Finite Difference Grid Research Articles

Modeling underwater noise propagation from marine hydrokinetic power devices through a time-domain, velocity-pressure solution.

Marine hydrokinetic (MHK) devices generate electricity from the motion of tidal and ocean currents, as well as ocean waves, to provide an additional source of renewable energy available to the United States. These devices are a source of anthropogenic noise in the marine ecosystem and must meet regulatory guidelines that mandate a maximum amount of noise that may be generated. In the absence of measured levels from in situ deployments, a model for predicting the propagation of sound from an array of MHK sources in a real environment is essential. A set of coupled, linearized velocity-pressure equations in the time-domain are derived and presented in this paper, which are an alternative solution to the Helmholtz and wave equation methods traditionally employed. Discretizing these equations on a three-dimensional (3D), finite-difference grid ultimately permits a finite number of complex sources and spatially varying sound speeds, bathymetry, and bed composition. The solution to this system of equations has been parallelized in an acoustic-wave propagation package developed at Sandia National Labs, called Paracousti. This work presents the broadband sound pressure levels from a single source in two-dimensional (2D) ideal and Pekeris wave-guides and in a 3D domain with a sloping boundary. The paper concludes with demonstration of Paracousti for an array of MHK sources in a simple wave-guide.

Two-phase pore-fluid distribution in fractured media: acoustic wave velocity vs saturation

The acoustic wave velocity varies with fluid saturation and pore-fluid distribution. We use a P-wave source and the staggered grid finite-difference method, with second-order accuracy in time and eighth-order accuracy in space, to simulate the acoustic wave field in a fractured medium that is saturated with a two-phase pore fluid (gas & water). Further, we analyze the variation of acoustic wave velocity with saturation for different pore-fluid distribution modes. The numerical simulation method is simple and yields accurate results.

Unified wave equation and numerical simulation of mechanical wave propagation in alloy solidification

A unified wave equation of mechanical wave propagation during solidification of an alloy was established and the numerical solution of the wave equation was obtained. A three-element model (KSLS) used to describe the stress–strain constitutive relation of the alloy in the mushy zone was established by the analysis of rheological characteristics of molten melt during solidification. Based on the KSLS model, we could describe the constitutive relation of the liquid alloy, a Maxwell medium, and of the solidified alloy, an elastic medium, by the various Lame coefficients. The wave propagation during solidification was identified by a unified wave equation coming from a unified integral constitutive equation, to adapt to a variety of viscoelastic media. The wave equations could be solved succinctly by introducing a "memory factor" and the staggered grid finite difference method. The analytical results demonstrated that the unified wave equation could be perfectly applied to the numerical simulation of wave propagation during solidification. The propagation of the P-wave in a one-dimensional alloy was simulated during solidification, obtaining the propagation law of the mechanical wave: the wave with variable wavelength and amplitude propagated with attenuation during solidification; otherwise, as the source excitation force was fixed, the fluctuations caused by the mechanical wave could be more serious when the vibration was applied in the melt directly, which is more conducive to grain refinement.

Unsteady double-diffusive natural convection in a two-sided lid-driven inclined porous enclosure with sinusoidal boundary conditions with Soret and Dufour effects

ABSTRACTUnsteady double-diffusive natural convection in an inclined porous enclosure with sinusoidal boundary conditions and Soret and Dufour parameters is studied. The unique aspects of the set-up are that the left vertical and bottom walls are heated and concentrated non-uniformly and uniformly, respectively, while the right vertical wall is well insulated and the top wall is maintained at a constant concentration and cold temperature. A staggered grid finite-difference method is used to solve the system of partial differential equations that model heat and mass transfer inside the enclosure. We demonstrate the effects of the Soret and Dufour parameters and the inclination angle on the unsteady double-diffusive natural convection in the inclined porous enclosure. With all the numerical studies, the least square curve fitting (exponential non-linear curve fitting) of average Nusselt number and average Sherwood number with respect to different Soret and Dufour numbers at the left vertical wall which is non-uniformly heated and concentrated is examined here. Comparison with previously published results shows an excellent agreement.

4D multi-physical stress modelling during shale gas production: A case study of Sichuan Basin shale gas reservoir, China

To investigate the time-lapse, three-dimensional (4D) stress during shale gas production, a multi-physical modeling method is proposed. This model enfranchises multiple years' scale dynamic production data. A finite difference method (FDM) reservoir simulator is used to couple thermal-hydrological-chemical (THC) processes, while a finite element method (FEM) geomechanical simulator takes on the role of a thermal-hydrological-mechanical (THM) coupling calculator. In order to couple the flow model with the geomechanical model, an interface (coupling) Python code is provided to communicate data between the finite difference (FD) and finite element (FE) grids. A set of FORTRAN routines are assigned to interpolate initial stress field and mechanical parameters into the geomechanical model based on field and experimental data. Ultimately, a 4D multi-physical model of Sichuan Basin shale gas reservoir (China) is proposed and analyzed. Based on the simulation results, the continuous reduction in reservoir pore pressure leads to evolving changes in the stress state and reservoir properties beyond the pre-consolidation pressure. Transversely isotropic characteristics have a considerable effect on the changes in the reservoir stresses and poro-elastic parameters. The results provide some fundamental insights for optimizing well patterns, ensuring wellbore stability, re-fracturing design and locating, drilling and completing infill wells.

Unsteady double-diffusive convection in a water-based Al2O3-nanofluid in a two-sided lid-driven porous cavity

ABSTRACTA numerical investigation of the flow and behaviour of properties of a water-based Al2O3-nanofluid inside a two-sided lid-driven inclined non-uniformly heated and concentrated porous cavity is made in this paper. The focus of the study is on determining how the buoyancy ratio and the inclination angles influence the unsteady double-diffusive natural convection in a cavity filled with a porous medium, and with non-uniform boundary conditions. We further consider different nanoparticle volume fractions of the nanofluid. It is assumed that the left and right vertical walls are insulated, while the bottom wall is heated and concentrated non-uniformly and the top wall maintained at a constant cold temperature. The top and bottom walls move from left to right and right to left with constant speed, respectively. The governing equations are solved numerically using a staggered grid finite-difference method for streamlines, isotherms, iso-concentrations, average Nusselt number and average Sherwood number for various values of nanoparticle volume fraction, inclination angle and buoyancy ratio. The change in the flow, temperature and concentration profile patterns with respect to time is depicted and described. The results are compared with previously published work and excellent agreement has been obtained.

Energy Density Limits of Multiphase Composites With Dielectric Nanoparticles

Analyzing electric field distributions of multiphase nanoparticle (NP) composites can provide physical insight into their energy density limits. A three-dimensional simulation platform that uses a control volume approach with a finite difference grid is created to efficiently calculate electric field patterns, effective permittivity, and energy density of multiphase composites. To estimate a maximum energy density metric for materials, a physically based percolation model is used to estimate the maximum voltage that can be applied to materials before a potential breakdown path occurs. The simulation predictions are compared to measured data from current work on BaTiO3 NPs infused in a PVDF matrix to help verify simulation results. The simulation platform is then used to explore the possible energy density limits for a variety of nanocomposite material configurations. For example, one such nanocomposite category that is highlighted is the one in which certain metal-oxide dielectric NPs can have localized and highly polarizable dipoles at their surfaces. Finally, simulation is used to explore the impact that an NP's average diameter, degree of clustering, and volumetric density can have on the maximum energy density for a variety of nanocomposite material types.

Study of the Nankai seismogenic fault using dynamic wave propagation modelling of digital rock from the Nobeoka Fault

To understand the characteristics of the Nankai seismogenic fault in the plate convergent margin, we calculated the P- and S-wave velocities (VP and VS) of digital rock models constructed from core samples of an ancient plate boundary fault at Nobeoka, Kyushu Island, Japan. We first constructed 3D digital rock models from microcomputed tomography images and identified their heterogeneous textures such as cracks and veins. We replaced the cracks and veins with air, water, quartz, calcite and other materials with different bulk and shear moduli. Using the Rotated Staggered Grid Finite-Difference Method, we performed dynamic wave propagation simulations and quantified the effective VP, VS and the ratio of VP to VS (VP/VS) of the 3D digital rock models with different crack-filling minerals. Our results demonstrate that the water-saturated cracks considerably decreased the seismic velocity and increased VP/VS. The VP/VS of the quartz-filled rock model was lower than that in the water-saturated case and in the calcite-filled rock model. By comparing the elastic properties derived from the digital rock models with the seismic velocities (e.g. VP and VP/VS) around the seismogenic fault estimated from field seismic data, we characterised the evolution process of the deep seismogenic fault. The high VP/VS and low VP observed at the transition from aseismic to coseismic regimes in the Nankai Trough can be explained by open cracks (or fractures), while the low VP/VS and high VP observed at the deeper coseismic fault zone suggests quartz-filled cracks. The quartz-rich fault zone characterised as low VP/VS and high VP in this study could partially relate to the coseismic behaviour as suggested by previous studies, because quartz exhibits slip-weakening behaviour (i.e. unstable coseismic slip).

A finite element and finite difference mixed approach for modeling fault rupture and ground motion

We combine a finite element (FE) code, Defmod, and a finite difference (FD) code, OpenSWPC, in modeling fault rupture and resulting ground motion. The rupturing process is modeled by the FE code, while the ground motion is modeled by the FD code. The FE-FD binding follows two steps: First, evaluate the FD grid points motion when executing the FE code; and then, impose those moving grid points to the FD domain to drive wave propagation. In order to avoid performance bottleneck, we implement this FE-FD binding in parallel. To verify this mixed approach, we evaluate two SCEC benchmark problems. The resulting synthetic waveforms agree well with the other benchmark participants. The waveform comparisons suggest that the FE-FD mixed model has less attenuation than the pure FE model.

Application of Streamline Simulation to Gas Displacement Processes

Performance evaluation of miscible and near-miscible gas injection processes is available through conventional finite difference (FD) compositional simulation. Streamline methods have also been developed in which fluid is transported along the streamlines instead of using the finite difference grid. In streamline-based simulation, a 3D flow problem is decoupled into a set of 1D problems solved along streamlines. This reduces simulation time relative to FD simulation, and suppresses the numerical dispersion errors that are present in FD simulations. Larger time steps and higher spatial resolution can be achieved in these simulations. Thus, streamline-based reservoir simulation can be orders of magnitude faster than the conventional finite difference methods. Streamline methods are traditionally only applied to incompressible flow processes. In this paper, the method is adopted and assessed for application to compressible flow processes. A detailed comparison is given between the results of conventional FD simulation and the streamline approach for gas displacement processes. Finally, some guidelines are given on how the streamline method can potentially be used to good effect for gas displacement processes.

Dispersion analysis of the [formula omitted] mixed finite element pair for atmospheric modelling

Mixed finite element methods provide a generalisation of staggered grid finite difference methods with a framework to extend the method to high orders. The ability to generate a high order method is appealing for applications on the kind of quasi-uniform grids that are popular for atmospheric modelling, so that the method retains an acceptable level of accuracy even around special points in the grid. The dispersion properties of such schemes are important to study as they provide insight into the numerical adjustment to imbalance that is an important component in atmospheric modelling. This paper extends the recent analysis of the P2−P1DG pair, that is a quadratic continuous and linear discontinuous finite element pair, to higher polynomial orders and also spectral element type pairs. In common with the previously studied element pair, and also with other schemes such as the spectral element and discontinuous Galerkin methods, increasing the polynomial order is found to provide a more accurate dispersion relation for the well resolved part of the spectrum but at the cost of a number of unphysical spectral gaps. The effects of these spectral gaps are investigated and shown to have a varying impact depending upon the width of the gap. Finally, the tensor product nature of the finite element spaces is exploited to extend the dispersion analysis into two-dimensions.

Accurate seabed modeling using finite difference methods

Finite difference is the most widely used method for seismic wavefield modeling. However, most finite-difference implementations discretize the Earth model over a fixed grid interval. This can lead to irregular model geometries being represented by ‘staircase’ discretization, and potentially causes mispositioning of interfaces within the media. This misrepresentation is a major disadvantage to finite difference methods, especially if there exist strong and sharp contrasts in the physical properties along an interface. The discretization of undulated seabed bathymetry is a common example of such misrepresentation of the physical properties in finite-difference grids, as the seabed is often a particularly sharp interface owing to the rapid and considerable change in material properties between fluid seawater and solid rock. There are two issues typically involved with seabed modeling using finite difference methods: firstly, the travel times of reflections from the seabed are inaccurate as a consequence of its spatial mispositioning; secondly, artificial diffractions are generated by the staircase representation of dipping seabed bathymetry. In this paper, we propose a new method that provides a solution to these two issues by positioning sharp interfaces at fractional grid locations. To achieve this, the velocity model is first sampled in a model grid that allows the center of the seabed to be positioned at grid points, before being interpolated vertically onto a regular modeling grid using the windowed sinc function. This procedure allows undulated seabed bathymetry to be represented with improved accuracy during modeling. Numerical tests demonstrate that this method generates reflections with accurate travel times and effectively suppresses artificial diffractions.

A staggered grid finite difference method for solving the gravity wave-model equations

The gravity wave-model equations are considered. The equations govern shallow water flows, where the gravity effect is significant. The equations form a system of partial differential equations with the hyperbolic type. When the gravity wave-model equations are solved using a finite difference method with staggered grids, artificial oscillations may occur in the numerical solution. In this paper, we propose a numerical treatment of a staggered grid finite difference method for solving the gravity wave-model equations. Using our proposed treatment, artificial oscillations can be eliminated, so the resulting numerical solution mimics the physics of the problem.

A staggered grid finite difference method for solving the elastic wave equations

We consider elasticity (elastic wave) equations for solving acoustics problems. We use the finite difference method on staggered grids to solve the elastic wave equations. On staggered grids, the pressure is computed at a set of spatial points, and the velocity is computed at another set of spatial points. An advantage of using staggered grids is that fluxes do not need to be approximated in the solving process, as the flux values are known. Numerical results confirm that our numerical method is reliable.

Entropy generation minimization for the sloshing phenomenon in half-full elliptical storage tanks

In this paper, the entropy generation in the sloshing phenomenon was obtained in elliptical storage tanks and the optimum geometry of tank was suggested. To do this, a numerical model was developed to simulate the sloshing phenomenon by using coupled Reynolds-Averaged Navier–Stokes (RANS) solver and the Volume-of-Fluid (VOF) method. The RANS equations were discretized and solved using the staggered grid finite difference and SMAC methods, and the available data were used for the model validation. Some parameters consisting of maximum free surface displacement (MFSD), maximum horizontal force exerted on the tank perimeter (MHF), tank perimeter (TP), and total entropy generation (Sgen) were introduced as design criteria for elliptical storage tanks. The entropy generation distribution provides designers with useful information about the causes of the energy loss. In this step, horizontal periodic sway motions as X=amsin(ωt) were applied to elliptical storage tanks with different aspect ratios namely ratios of large diameter to small diameter of elliptical storage tank (AR). Then, the effect of am and ω was studied on the results. The results show that the relation between MFSD and MHF is almost linear relative to the sway motion amplitude. Moreover, the results show that an increase in the AR causes a decrease in the MFSD and MHF. The results, also, show that the relation between MFSD and MHF is nonlinear relative to the sway motion angular frequency. Furthermore, the results show that an increase in the AR causes that the relation between MFSD and MHF becomes linear relative to the sway motion angular frequency. In addition, MFSD and MHF were minimized in a sway motion with a 7 rad/s angular frequency. Finally, the results show that the elliptical storage tank with AR=1.2–1.4 is the optimum section.

Eventual periodicity of the forced oscillations for a Korteweg–de Vries type equation on a bounded domain using a sinc collocation method

We demonstrate numerically the eventual time periodicity of solutions u ( . , t ) to the Korteweg–de Vries type equation with periodic forcing at one end using the sinc-collocation method. This method approximates the space dimension of the solution with a cardinal expansion of sinc functions, thus allowing the avoidance of a costly finite difference grid for a third order boundary value problem. The first order time derivative is approximated with a θ -weighted finite difference method. The sinc-collocation method was found to be more robust and more efficient than other numerical schemes when applied to this problem.

Modeling effective elastic properties of digital rocks using a new dynamic stress-strain simulation method

We have developed a dynamic stress-strain simulation methodology to compute effective elasticity on digital rocks in a wide range of frequency based on the rotated staggered grid finite-difference method. The primary advantage of this simulator lies in characterizing the anisotropic behavior of complex porous rocks by setting specified boundary conditions along specified directions: The edges perpendicular to the propagating wave are applied with a strain boundary condition, and the edges parallel to the propagating wave are applied with a periodic boundary condition. The accuracy of the simulator is validated by comparing the simulating results in microinhomogeneous porous media containing randomly distributed inclusions and aligned oriented cracks, with that obtained by effective medium theories of self-consistent approximation in isotropic and anisotropic domains. This dynamic simulator can successfully capture anisotropic magnitude of rocks containing needle-like inclusions with different degrees of alignment. For real digital rock saturated with viscoelastic fluids, it is able to predict the dependence of velocities and attenuating factors on the frequency. We found that the magnitude of dispersion increases with the increase of pore fluid viscosity. Therefore, this method offers a robust and effective tool to compute effective elastic properties and anisotropy for real complex porous rocks.

Analysis of Hexagonal Grid Finite Difference Methods for Anisotropic Laplacian Related Equations

AbstractHexagonal grids are valuable in two-dimensional applications involving Laplacian. The methods and analysis are investigated in current work in both linear and nonlinear problems related to anisotropic Laplacian. Ordinary and compact hexagonal grid finite difference methods are developed by elementary arguments, and then analyzed by perturbation for standard Laplacian. In the anisotropic case, analysis is done through reduction to the standard one by using Fourier vectors of mixed types. These hexagonal seven-point methods, with established theoretic stabilities and accuracies, are numerically confirmed in linear and semi-linear anisotropic Poisson problems, and can be applied also in time-dependent problems and in many applications in two-dimensional irregular domains.

Numerical simulation of 2-D seismic wave propagation in the presence of a topographic fluid–solid interface at the sea bottom by the curvilinear grid finite-difference method

Numerical simulation of 2-D seismic wave propagation in the presence of a topographic fluid–solid interface at the sea bottom by the curvilinear grid finite-difference method

Modelling of Fluidised Geomaterials: The Case of the Aberfan and the Gypsum Tailings Impoundment Flowslides.

The choice of a pure cohesive or a pure frictional viscoplastic model to represent the rheological behaviour of a flowslide is of paramount importance in order to obtain accurate results for real cases. The principal goal of the present work is to clarify the influence of the type of viscous model—pure cohesive versus pure frictional—with the numerical reproduction of two different real flowslides that occurred in 1966: the Aberfan flowslide and the Gypsum tailings impoundment flowslide. In the present work, a depth-integrated model based on the Biot–Zienkiewicz formulation, enhanced with a diffusion-like equation to account for the pore pressure evolution within the soil mass, is applied to both 1966 cases. For the Aberfan flowslide, a frictional viscous model based on Perzyna viscoplasticity is considered, while a pure cohesive viscous model (Bingham model) is considered for the case of the Gypsum flowslide. The numerical approach followed is the SPH method, which has been enriched by adding a 1D finite difference grid to each SPH node in order to improve the description of the pore water evolution in the propagating mixture. The results obtained by the performed simulations are in agreement with the documentation obtained through the UK National Archive (Aberfan flowslide) and the International Commission of large Dams (Gypsum flowslide).