Coarse Grid Research Articles

A 3-D extension of the Multiscale Control Volume method for the simulation of the single-phase flow in anisotropic and heterogeneous porous media

The level of detail on modern geological models requires higher resolution grids that may render the simulation of multiphase flow in porous media intractable. Moreover, these models may comprise highly heterogeneous media with phenomena taking place in different scales. Scale transferring techniques allow for the solution of such problems in a lower resolution scale at reduced computational cost. Among these techniques, the Multiscale Finite Volume (MsFV) method constructs a set of numerical operators in order to map quantities from the fine-scale mesh to a coarser one and vice-versa while maintaining flux conservation on both scales. However, the MsFV formulation, as originally stated, is only consistent on k-orthogonal grids since it uses a linear Two-point Flux Approximation (TPFA) method and may struggle to generate consistent primal-dual coarse grids pairs on unstructured grids. The Multiscale Restriction Smoothed-Basis method (MsRSB) improves on the MsFV by introducing a new iterative procedure to find the multiscale operators and the concept of support regions which reduces the method's complexity when applied to unstructured fine and coarse grids. The original version was only consistent on k-orthogonal fine grids due to the TPFA discretization, but filtering methods have been developed to also enable consistent multipoint schemes on the fine scale. Meanwhile, the Multiscale Control Volume method (MsCV) replaces the TPFA by the Multipoint Flux Approximation with a Diamond stencil (MPFA-D) scheme on the fine-scale while further enhancing the generation of the geometric entities to allow unstructured grids on the fine and coarse scales for two-dimensional simulations. In this work we propose an extension to three-dimensional geometries of both the MsCV and the algorithm to obtain the multiscale geometric entities based on the concept of a background grid, a coarser grid used as a proxy for the primal coarse grid. We modify the original MPFA-D method to use the very robust Global Least Squares (GLS) interpolation technique to obtain the required auxiliary nodal unknowns. We also introduce an enhanced version of the 3-D MsCV with the incorporation of the enhanced MsRSB (E-MsRSB) to enforce M-matrix properties and improve convergence. Finally, we employ the MsCV operators in a two-stage smoother to show how it can be used as a good iterative procedure to recover the fine-scale solution. We show that the 3-D MsCV method produces good results employing unstructured grids on both scales to handle the simulation of the single-phase flow in anisotropic, and heterogeneous porous media and that the smoothing procedure is able to accurately retrieve the fine-scale solution within a few iterations.

Time two-grid fitted scheme for the nonlinear time fractional Schrödinger equation with nonsmooth solutions

In this paper, we delve into the regularity properties and the development of an efficient numerical strategy for addressing the nonlinear time-fractional Schrödinger equation. Initially, we embark on an examination of the solution’s regularity, followed by an investigation into regularity enhancement through the application of the Laplace and finite Fourier sine transforms. Subsequently, after the decomposition of u-u0, we introduce an innovative time two-grid fitted scheme designed for the equation’s resolution, which demonstrates unconditional stability and convergence in the L2 norm, achieving the global convergence order of O(τF2+τC4+h2). Here, τF and τC denote the temporal step sizes on the fine and coarse grids, respectively, while h represents the spatial step size. Notably, this is the inaugural application of such an approach for solving the nonlinear time fractional Schrödinger equation. Ultimately, our numerical experiments validate that this method not only retains computational efficiency but also significantly enhances convergence accuracy.

Seismic modeling by combining the finite-difference scheme with the numerical dispersion suppression neural network

Seismic finite-difference (FD) modeling suffers from numerical dispersion including both the temporal and spatial dispersion, which can decrease the accuracy of the numerical modeling. To improve the accuracy and efficiency of the conventional numerical modeling, I develop a new seismic modeling method by combining the FD scheme with the numerical dispersion suppression neural network (NDSNN). This method involves the following steps. First, a training data set composed of a small number of wavefield snapshots is generated. The wavefield snapshots with the low-accuracy wavefield data and the high-accuracy wavefield data are paired, and the low-accuracy wavefield snapshots involve the obvious numerical dispersion including both the temporal and spatial dispersion. Second, the NDSNN is trained until the network converges to simultaneously suppress the temporal and spatial dispersion. Third, the entire set of low-accuracy wavefield data is computed quickly using FD modeling with the large time step and the coarse grid. Fourth, the NDSNN is applied to the entire set of low-accuracy wavefield data to suppress the numerical dispersion including the temporal and spatial dispersion. Numerical modeling examples verify the effectiveness of my proposed method in improving the computational accuracy and efficiency.

A two-grid characteristic block-centered finite difference method for Darcy-Forchheimer slightly compressible miscible displacement problem

In this paper, a two-grid characteristic block-centered finite difference method is developed for solving a nonlinear Darcy-Forchheimer slightly compressible miscible displacement problem in porous media. The method is proposed for solving the nonlinear system in two steps. In the first step, the nonlinear equations are approximated on a coarse grid using the characteristic block-centered finite difference method. In the second step, the nonlinear system is linearized using Newton's method with a small positive parameter to ensure the differentiability of the nonlinear term |u|u in the Darcy-Forchheimer equation. The proposed two-grid method is rigorously analyzed, in which a priori error estimates of the velocity, pressure, concentration and its flux are provided, and the rates of convergence are O(ε+Δt+H3+h2). Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods by comparing the efficiency, especially in terms of CPU time.

A comprehensive analysis of uncertainties in warm rain parameterizations in climate models based on in situ measurements

Abstract Because of the coarse grid size of Earth system models (ESM), representing warm-rain processes in ESMs is a challenging task involving multiple sources of uncertainty. Previous studies evaluated warm-rain parameterizations mainly according to their performance in emulating collision-coalescence rates for local droplet populations over a short period of a few seconds. The representativeness of these local process rates comes into question when applied in ESMs for grid sizes on the order of 100 kilometers and time steps on the order of 20-30 minutes. We evaluate several widely used warm-rain parameterizations in ESM application scenarios. In the comparison of local and instantaneous autoconversion rates, the two parameterization schemes based on numerical fitting to stochastic collection equation (SCE) results perform best. However, because of Jessen’s inequality, their performance deteriorates when grid-mean, instead of locally-resolved, cloud properties are used in their simulations. In contrast, the effect of Jessen’s inequality partly cancels the overestimation problem of two semi-analytical schemes, leading to an improvement in the ESM-like comparison. In the assessment of uncertainty due to the large time step of ESMs, it is found that the rain-water tendency simulated by the SCE is roughly linear for time steps smaller than 10 minutes, but the nonlinearity effect becomes significant for larger time steps, leading to errors up to a factor of 4 for a time step of 20 minutes. After considering all uncertainties, the grid-mean and time-averaged rain-water tendency based on the parameterization schemes are mostly within a factor of 4 of the local benchmark results simulated by SCE.

A dual-grid approach to speed up large-scale CFD-DEM simulations

With the development of heterogeneous computing, the coupling of discrete element method (DEM) and computational fluid dynamics (CFD) has become an effective approach to simulate large-scale particle–fluid systems. With various models and algorithms accelerating the DEM part significantly, the CFD part becomes an efficiency bottleneck, especially for large-scale simulations. A dual-grid approach is proposed to address this issue. On the fine grid, the fluid velocity and particle–fluid interaction are solved without compromising accuracy in capturing flow details, and the pressure is solved on the coarse grid to ensure mass conservation to boost the CFD part. This approach is validated by simulating several gas–solid fluidized beds, demonstrating a performance speedup of 2.6 for low workload and 10.1 for high workload.

High fidelity FEM based on deep learning for arbitrary composite material structure

Due to the outstanding performance, composite materials are widely used and analyzing their properties and designing them based on performance has become a crucial task in the field of many manufacturing industries. Composite materials possess complex multiscale structures, and traditional fine-scale finite element modeling and analysis may lead to severe computational resource challenges. To overcome this difficulty, breakthroughs in key technologies of multiscale accelerated analysis algorithms are required. In this study, an innovative approach based on artificial intelligence and multiscale finite element method is presented. This approach involves partitioning the entire composite material structure into coarse grids that resemble homogenous structures of similar size, providing results consistent to fine-grid finite element analysis. By utilizing CNN for image feature recognition and employing the CGAN adversarial method, coarse-grid equivalent stiffness matrices and multiscale shape functions from completely random microstructures of composite materials can be obtained. Consequently, this enables a rapid response process from microstructure to low-resolution grid to high-resolution physical field, with remarkably accurate physical field results. Moreover, compared to traditional fine-grid finite element methods, this approach significantly reduces memory usage and computation time. This method is applicable to composite materials with varying shaped inclusions, different component properties, and diverse geometric distributions, allowing these materials to perform high-fidelity finite element calculations on coarse grids and predict their mechanical behavior. Furthermore, this breakthrough opens avenues for accelerating the optimization design of composite materials with diverse mechanical functionalities, by employing a bottom-up approach.

Centralized gradient-based reconstruction for wall modeled large eddy simulations of hypersonic boundary layer transition

In this study, we introduce a robust central Gradient-Based Reconstruction (GBR) scheme for the compressible Navier-Stokes equations. The method leverages transformation to characteristic space, allowing selective treatment of waves from the compressible Euler equations. By averaging left- and right-biased state interpolations, a central scheme is achieved for all but the acoustic waves, which require upwinding for stability. Distinct differences were observed between transformations using either primitive or conservative variables. We evaluated the method's robustness and superiority using benchmark problems, including the two-dimensional shock entropy problem, two-dimensional viscous shock tube, and three-dimensional inviscid Taylor-Green vortex. Subsequently, we assessed the method in the context of Wall Modeled Large Eddy Simulations (WMLES), where coarse grids are used to reduce computational cost but also introduce substantial numerical dissipation. Using WMLES, we simulated oblique shock impingement on a Mach 6 disturbed boundary layer and a Mach 7.7 flow over a 15∘ compression ramp. Our findings reveal that: 1) transformation to characteristic space using conservative variables leads to more accurate results; 2) minimizing numerical dissipation through centralized interpolation is crucial. In the compression ramp case, boundary layer separation was shifted slightly upstream, and there was an over-prediction of wall heating, likely attributable to the equilibrium-assuming wall model. Overall, this work showcases the method's potential in accurately capturing complex flow dynamics with reduced numerical dissipation.

Accelerating the force-coupling method for hydrodynamic interactions in periodic domains

The efficient simulation of fluid-structure interactions at zero Reynolds number requires the use of fast summation techniques in order to rapidly compute the long-ranged hydrodynamic interactions between the structures. One approach for periodic domains involves utilising a compact or exponentially decaying kernel function to spread the force on the structure to a regular grid where the resulting flow and interactions can be computed efficiently using an FFT-based solver. A limitation to this approach is that the grid spacing must be chosen to resolve the kernel and thus, these methods can become inefficient when the separation between the structures is large compared to the kernel width. In this paper, we address this issue for the force-coupling method (FCM) by introducing a modified kernel that can be resolved on a much coarser grid, and subsequently correcting the resulting interactions in a pairwise fashion. The modified kernel is constructed to ensure rapid convergence to the exact hydrodynamic interactions and a positive-splitting of the associated mobility matrix. We provide a detailed computational study of the methodology and establish the optimal choice of the modified kernel width, which we show plays a similar role to the splitting parameter in Ewald summation. Finally, we perform example simulations of rod sedimentation and active filament coordination to demonstrate the performance of fast FCM in application.

Numerical Diffusion and Turbulent Mixing in Convective Self‐Aggregation

AbstractConvective Self‐Aggregation (CSA) is a common feature of idealized numerical simulations of the tropical atmosphere in Radiative‐Convective Equilibrium (RCE). However, at coarse grid resolution where deep convection is not fully resolved, the occurrence of this phenomenon is extremely sensitive to subgrid‐scale processes. This study examines the role of mixing and entrainment, provided by the turbulence model and the implicit numerical diffusion. The study compares the results of two models, WRF and SAM, by varying turbulence models, initial conditions, and horizontal spatial resolution. At a coarse grid resolution of 3 km, the removal of turbulent mixing prevents CSA in models with low numerical diffusivity but is preserved in models with high numerical diffusivity. When the horizontal grid resolution is refined to 1 km, CSA can only be achieved by increasing explicit turbulent mixing, even with a small amount of shallow clouds. Therefore, the sensitivity of CSA to horizontal grid resolution is not primarily caused by the decrease in shallow clouds. The analysis of the total water path spectrum suggests that the amplitude of initial humidity perturbations introduced by convection in the free troposphere is the key factor. This amplitude is regulated by turbulent mixing and diffusion at small scales. Prior to the onset of CSA, increased mixing makes updrafts more sensitive to the dryness of the free troposphere, which strengthens the moisture‐convection feedback. This leads to an increased distance between convective cores and a stronger humidity perturbation in the free troposphere, which can destabilize the RCE state.

Generalized multiscale finite element method for language competition modeling II: Online approach

In this paper, we consider language competition in a heterogeneous medium. The mathematical model considers bilingualism, population growth, and social pressure to use specific languages (diglossia). The transition from one language group to another is determined by probabilities that depend on status, group size, and spatial location. The model considers not only diffusive spread but also convective spread, which models seasonal migrations of speaker populations. We develop multiscale approaches for coarse grid modeling while maintaining reasonable accuracy. We present offline and online approaches based on the Generalized Multiscale Finite Element Method. We consider a model problem in a heterogeneous environment to test these approaches. The results show that both approaches can significantly reduce the discrete problem’s size. However, the online approach achieves better accuracy.

An advanced numerical radial sub-resolution technique to correct heat fluxes in the vicinity of bubbles at saturation

This work explores the use of a thermal sub-resolution technique applied on coarse grids to better estimate the temperature gradient in the vicinity of a bubble at saturation as part of a DNS-Front-Tracking framework. By refining locally the solution and realigning it with the bubble’s interface normal direction, the temperature and fluxes prediction are notably enhanced. The refined profiles computed by Finite-Difference in a local spherical frame of reference are then transmitted to the fixed grid using several coupling methods both identified from the literature. Coupling methods overriding the fixed grid temperature are interesting in terms of ease of implementation and robustness whereas face fluxes methods are conservative, but they may cause underestimation of quantities and may require local smoothing like neighbouring averaging. In pure diffusion, these coupling approaches show a good agreement with the theoretical Nusselt number profile. However, the quasi-static hypothesis is difficult to satisfy at the beginning of the simulation which causes a permanent bias. Results are promising on a rising bubble case, reducing the error on the heat transfer prediction below 6.5% with a reduction in computation cost by a factor of 1300 compared to a reference simulation.

A Two‐Column Model Parameterization for Subgrid Surface Heterogeneity Driven Circulations

AbstractEarth system models currently struggle to account for the complex effects that land surface heterogeneity can have on land‐atmosphere interactions. There have been attempts to include the impact of this heterogeneity on the atmosphere, but they ignore the development of coherent circulations that can be driven by spatial differential surface heating. A wealth of literature, particularly large‐eddy simulation (LES) based studies, shows that these circulations have significant impacts on the development and organization of clouds. In this work, we describe a two‐column model with a parameterized circulation driven by atmospheric virtual potential temperature profiles, differences in near surface temperature between the two columns, patterns of surface heterogeneity, and the mean background wind. Key aspects of the proposed model structure are compared with LES output, and the model is then implemented between two otherwise independent single column models. While some avenues for improvement exist, when the circulations are parameterized, we see increased cloud development and realistic changes to the mean profiles of temperature and moisture. The proposed model qualitatively matches expectations from the literature and LES, and points to the potential success of its future implementation in coarse grid models.

Addressing equifinality in agent-based modeling: a sequential parameter space search method based on sensitivity analysis

This study addresses the challenge of equifinality in agent-based modeling (ABM) by introducing a novel sequential calibration approach. Equifinality arises when multiple models equally fit observed data, risking the selection of an inaccurate model. In the context of ABM, such a situation might arise due to limitations in data, such as aggregating observations into coarse spatial units. It can lead to situations where successfully calibrated model parameters may still result in reliability issues due to uncertainties in accurately calibrating the inner mechanisms. To tackle this, we propose a method that sequentially calibrates model parameters using diverse outcomes from multiple datasets. The method aims to identify optimal parameter combinations while mitigating computational intensity. We validate our approach through indoor pedestrian movement simulation, utilizing three distinct outcomes: (1) the count of grid cells crossed by individuals, (2) the number of people in each grid cell over time (fine grid) and (3) the number of people in each grid cell over time (coarse grid). As a result, the optimal calibrated parameter combinations were selected based on high test accuracy to avoid overfitting. This method addresses equifinality while reducing computational intensity of parameter calibration for spatially explicit models, as well as ABM in general.

Computable turbulence modeling of laminar-turbulent transition characterized boundary layer flows with the aid of artificial neural network

The continuous development of machine learning algorithms has stimulated the technological revolution on turbulence modeling for Reynolds-averaged Navier–Stokes (RANS) simulations. In this paper, a computable transition-enabled turbulence model is developed with the aid of an artificial neural network (ANN), which maps the relation between the mean flow variables from the shear stress transport (SST) model on coarse grid and the turbulent viscosity from SST-γ model on fine grid. It turns out that the ANN model can predict the aerodynamic integral quantities, transition onset location, laminar separation bubble structure, streamwise velocity distribution, etc. with superior accuracy and robustness for subsonic and moderate transonic airfoil flows. Meanwhile, the ANN model can significantly improve the convergence property with a much higher convergence speed of the model equation in comparison with traditional transition-enabled RANS model. Moreover, due to the lack of solving RANS model equations and use of grid interpolation technology, the computation speed of this ANN model is improved by a factor of about six over the conventional benchmark model. Therefore, the present computable ANN model provides a new perspective for high-efficiency simulation of transition-characterized flows in engineering applications.

SR-M−GAN: A generative model for high-fidelity stress fields prediction of the composite bolted joints

Predicting the stress fields induced by composite bolted joints constitutes a valuable research endeavor aimed at enhancing the structural integrity of equipment in the aerospace and maritime fields. The stress field characteristics of composite bolted joints may undergo changes due to potential mechanical behaviors, such as damage, constitutive relationships and so on, which exhibit complexity and pose challenges for prediction. Nevertheless, modeling the intricate stress field distribution directly from experimental data poses a formidable challenge, necessitating a substantial investment in materials and resources. Traditional finite element numerical methods have the capability to model the distribution of the stress fields. However, to obtain a high-fidelity solution of this distribution, it is imperative to generate a fine-grid mesh and allocate the significant computational resources. The incorporation of computational techniques in coarse −grid would enhance the efficiency of computations, but these approaches compromise the accuracy of the calculations. In this paper, an innovative deep learning framework is introduced. The network named as the SR-M−GAN is drew inspiration from the conditional generative adversarial network (cGAN) used in the image super-resolution. The primary aim of SR-M−GAN is to mitigate the intrinsic trade-off between result fidelity and computational complexity. The model proposed in this paper effectively reconstructs high-fidelity stress field data from calculations on coarse grids and exhibits excellent reconstruction metrics. Simultaneously, the model only incurs additional computational costs during the deployment of the network model and training. When compared to experimental data, the model continues to demonstrate superior performance in both accuracy and computational efficiency. Therefore, this method offers a high-fidelity data augmentation approach for stress analysis in composite bolted joints, laying the groundwork for large-scale analysis of stress field characteristics in composite bolted joints.

Accurate Nonstandard Path Integral Models for Arbitrary Dielectric Boundaries in 2-D NS-FDTD Domains.

An efficient path integral (PI) model for the accurate analysis of curved dielectric structures on coarse grids via the two-dimensional nonstandard finite-difference time-domain (NS-FDTD) technique is introduced in this paper. In contrast to previous PI implementations of the perfectly electric conductor case, which accommodates orthogonal cells in the vicinity of curved surfaces, the novel PI model employs the occupation ratio of dielectrics in the necessary cells, providing thus a straightforward and instructive means to treat an assortment of practical applications. For its verification, the reflection from a flat plate and the scattering from a cylinder using the PI model are investigated. Results indicate that the featured methodology can enable the reliable and precise modeling of arbitrarily shaped dielectrics in the NS-FDTD algorithm on coarse grids.

Fast Two-Grid Finite Element Algorithm for a Fractional Klein- Gordon Equation

In this article, we propose a spatial two-grid finite element algorithm combined with a shifted convolution quadrature (SCQ) formula for solving the fractional Klein-Gordon equation. The time direction at tn − θ is approximated utilizing a second-order SCQ formula, where θ is an arbitrary constant. The spatial discretization is performed using a two-grid finite element method involving three steps: calculating the numerical solution by solving a nonlinear system iteratively on the coarse grid, obtaining the interpolation solution based on the computed solutions in the first step, and solving a linear finite element system on the fine grid. We present a numerical algorithm, validate the two-grid finite element method’s effectiveness, and demonstrate the computational efficiency for our method by the comparison of the computing results between the two-grid finite element method and the standard finite element method.

Large Eddy Simulation of Separated Flows on Unconventionally Coarse Grids

Abstract We examine and benchmark the emerging idea of applying the large eddy simulation (LES) formalism to unconventionally coarse grids where Reynolds-averaged Navier–Stokes (RANS) would be considered more appropriate at first glance. We distinguish this idea from very large eddy simulation and detached eddy simulation, which require switching between RANS and LES formalism. LES on RANS grid is appealing because first, it requires minimal changes to a production code; second, it is more cost-effective than LES; third, it converges to LES; and most importantly, it accurately predicts flows with separation. This work quantifies the benefit of LES on RANS-like grids as compared to RANS on the same grids. Three canonical cases are considered: periodic hill, backward-facing step, and jet in cross flow. We conduct direct numerical simulation (DNS), proper LES on LES grids, LES on RANS-quality grids, and RANS. We show that while the LES solutions on the RANS-quality grids are not grid converged, they are twice as accurate as the RANS on the same grids.

Numerical homogenization of the gas filtration problem

The non-stationary gas flow equation in homogeneous and heterogeneous media is considered. Areas of this type are often encountered when considering gas production processes from a gas-bearing reservoir. The heterogeneous structure of the reservoir can strongly affect the extraction processes. For the numerical solution of the problem under consideration, we construct an approximation of the equations on a coarse grid using the numerical averaging method. The method allows us to solve the problem using less computational power and in a shorter time. A numerical comparison of the results of solving the model problem is carried out in a two-dimensional domain for the cases of linear and nonlinear variants of the equations, as well as in a homogeneous and heterogeneous medium. The finite element solution on a fine mesh was taken as a reference solution. The computational realization was performed using the FEniCS library. The construction of the geometric computational domain was performed in the program Gmsh. Paraview and the Matplotlib library in Python were used for data visualization.