Spatial Discretization Method Research Articles

CFD-DEM solution verification: Fixed-bed studies

Fixed (frozen) particle configurations are used to study CFD-DEM convergence behavior in the limit of infinite grid resolution. A filtering scheme based on solving a diffusion equation, i.e., a Gaussian kernel, is used to interpolate particle information to the continuum grid, free of cell size limitations. A regression method is used for Richardson extrapolation of the discrete solutions to the grid-free solution. The regression method is shown to predict accurate extrapolation curves in the presence of the observed oscillatory convergence behavior. The extrapolation curve is used to complete the solution verification process by quantifying the numerical uncertainty present in the calculations. Several numerical and model parameters are studied including the spatial discretization method, the kernel width and the mean drag law. A manageable level of discretization error is found with relative discretization errors generally <4%.

Non-normal stability of embedded boundary methods through pseudospectra

We present non-normal linear stability of embedded boundary (EB) methods employing pseudospectra and resolvent norms. We analyze the stability of the linear wave equation in a domain with an embedded boundary in one and two dimensions. In particular, we characterize the stability in terms of the normalized distance of the EB to the nearest ghost node (α). An important objective is that the CFL condition remains unaffected by the EB which is taken to be that of the underlying grid (as if the EB is absent or aligned with the regular grid). We consider various spatial and temporal discretization methods including both central and upwind-biased schemes. Stability is guaranteed when α≤αmax where αmax ranges between 0.5 and 0.77 depending on the discretization scheme. The stability analysis is also briefly examined for the one-dimensional advection–diffusion equation. Sharper limits on the sufficient conditions for stability are obtained based on the pseudospectral radius (the Kreiss constant) than the restrictive limits based on the usual singular value decomposition analysis. A resulting sufficient condition for stability is that the intersection of the EB with the surface normal passing through a ghost node (x→g) must lie within the ghost cell (0.5Δx distance from x→g). This condition leads us to propose a simple and robust reclassification scheme for the ghost cells (dubbed “hybrid ghost cells”) to ensure Lax stability of the discrete system. This has been tested successfully for both low and high order discretization schemes with transient growth of at most O(1). Moreover, we present a fourth order EB reconstruction scheme taking into account both accuracy and stability.

Numerical simulations of nonlocal phase-field and hyperbolic nonlocal phase-field models via localized radial basis functions-based pseudo-spectral method (LRBF-PSM)

In this paper we consider the two-dimensional nonlocal phase-field and hyperbolic nonlocal phase-field models to obtain their numerical solutions. For this purpose, we propose a localized method based on radial basis functions (RBFs), namely localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for spatial discretization. The basic idea of the LRBF-PSM is to construct a set of orthogonal functions by RBFs on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. This approach does not require meshing in spatial domain and hence inherits the meshless and spectral convergence properties of the global radial basis functions collocation method (GRBFCM). Some numerical results indicate that the obtained simulations via the LRBF-PSM is effective and stable for approximating the solution of nonlocal models investigated in the current paper.

Decoupled, Energy Stable Scheme for Hydrodynamic Allen-Cahn Phase Field Moving Contact Line Model

In this paper, we present an efficient energy stable scheme to solve a phase field model incorporating contact line condition. Instead of the usually used Cahn-Hilliard type phase equation, we adopt the Allen-Cahn type phase field model with the static contact line boundary condition that coupled with incompressible Navier-Stokes equations with Navier boundary condition. The projection method is used to deal with the Navier-Stokes equa- tions and an auxiliary function is introduced for the non-convex Ginzburg-Landau bulk potential. We show that the scheme is linear, decoupled and energy stable. Moreover, we prove that fully discrete scheme is also energy stable. An efficient finite element spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed scheme is very efficient and accurate

Understanding methanol synthesis from CO/H2 feeds over Cu/CeO2 catalysts

Understanding copper-based methanol synthesis catalysts as a function of catalyst type and applied reaction conditions is an area of active industrial and academic research. In this work, results of a methanol synthesis study over a Cu/CeO2 catalyst using CO/H2 feeds are presented, using catalyst performance data combined with catalyst characterisation information (DRIFTS, XPS, TPR, XRD) obtained during start-up and under steady state methanol synthesis conditions. The results indicate that the active site and reaction mechanism for methanol synthesis over Cu/CeO2 are different from the conventional Cu/ZnO/Al2O3 catalyst, with CO, rather than CO2, being the carbon source for methanol. Fixed-bed micro reactors were employed to obtain catalyst performance data in discrete bed sectors using experimental spatial discretisation methods. Methanol synthesis activity over Cu/CeO2 from CO/H2 is preceded by transient CO2 formation, with the onset of methanol synthesis activity observed when the CO2 formation reaches its peak. Considerable differences between reactor bed sectors are observed during start-up, with CO2 formation, CO2 re-adsorption (as surface carbonates and formates), methanol formation and transient methanol decomposition occurring to different degrees and at different time scales.The interfaces of defective CeO2−x in contact with highly dispersed copper particles form the active site of the catalyst under steady-state methanol synthesis conditions, with the copper oxidation state being a combination of 0 and +1 states. The CeO2−x defect sites are formed during start-up upon reaction of CeO2 with CO. The CO2 thus formed leads to catalyst deactivation due to build-up of carbonate and formate species on the catalyst surface. A gradient of progressive poisoning/deactivation is observed going further down the catalyst bed, where the level of deactivation can be controlled by modifying the CO content in the feed. CO2 thus acts as a poison for the Cu/CeO2 catalyst, and addition of even low levels of CO2 to the feed leads to an evenly deactivated catalyst. During start-up in CO/H2, transient CO2 and H2O formation is also observed in the case of Cu/ZnO/Al2O3, ascribed to partial reduction of ZnO, similar to the partial reduction of CeO2 as seen in Cu/CeO2. The key difference between the two systems is the absence of CO2 and H2O re-adsorption on Cu/ZnO/Al2O3, resulting in stable methanol production distributed equally over the catalyst bed. Introduction of CO2 to the CO/H2 feed in case of Cu/ZnO/Al2O3 leads to reduction of Cu+ to Cu0 and a change in methanol synthesis mechanism from CO to CO2 hydrogenation.

Gauss‐Lobatto‐Legendre‐Birkhoff pseudospectral approximations for the multi‐term time fractional diffusion‐wave equation with Neumann boundaryconditions

A second‐order finite difference/pseudospectral scheme is proposed for numerical approximation of multi‐term time fractional diffusion‐wave equation with Neumann boundary conditions. The scheme is based upon the weighted and shifted Grünwald difference operators approximation of the time fractional calculus and Gauss‐Lobatto‐Legendre‐Birkhoff (GLLB) pseudospectral method for spatial discretization. The unconditionally stability and convergence of the scheme are rigorously proved. Numerical examples are carried out to verify theoretical results.

AFiD-GPU: A versatile Navier–Stokes solver for wall-bounded turbulent flows on GPU clusters

The AFiD code, an open source solver for the incompressible Navier–Stokes equations (http://www.afid.eu), has been ported to GPU clusters to tackle large-scale wall-bounded turbulent flow simulations. The GPU porting has been carried out in CUDA Fortran with the extensive use of kernel loop directives (CUF kernels) in order to have a source code as close as possible to the original CPU version; just a few routines have been manually rewritten. A new transpose scheme has been devised to improve the scaling of the Poisson solver, which is the main bottleneck of incompressible solvers. For large meshes the GPU version of the code shows good strong scaling characteristics, and the wall-clock time per step for the GPU version is an order of magnitude smaller than for the CPU version of the code. Due to the increased performance and efficient use of memory, the GPU version of AFiD can perform simulations in parameter ranges that are unprecedented in thermally-driven wall-bounded turbulence. To verify the accuracy of the code, turbulent Rayleigh–Bénard convection and plane Couette flow are simulated and the results are in excellent agreement with the experimental and computational data that have been published in literature. Program summaryProgram Title: AFiD-GPUProgram Files doi:http://dx.doi.org/10.17632/rwjdg7ry66.1Licensing provisions: MITProgramming language: Fortran 90, CUDA Fortran, MPIExternal routines: PGI, CUDA Toolkit, FFTW3, HDF5Nature of problem: Solving the three-dimensional Navier–Stokes equations coupled with a scalar field in a cubic box bounded between two walls and with periodic boundary conditions in the horizontal directions.Solution method: Second order finite difference method for spatial discretization, third order Runge–Kutta scheme in combination with Crank–Nicolson for the implicit terms for time advancement, two dimensional pencil distributed MPI parallelization, GPU accelerated routines.Additional comments including restrictions and unusual features: The code is available and supported on https://github.com/PhysicsofFluids/AFiD_GPU_opensource.

Numerical Hydrodynamic Wave Modelling Using Spatial Discretization in Brebes Waters, Central Java, Indonesia

When the wave is moving toward the shoreline, hydrodynamic processes occurred in the waves that give an effect on the shoreline and its surrounding buildings. The process of erosion and accretion that occurred on Brebes coastal area induces certain impacts. The purpose of this study was to determine the pattern of hydro-oceanography and seabed morphology in coastal areas. Primary data obtained during field survey was employed in this study, which was processed using simulation models by applying spatial discretization method that is based on the finite volume equations to obtain the condition of waves, currents, sediment transports and tides. The results showed Brebes waters included in the category of shallow waters with slope topography, while the tidal type is a mixed tide prevailing semidiurnal. Waves were generated in the Northern Brebes then were moved towards the mainland until it deformed, leaving energy in the form of longshore current that goes into the area around the beach ridge. The time difference between high and low tide was influenced by the mass of estuary water supply from the sea as well as the mass of water from upstream to downstream. The high level of vulnerability in the coastal area requires the more effective and efficient treatment to prevent a negative impact on physical and social conditions.

An optimised logarithmic discretisation approach for accurate and efficient compact thermal models

The accuracy of building energy simulations is of considerable interest as discrepant results can elicit adverse financial and environment consequences. The physical and temporal scales considered within building energy applications necessitate compact modelling approaches. The prediction accuracy of such simulations is intrinsically linked with the ability to predict the thermal responses of structural elements. The optimal means of representing these components such that accurate solutions are ensured at minimal computational cost remains unclear. The current study seeks to optimise the spatial placement of nodes through assessing and reporting results pertaining to a logarithmic spatial discretisation method. Contour plots are presented to intuitively determine optimal discretisation levels and time steps required for accurate thermal response predictions. This is achieved by comparing numerical solutions of varying discretisation levels with benchmark analytical solutions. Results are reported in terms of governing dimensionless parameters, Biot and Fourier numbers, to ensure generality of findings. Furthermore, spatial and temporal discretisation errors are separated and assessed independently. Finally, models derived using the proposed guidance achieve high levels of prediction accuracy for typically encountered boundary conditions with buildings.

The influence of the numerical solver selection on the nozzle impulse flow simulation results

Numerical simulations are currently used for different applications in a various fields of science. Certain solutions are not as obvious as the others while the results can give very valuable conclusions. Computational Fluid Dynamics (CFD) is one of the tools that can be used to solve different problems related with the mass and heat transfer. Nowadays it is already known that the impulse flow simulation allows to determine pressure pulsation attenuation parameters by a given geometry. However, the nozzle shape optimization method strongly depends on the numerical results obtained from the impulse flow simulation. In commercial CFD software Ansys-Fluent the obtained results depends strongly on the chosen numerical methods, especially the spatial discretization method. This is the reason to use other software as a benchmark. Alternative software FlowVision was used to perform the impulse flow simulation for the same geometries to compare the results. As there is a different problem definition in both systems the calculations, accuracy and results differ from each other. The paper describes the numerical differences between solvers. Article contains discussion about obtained results and includes hints how to avoid mistakes when user change software, especially in solving unusual CFD problems.

A Padé Approximate Model of Lithium Ion Batteries

We present a reduced order model for a lithium ion battery in which Padé approximants are used to simplify complex, transcendental, transfer functions associated with the linearized, pseudo 2-dimensional (P2D) electrochemical model of the battery. The resulting transfer functions take the form of simple, rational polynomial functions, which can be used to compute any variable at any location within a one-dimensional representation of the battery domain. Corrections for nonlinear behavior are also incorporated into the reduced model. The results obtained using the reduced model compare favorably to those from the full (nonlinear) P2D model and the computational time required to produce these results is significantly reduced. Importantly, the form of the reduced model means that variables can be evaluated at specific discrete locations within the cell domain, without the need to compute all values of the variable at all discrete locations, as is the case with the spatial discretization methods most commonly used to implement the P2D model. We show that this results in further significant time savings and enhances the suitability of the model for variety of applications.

Implicit-Modal Discontinuous Galerkin Scheme for Two-Phase Flow with Discontinuous Capillary Pressure

In this paper, we propose an implicit-modal discontinuous Galerkin scheme for solving an incompressible and immiscible oil-water flow through a heterogeneous reservoir with different rock types. The scheme uses the modal discontinuous Galerkin method for spatial discretization of both pressure and saturation equations and the backward Euler method for temporal discretization. In the modal discontinuous Galerkin method, we apply the well-localized bases that are constructed by both the nodal and modal basis functions. To find the approximate solution for the saturation equation, we propose the upwind technique. In the stability analysis of the scheme, we obtain the constant of the trace inequality for the approximate function and its first derivatives using the modal basis functions. In this way, we give suitable ranges for penalty terms. To guarantee the convergence of the approximate solution, we find suitable ranges of the time step. To demonstrate the applicability of the proposed scheme, we apply it to the dense nonaqueous phase liquids and the Versuchseinrichtung zur Grundwasser und Altlastensanierung infiltration problems. Numerical results are presented.

A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

A numerical framework for the direct simulation of dense particulate flow under explosive dispersal

In this paper, we present a Cartesian grid-based numerical framework for the direct simulation of dense particulate flow under explosive dispersal. This numerical framework is established through the integration of the following numerical techniques: (1) operator splitting for partitioned fluid–solid interaction in the time domain, (2) the second-order SSP Runge–Kutta method and third-order WENO scheme for temporal and spatial discretization of governing equations, (3) the front-tracking method for evolving phase interfaces, (4) a field function proposed for low-memory-cost multimaterial mesh generation and fast collision detection, (5) an immersed boundary method developed for treating arbitrarily irregular and changing boundaries, and (6) a deterministic multibody contact and collision model. Employing the developed framework, this paper further studies particle jet formation under explosive dispersal by considering the effects of particle properties, particulate payload morphologies, and burster pressures. By the simulation of the dispersal processes of dense particle systems driven by pressurized gas, in which the driver pressure reaches $$1.01325\times 10^{10}\,\,\mathrm {Pa}$$ ( $$10^5$$ times the ambient pressure) and particles are impulsively accelerated from stationary to a speed that is more than $$12000\,\,\mathrm {m/s}$$ within $$15\,\,\mathrm {\upmu s}$$ , it is demonstrated that the presented framework is able to effectively resolve coupled shock–shock, shock–particle, and particle–particle interactions in complex fluid–solid systems with shocked flow conditions, arbitrarily irregular particle shapes, and realistic multibody collisions.

A parallel framework for a multipoint flux mixed finite element equation of state compositional flow simulator

Mathematical models of physical problems are becoming increasingly complex and computationally intensive. At the same time, computing hardware is becoming more parallelized with an increasing number of cores promoting simultaneous tasks. In this work, we present a parallel, equation of state (EOS), compositional flow simulator for evaluating CO2 sequestration, enhanced oil recovery techniques such as gas flooding, and other subsurface porous media applications. Using the multipoint flux mixed finite element (MFMFE) method for spatial discretization, it can handle complex reservoir geometries using general distorted hexahedral grid elements, as well as satisfy local mass conservation and compute accurate phase fluxes. A parallel framework for the MFMFE is presented that has been extended to the highly non-linear, EOS, compositional flow model. Much of the non-linearity is due to the local flash and stability calculations associated with interphase mass transfer and phase behavior. Parallel multigrid linear solver libraries such as HYPRE are utilized to solve the algebraic problems on each Newton step. We perform a variety of strong and weak parallel scaling studies up to 10 million elements and 1024 processors, and discuss possible load balancing issues.

A shallow water equation based on displacement and pressure and its numerical solution

The primary purpose of this paper is to develop an efficient numerical scheme for solving the shallow water wave problem with a sloping water bottom and wet-dry interface. For this purpose, the Lagrange method and the constrained Hamilton variational principle are used to solve the shallow water wave problem. According to the constrained Hamilton variational principle, a shallow water equation based on the displacement and pressure (SWE-DP) is derived. Based on the discretized constrained Hamilton variational principle, a numerical scheme is developed for solving the SWE-DP. The proposed scheme combines the finite element method for spatial discretization and the simplectic Zu-class method for time integration. The correctness of the SWE-DP and the effectiveness of the proposed scheme are verified by three classical numerical examples. Numerical examples show that the proposed method performs well in the simulation of the shallow water problem with a sloping water bottom and wet-dry interface.

Dynamic responses of an engine front-end accessory belt drive system with pulley eccentricities via two spatial discretization methods

In this study, a generic mathematical model for calculating the natural frequencies and the dynamic responses of a typical front-end accessory drive system with any number of pulleys and arbitrary configurations of the tensioner and pulleys is established. The belt bending stiffness and the pulley eccentricities are considered in this model, and their influences on the natural frequency and the dynamic responses of the front-end accessory drive system are examined. A generic spatial discretization method and a Galerkin discretization method, which uses Lagrange multipliers to enhance the boundary conditions, are presented to discretize the continuous belt spans and to transform the governing partial differential equations into ordinary differential equations. The accuracies of the generic spatial discretization method and the Galerkin discretization method are validated by modal tests, and the advantages of the generic spatial discretization method with respect to the efficiency and the convenience of implementation are assessed by comparing the generic spatial discretization method with the Galerkin discretization method and the two-layer iteration approach. The dynamic responses of the typical front-end accessory drive system at different operational velocities are calculated from the governing ordinary differential equations derived from these two methods. It is shown that large vibration amplitudes occur in certain belt spans owing to the resonance conditions or the beat phenomena in certain operational conditions and that the belt bending stiffness has a negligible influence on the vibrations of the belt drive system because its value is small.

A Discrete-Continuous Method of Mechanical System Modelling

Abstract The paper describes a discrete-continuous method of dynamic system modelling. The presented approach is hybrid in its nature, as it combines the advantages of spatial discretization methods with those of continuous system modelling methods. In the proposed method, a three-dimensional system is discretised in two directions only, with the third direction remaining continuous. The thus obtained discrete-continuous model is described by a set of coupled partial differential equations, derived using the rigid finite element method (RFEM). For this purpose, firstly the general differential equations are written. Then these equations are converted into difference equations. The derived equations, expressed in matrix form, allow to create a global matrix for the whole system. They are solved using the distributed transfer function method. The proposed approach is illustrated with the examples of a simple beam fixed at both ends and a simply supported plate.

Pathwise convergence of a numerical method for stochastic partial differential equations with correlated noise and local Lipschitz condition

In this paper we obtain a general statement concerning pathwise convergence of the full discretization of certain stochastic partial differential equations (SPDEs) with non-globally Lipschitz continuous drift coefficients. We focus on non-diagonal colored noise instead of the usual space–time white noise. By applying a spectral Galerkin method for spatial discretization and a numerical scheme in time introduced by Jentzen, Kloeden and Winkel we obtain the rate of path-wise convergence in the uniform topology. The main assumptions are either uniform bounds on the spectral Galerkin approximation or uniform bounds on the numerical data. Numerical examples illustrate the theoretically predicted convergence rate.

Hybrid [formula omitted] with Lagrange multiplier and upwinding for the multiscale transport capability in Rattlesnake

Two interface conditions, the Lagrange multiplier method and the upwinding method, for hybrid PN-SN calculations are proposed for the self-adjoint angular flux (SAAF) formulation of the transport equation using the continuous finite element method (FEM) for spatial discretization. These interface conditions are implemented in Rattlesnake, the radiation transport application built on MOOSE, for the on-going multiscale transport simulation effort at INL. For smoothing the solution at the interface for the Lagrange multiplier method, a SN Lagrange interpolation method on the unit sphere is proposed. The discrete PN interface condition with upwinding method for coupling the SN solution is desired. Numerical results indicate that the interface conditions give the expected convergence.