Coarse Grid Research Articles

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.

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.

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.

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.

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

With the development of heterogeneous computing, the discrete element method (DEM) and computational fluid dynamics (CFD) coupling 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, a performance demonstrating speedup of 2.6 for low workload and 10.1 for high workload.

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.

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.

Modeling of flow and transport in multiscale digital rocks aided by grid coarsening of microporous domains

Many subsurface porous media such as soils, carbonate rocks, and mudstones possess multiscale porous structures that play an important role in regulating fluid flow and transport therein. A pore-network-continuum hybrid model is promising for numerical studies of a multiscale digital rock. It is, however, still prohibitive to the REV-size modeling because tens of millions of microporosity voxels may exist. In this work, we develop a novel and robust algorithm for coarsening microporosity voxels of a multiscale digital rock. Then, we combine coarsened microporosity grids with the pore network of resolved macropores to form efficient computational meshes. Furthermore, a pore-network-continuum simulator is developed to simulate flow and transport in both a synthesized multiscale digital rock and a realistic Estaillades carbonate rock. We show that the coarsening algorithm can reduce computational grids by about 90%, which substantially reduces computational costs. Meanwhile, coarsening microporosity has a minor impact on the predictions of absolute permeability, gas production curves, and breakthrough curves of solute transport. We illustrate the mechanisms of flow and transport in multiscale porous media induced by microporosity. Finally, the efficient hybrid model is used to predict the absolute permeability of an Estaillades digital rock. The numerical prediction matches well with the reported experimental data. We highlight the importance of characterizing mean pore-size distributions in microporosity for the prediction of rock permeability and local flow fields. The developed pore-network-continuum hybrid model aided by grid coarsening of microporosity serves as a useful numerical tool to study flow and transport in multiscale porous media.

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.

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.

A hybrid numerical framework of potential and viscous flows for simulations of free running surface ship maneuvering in waves

This study presents a hybrid numerical framework for the accurate and efficient simulation of free-running surface ship maneuvering in waves. The proposed hybrid framework combines the MOUM (Modified Osaka University Method) body-force method to model the propeller and a hybrid approach of potential and viscous flows for wave-ship interactions. The MOUM body-force method, based on BEMT (Blade Element Momentum Theory), considers the three-dimensional viscous effects of propellers. The hybrid approach of potential and viscous flows decomposes the total physical field into the incident field and the complementary field, where the incident wave is solved by the potential flow theory to ensure the accuracy of wave propagation with coarse grids, and viscous and nonlinear effects are incorporated through the adapted URANS (Unsteady Reynolds Averaged Navier-Stokes) with refined grids. The 6DOF (Degrees of Freedom) motion equations of the rigid body and the dynamic structured grid with the overset technology are adopted to simulate the 6DOF motions of ship maneuvering in waves. Firstly, the applicability of the computational domain, domain-moving strategy, and ship motion-solving method for effectively simulating ship maneuvering in wave conditions are thoroughly examined. Then the reliability of present hybrid approach is validated using turning benchmark model tests of KCS model in regular waves. Simulations of turning circle and zig-zag maneuvers in regular waves are carried out by the traditional viscous CFD (Computational Fluid Dynamics) based on the boundary wave-generation approach and the proposed hybrid framework. The trajectories, motions and forces of ship maneuvering in waves predicted by both methods, as well as the computational efficiency, are compared. Numerical results indicated that the proposed method offers high accuracy and efficiency for predicting ship maneuvering in waves and could serve as a highly credible alternative to existing methods.

A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2−1σ formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(τF2+τC4+hx2+hy2), where τF and τC are the time step sizes of fine grid and coarse grid respectively, while hx and hy are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.

Improved matrix model of sequence grid partition based on vector space sampling

In information technology and data science predictive analysis work, the goal is to infer guess values as accurately as possible. A more densely spaced grid of points should be set as a tool for operating measurement models. It has a wide range of applications in psychology, education and other social science research fields. Although the grid seems to meet the requirements of further improving the accuracy of reasoning and guessing from a certain point of view, it also means complex calculations. When multiple false values fall between another grid point, the traditional Data Oriented Architecture fails. To guarantee the accuracy and efficiency of the calculation, it is necessary to solve the problem of excessive noise in the prediction of samples in meta-learning to solve the uncertainty caused by the noise database data or the assumption of the matrix model. This paper proposes a grid partition movable inference least squares method, that is, the sparse Bayesian least squares method based on the grid minimum matrix. The method is used for solving the problem that the prediction noise of the sample in the meta-learning is too large, to solve the uncertainty brought by the generated noise database data or the array matrix model assumption. According to the Symplectic Bayesian learning of the sequence matrix model, the basic parameters for processing the input sample database data are determined. The sparse Bayesian algorithm solution is used on the grid, combined with the off-grid inference guess of the sequence matrix model and the grid division. We set a coarse grid of points around the grid of false values. The rough point grid division is set, and the grid division around the false value is divided in detail. Then select more appropriate meta-learning parameters to clean up data. In the experiment analysis, we take the small sample study as the scene and carry on the concrete analysis to the experiment. The test proof in the fitting degree of the algorithm adopted in the paper surpasses the Support Vector Machine (SVM) and the k-Nearest Neighbor (KNN) algorithm by 3.5% and 6.4% respectively. The convergence effect surpasses the contrast plan above 10 and has a bigger superiority in the inference factor thrust. It proves that the optimization method in this paper has a strong promotion effect for the application of data forecasting systems in various industries, and has theoretical value as well as practical significance.

Enforcing accurate volume conservation in VOF‐based long‐term simulations of turbulent bubble‐laden flows on coarse grids

AbstractThis study proposes two different strategies to enforce accurate volume conservation in volume‐of‐fluid (VOF)‐based simulations of turbulent bubble‐laden flows on coarse grids. It is demonstrated that, without a correction, minimal volume errors on a time‐step level, caused by the under‐resolution of the interface, can accumulate to significant deviations from the intended flow conditions despite the comparably good volume conservation properties of the geometric VOF method. In particular, large volume errors are observed for challenging setups combining coarse grid resolutions and comparably high Reynolds and Eötvös numbers. The problem is reinforced for long‐term simulations in periodic domains, which are often performed to collect flow statistics of bubbly flows. The first proposed volume conservation method simply corrects the volume error of a bubble by uniformly adding or removing the respective amount of gas volume in the interface cells. The second proposed method performs an additional reconstruction and advection step of the VOF field using a non‐divergence‐free velocity field, which can be interpreted as a slight dilatation or contraction of the bubble. A comparison between the global flow statistics as well as the individual bubble dynamics for both volume conservation methods reveals that the results are quasi‐identical for a number of challenging test cases, while the gas volume is accurately conserved. The proposed methods allow to perform numerical simulations of freely deformable bubbles in turbulent flows for setups that have previously been out of reach for this numerical framework.

Hybrid RANS/LES modeling of hypersonic turbulent boundary layers with cold walls

Various strategies for hybrid RANS/LES modeling of hypersonic turbulent boundary layers subjected to strong wall-cooling are developed and analyzed. By defining a turbulent thermal length scale, the eddy viscosity and diffusivity of the hybrid model are calculated by the same formula, ℓ2ω. This allows a similar blending of the length scales and the development of analogous dynamic models for turbulent stress and heat flux. The dynamic procedure is applied to heat transfer by equating norms, rather than invoking least-squares with a lower bound, which improves robustness. The shielding function, as used in detached eddy simulations models, is revised to prevent a spurious early switch from the RANS to the LES mode. This revision significantly improves the accuracy of hybrid models on coarse grids. Finally, the proposed models are tested and validated for Mach numbers between 2.5 and 14, and wall-to-recovery temperature ratios between 0.18 and 1.

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

This article presents and studies a two-level grad-div stabilized finite element discretization method for solving numerically the steady incompressible Navier–Stokes equations. The method consists of two steps. In the first step, we compute a rough solution by solving a nonlinear Navier–Stokes system on a coarse grid. And then, in the second step, we pass the coarse grid solution to a fine grid to linearize the nonlinear term, update the solution by solving a linearized problem based on Newton iterations. In both steps, a grad-div stabilization term is incorporated into the system to reduce the influence of pressure on the approximate velocity. We analyze stability and asymptotic convergence of the approximate solutions, derive explicit dependence of the solution errors on the grad-div stabilization parameter and viscosity. We perform also some numerical tests to validate the theoretical analysis and illustrate the efficiency of the proposed method. Compared with the standard two-level method without stabilizations, the grad-div stabilization term added in present method improves the accuracy of the approximate velocity, accelerates the convergence of the nonlinear iterations for the coarse mesh nonlinear system, and reduces the computational time.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.