (EM) wave propagation in anisotropic and dispersive media. RSG-FDTD places electric field components on the same grid system and magnetic field components on the same grid system, however, half a spacing away. The RSG-FDTD method was first used in elastic wave modeling. It requires no interpolations of wave fields and is advantageous for modeling wave propagation in anisotropic media. In this paper, the 3D anisotropic and dispersive EM wave equations are presented, and the RSG-FDTD approximations are derived. The application of the new method is tested using homogeneous media models. To validate the proposed method, results are compared with those of GprMax, and good agreement is achieved. The common offset sections of complex cases with anisotropy and dispersion are also tested and are compared with the corresponding isotropic case.

Superposition method for modelling boundaries between media in viscoelastic finite difference time domain simulations.

Finite-difference time domain (FDTD) techniques are widely used to model the propagation of viscoelastic waves through complex and heterogeneous structures. However, in the specific case of media mixing liquid and solid, attempts to model continuous media onto a Cartesian grid produces errors when the liquid-solid interface between different media do not align precisely with the Cartesian grid. The increase in spatial resolution required to eliminate this grid staircasing effect can be computationally prohibitive. Here, a modification to the Virieux staggered-grid FDTD scheme called the superposition method is presented. This method is intended to reduce this staircasing effect while keeping a manageable computational time. The method was validated by comparing low-spatial-resolution simulations against simulations with sufficiently high resolution to provide reasonably accurate results at any incident angle. The comparison of the root-mean-square of the stress amplitude maps showed that the amplitude of artifactual waves could be reduced by several orders of magnitude when compared to the Virieux staggered-grid FDTD method and that the superposition method helped to significantly reduce the staircasing effect in FDTD simulations.

Seiches in a Closed Basin of Various Geometric Shape

In this paper, we will observe wave’s profile when the resonance occur in a closed basin of various geometric shape, which are rectangular, triangular, and semi-parabolic. The governing equation is shallow water equation. We solve the governing equation analytically to obtain the value of natural resonant period of the basin, which causes resonance phenomena. The analytical results were compared to the experimental results to see how good our model is. The equation will also be solved numerically using finite volume on a staggered grid method. Then, the numerical scheme will be validated against the analytical solutions we derived before.

A Robust Optimal Finite Difference Scheme for the Three-Dimensional Helmholtz Equation

We propose a robust optimal 27-point finite difference scheme for the Helmholtz equation in three-dimensional domain. In each direction, a special central difference scheme with 27 grid points is developed to approximate the second derivative operator. The 27 grid points are divided into four groups, and each group is involved in the difference scheme by the manner of weighted combination. As for the approximation of the zeroth-order term, we use the weighted average of all the 27 points, which are also divided into four groups. Finally, we obtain the optimal weights by minimizing the numerical dispersion with the least-square method. In comparison with the rotated difference scheme based on a staggered-grid method, the new scheme is simpler, more practical, and much more robust. It works efficiently even if the step sizes along different directions are not equal. However, rotated scheme fails in this situation. We also present the convergence analysis and dispersion analysis. Numerical examples demonstrate the effectiveness of the proposed scheme.

Phase Prediction of Supercritical Carbon Dioxide and its Application in Fracturing Oil Wellbores

In recent oil and gas exploration, the most reservoirs are low permeability with abundant reserves. Conventional mining of low permeability reservoir is commonly utilizing the hydraulic fracturing technology, whereas, it encounters various technical issues, such as clay expansion and water lock damage. Using the fluid of supercritical carbon dioxide (S-CO2) to exploit the low permeability oil and gas reservoirs is attracting more attention. The implementation of S-CO2, without liquid phase, can help avoid the aforementioned problems. Nevertheless, the phase change of CO2 during fracturing is complicate, and it is difficult to accurately predict the CO2 phase transition. In this work, first, the physical properties of S-CO2 were analyzed by the Span-Wagner model and Vesovic model. Next, S-CO2 was applied to a typical oilfield, and an unsteady coupling model of heat transfer and pressure drop was developed. Then the staggered grid method and iteration procedures were used for numerical solutions, and the temperature and pressure distributions of wellbores were investigated. The results indicate that the temperature control of a wellbore is the key to the phase prediction of S-CO2; CO2 within the single-diameter pipeline below 2300 m can maintain the supercritical state, while CO2 within the stepped pipeline can maintain the supercritical state at the depth of 2280 m. Moreover, compared with the single-diameter pipeline, the bottom pressure of the stepped pipeline is lower and the bottom temperature is higher. By analyzing the flow and heat transfer of S-CO2 in the wellbores, the phase state of S-CO2 was well predicted, which is helpful to improve the exploring performance of low permeability oil and gas reservoirs.