Discrete Grid Research Articles

A data-driven strategy for phase field nucleation modeling

We propose a data-driven strategy for parameter selection in phase field nucleation models using machine learning and apply it to oxide nucleation in Fe-Cr alloys. A grand potential-based phase field model, incorporating Langevin noise, is employed to simulate oxide nucleation and benchmarked against the Johnson-Mehl-Avrami-Kolmogorov model. Three independent parameters in the phase field simulations (Langevin noise strength, numerical grid discretization and critical nucleation radius) are identified as essential for accurately modeling the nucleation behavior. These parameters serve as input features for machine learning classification and regression models. The classification model categorizes nucleation behavior into three nucleation density regimes, preventing invalid nucleation attempts in simulations, while the regression model estimates the appropriate Langevin noise strength, significantly reducing the need for time-consuming trial-and-error simulations. This data-driven approach improves the efficiency of parameter selection in phase field models and provides a generalizable method for simulating nucleation-driven microstructural evolution processes in various materials.

Optimised use of interferometry, spectroscopy, and stellar atmosphere models for determining the fundamental parameters of stars

Thanks to recent progress in the field of optical interferometry, instrument sensitivities have now reached the level achieved in the domain of new space missions dedicated to exoplanet and stellar studies. Combining interferometry with other observational approaches enables the determination of stellar parameters and helps improve our understanding of stellar physics. In this paper, we aim to demonstrate a new way of using stellar atmosphere models for a joint interpretation of spectroscopic and interferometric observations. Starting from a discrete grid of one-dimensional (1D) stellar atmosphere models, we developed a training algorithm, based on an artificial neural network, capable of estimating the spectrum and intensity profile of a star over a range of wavelengths and viewing angles. A minimisation algorithm based on the trained function allowed for the simultaneous fitting of the observational spectrum and interferometric complex visibilities. As a result, coherent and precise stellar parameters can be extracted. We show the ability of the trained function to match the modelled intensity profiles of stars in the effective temperature range of $4500$ K to $7000$ K and surface gravity range of $3$ to $5$ dex, with a relative precision to the model that is better than $0.05<!PCT!>$. Using simulated interferometric data and actual spectroscopic measurements, we demonstrated the performance of our algorithm on a sample of five benchmark stars. Using this method, we achieved an accuracy within $0.5<!PCT!>$ for the angular diameter, radius, and surface gravity, and within $20$ K for the effective temperature. This paper demonstrates a new method of using interferometric data combined with spectroscopic observations. This approach offers an improved determination of the radius, effective temperature, and surface gravity of stars.

HierGP: Hierarchical Grid Partitioning for Scalable Geospatial Data Analytics

Application domains such as environmental health science, climate science, and geosciences—where the relationship between humans and the environment is studied—are constantly evolving and require innovative approaches in geospatial data analysis. Recent technological advancements have led to the proliferation of high-granularity geospatial data, enabling such domains but posing major challenges in managing vast datasets that have high spatiotemporal similarities. We introduce the Hierarchical Grid Partitioning (HierGP) framework to address this issue. Unlike conventional discrete global grid systems, HierGP dynamically adapts to the data’s inherent characteristics. At the core of our framework is the Map Point Reduction (MPR) algorithm, designed to aggregate and then collapse data points based on user-defined similarity criteria. This effectively reduces data volume while preserving essential information. The reduction process is particularly effective in handling environmental data from extensive geographical regions. We structure the data into a multilevel hierarchy from which a reduced representative dataset can be extracted. We compare the performance of HierGP against several state-of-the-art geospatial indexing algorithms and demonstrate that HierGP outperforms the existing approaches in terms of runtime, memory footprint, and scalability. We illustrate the benefits of the HierGP approach using two representative applications: analysis of over 289 million location samples from a registry of participants and efficient extraction of environmental data from large polygons. While the application demonstration in this work has focused on environmental health, the methodology of the HierGP framework can be extended to explore diverse geospatial analytics domains.

A Priori Test of Scale-Similarity Parameterizations for Subgrid-Scale Fluxes in Convective Storms

Abstract This study evaluates a parameterization scheme for subgrid-scale (SGS) fluxes based on the scale-similarity assumption and employing a large-eddy simulation of an idealized back-building convective system. In this parameterization, the SGS fluxes are decomposed into the “Leonard term,” which depends only on the resolved scale components, the “Reynolds term,” which depends only on the SGS components, and the “cross term,” which corresponds to the interaction between the resolved scale and SGS components. Assuming a linear relationship between the Leonard term and the Reynolds and cross terms, SGS fluxes are expressed as the product of an empirical coefficient and the Leonard term. The Leonard term reasonably represents the SGS flux derived by a smooth filter operation, including the countergradient vertical SGS transport of potential temperature, which cannot be represented by a traditional eddy-diffusivity model. The dependence of the empirical coefficient on filter width is also evaluated. This dependence is related mainly to the Reynolds term, the magnitude of which varies widely with the filter width. The estimation based on the spectral decomposition of the Reynolds term explains the obtained dependence of the empirical coefficient for the vertical flux on filter width. In contrast, the variation of the empirical coefficient with filter width is not required to obtain the horizontal flux. For the parameterization of SGS fluxes in kilometer-scale models that use finite difference or volume methods, the Leonard term is expressed by the horizontal gradient of variables on a discrete grid. The Leonard term on a discrete grid also accurately represents the amplitude and spatial pattern of the SGS flux. Significance Statement Kilometer-scale numerical weather prediction models can reproduce deep convection explicitly. However, they cannot reproduce smaller inner structures within this deep convection. Therefore, a parameterization scheme that can express the nature of the transport associated with such structures is required. Using a large-eddy simulation with a horizontal resolution sufficient to express such inner structures, we evaluated a parameterization based on the scale similarity assumption, including an empirical coefficient. Based on spectral decomposition, we quantitatively estimated the relationship between the empirical coefficient and grid spacing. Our results provide guidance in selecting the value of the empirical coefficient in this parameterization.

Direct numerical simulation of droplet impact onto dry stationary and moving walls at low to high Weber numbers

The accurate prediction of the drop-wall impact dynamics and regime are of interest in many engineering applications. Considerable progress has been made in different elements of Direct Numerical Simulation (DNS) approaches. However, identifying the optimal combination of improved submodels that perform well under a wide range of operating conditions, covering low to high Weber numbers with varying surface wettabilities and wall motions, still requires further investigation. Here, we conduct a comparative study on the performance of key DNS elements, including the Interface Tracking (IT) algorithm for the widely-used volume-of-fluid approach, Contact Angle (CA), Contact-Line Velocity (CLV), Instability Actuation (IA), and numerical discretization. The study spans a broad range of Weber numbers, encompassing various impact phenomena such as drop oscillation, partial rebound, fingering, and splash. It is concluded that the Kistler CA model provides the most accurate predictions among the models considered here. In terms of IT, the Multidimensional Universal Limiter with Explicit Solution (MULES) algorithm is the most efficient one, especially for moderate to high Weber numbers. For high Weber numbers, involving the fingering instability and splash events, while applying the IA mechanism slightly improves the results, using a high-quality fine-enough grid and appropriate numerical discretization scheme to control the dispersion and dissipation errors are more important ingredients. It is shown that the recommended model combination is also able to predict the important features of drop impact on moving walls with reasonable accuracy.

Non-steady flow decomposition behaviors of heat and mass transfer on intrinsic kinetics for methane hydrate particles transported in pipelines

The differential equations governing the decomposition rate and activation energy, convective mass transfer, methane diffusion and fugacity, as well as heat conduction and enthalpy change were derived by integrating thermal convection, heat conduction, mass transfer and intrinsic kinetics. Mathematical models for decomposition kinetics and thermodynamics in a non-steady field were solved using a radial logarithmic grid for node division and discretization of the equations. The study elucidated the heat and mass transfer processes as well as decomposition kinetics in pipelines by analyzing the impacts of particle size, temperature, pressure, flow rate and activation energy on decomposition rate. The results indicated that compared with the isothermal decomposition of the intrinsic kinetic model and static experimental decomposition, the complex non-steady multiphase flow model offers a more accurately simulation of hydrate decomposition as well as associated heat and mass transfer processes. Hydrate decomposition is characterized as a non-isothermal dynamic ablation process with heat transfer rate and intrinsic kinetics dominating during the preliminary stage and the stable stage, respectively. An increasing in the slurry temperature, temperature fluctuation and flow rate accelerated the heat transfer rate capacity, enhanced convective heat transfer capacity and facilitated hydrate decomposition, with the mass transfer rate exhibiting lesser influence. The particle surface area and slurry temperature exhibited the most significant influence on hydrate decomposition, which was followed by flow rate. Upon increasing particle size from 1 mm to 5 mm and the temperature from 293 K to 313 K, the decomposition rate increased by 20.7 times and 13.5 times, respectively. Furthermore, enhancing temperature and flow rate while decreasing particle size and pressure decreased both the decomposition time and the duration required for the decomposition rate to reach its peak. Additionally, the decomposition rate and its constant vary with the activation energy.

Spherical Magnetic Vector Forwarding of Isoparametric DGGS Cells with Natural Superconvergent Points

With the rapid advancement of satellite remote sensing technology, many scientists and organizations, including NASA, ESA, NAOC, and Roscosmos, observe and study significant changes in the geomagnetic field, which has greatly promoted research on the geomagnetic field and made it an important research direction in Earth system science. In traditional geomagnetic field research, tesseroid cells face degradation issues in high-latitude regions and accuracy limitations. To overcome these limitations, this paper introduces the Discrete Global Grid System (DGGS) to construct a geophysical model, achieving seamless global coverage through multi-level grid subdivision, significantly enhancing the processing capability of multi-source and multi-temporal spatial data. Addressing the challenges of the lack of analytical solutions and clear integration limits for DGGS cells, a method for constructing shape functions of arbitrary isoparametric elements is proposed based on the principle of isoparametric transformation, and the shape functions of isoparametric DGGS cells are successfully derived. In magnetic vector forwarding, considering the potential error amplification caused by Poisson’s formula, the DGGS grid is divided into six regular triangular sub-units. The triangular superconvergent point technique is adopted, and the positions of integration points and their weight coefficients are accurately determined according to symmetry rules, thereby significantly improving the calculation accuracy without increasing the computational complexity. Finally, through the forward modeling algorithm based on tiny tesseroid cells, this study comprehensively compares and analyzes the computational accuracy of the DGGS-based magnetic vector forwarding algorithm, verifying the effectiveness and superiority of the proposed method and providing new theoretical support and technical means for geophysical research.

A numerical study of transformed mixed-integer optimal control problems

AbstractTime transformation is a ubiquitous tool in theoretical sciences, especially in physics. It can also be used to transform switched optimal control problems into control problems with a fixed switching order and purely continuous decisions. This approach is known either as enhanced time transformation, time-scaling, or switching time optimization (STO) for mixed-integer optimal control. The approach is well understood and used widely due to its many favorable properties. Recently, several extensions and algorithmic improvements have been proposed. We use an alternative formulation, the partial outer convexification (POC), to study convergence properties of (STO). We introduce the open-source software package _ (Sager et al., czeile/ampl_mintoc: Math programming c release, 2024, https://doi.org/10.5281/zenodo.12520490). It is based on AMPL, designed for the formulation of mixed-integer optimal control problems, and allows to use almost identical implementations for (STO) and (POC). We discuss and explain our main numerical result: (STO) is likely to result in more local minima for each discretization grid than (POC), but the number of local minima is asymptotically identical for both approaches.

Improved phase-state identification bypass approach of the hydrocarbons-CO2-H2O system for compositional reservoir simulation

CO2 injection is a highly effective technique to enhance oil recovery, achieved through continuous or alternative injection. However, the intricate interactions between different phases within porous media present significant challenges when predicting the performance of CO2 injection. To address this, it is crucial to employ compositional simulation, which accounts for the multiphase multicomponent transport. Nonetheless, conventional multiphase flash calculations can be computationally inefficient for large-scale reservoir simulations. Therefore, it is necessary to accelerate the Equation-of-State (EoS)-based compositional simulation, given the widespread use of CO2 enhanced oil recovery (CO2-EOR) in recent years. The phase-state identification bypass method has proven to be superior to other methods in terms of efficiency. However, this approach struggles with regions near phase boundaries, resulting in reduced computational efficiency in those areas.In this study, an enhanced phase-state identification bypass approach is developed to address this limitation. The first step involves discretising the pressure-temperature space using rectangular grids. Additionally, the tie-simplexes, which represent regions defined by the maximum number of phases formed by the fluid under consideration, are discretized in the phase-fraction space at the pressure and temperature of each discretization node. Subsequently, the discretization grid associated with the given point (the overall composition, pressure, and temperature) is located, and the phase states of the grid nodes are determined using the conventional multiphase flash method. If all nodes exhibit the same phase state, that phase state is assigned to the given point. However, if multiple phase states are obtained, a novel process is proposed to determine the phase state of the given point. To validate this improvement to the phase-state identification bypass method, phase diagram calculations and simulation cases are conducted, and the results demonstrate the robustness of the proposed method and its superior computational efficiency compared to the previous method.

Arctic seas data cube based on discrete global grid system

Arctic seas data cube based on discrete global grid system

Steady and transient analysis on thermal hydraulic characteristic of helical coiled once-through steam generator of liquid metal fast reactor

A transient model for thermal–hydraulic characteristics of helical coiled once-through steam generator (HCOTSG) of liquid metal fast reactor was established based on global discrete grid method. The model meshed the liquid metal circuit and water-steam circuit. Meanwhile, unsteady heat conduction equations were built to accurately simulate coupled heat transfer between two sides. Four-equation drift-flux method was adopt for water-steam flow. Taking lead–bismuth fast reactor as example, thermal–hydraulic characteristics of HCOTSG were first calculated under steady conditions. It was found that heat flux distribution along steam generator is extremely uneven, and the difference between maximum and minimum wall heat flux reaches hundreds of times. Then, the transient response was simulated when inlet conditions of primary side were perturbed by step changes, and it was found that maximum wall heat flux increases from 1175.5 kW/m2 to 1365 kW/m2, increasing by nearly 16 %, when inlet lead–bismuth temperature step increases from 450°C to 480 °C.

Hydrodynamic Model of the Area of the Żelazny Most Mining Waste Storage Facility to Reconstruct the Migration of Saline Groundwater

This paper presents the construction of a numerical three-dimensional model of the area of the Żelazny Most Mining Waste Storage Facility (MWSF). In the study area, the difficult geological conditions associated with glaciotectonics are accompanied by a complex hydrotechnical system of sediment deposition and sedimentary water drainage. In order to effectively reflect the water flow paths, a detailed schematization was carried out, using 700,000 boreholes and more than 300 hydrogeological cross-sections. In addition, numerous drainage sections, streams, and ditches were included to reliably assess the amount of saline water entering the underlying aquifers. This research was supported by magnetic resonance sounding (MRS) studies of the reservoir’s sediments. The MWSF is currently being expanded, so the work primarily focuses on illustrating changes in the hydrodynamic field resulting from the inclusion of the new southern section. Models of similar facilities have been implemented before, but in the current one, the combination of meticulous analysis of the hydro-structural system, the water balance, a significant amount of data, the size of the facility, and the use of an unstructured discretization grid in the calculations is undoubtedly innovative and will be an important contribution to the development of analogous solutions around the world.

A novel successive convexification framework with symplectic pseudospectral solutions for nonlinear optimal control problems of constrained pantograph delay systems

This paper proposes a novel symplectic pseudospectral iteration framework for the nonlinear optimal control problem (OCP) of a pantograph delay system with inequality constraints. Traditional numerical algorithms for OCPs with pantograph delay systems have predominantly focused on unconstrained scenarios, neglecting constrained problems. Therefore, we propose a novel symplectic pseudospectral iteration framework for constrained problems. First, we employ the successive convexification (SCvx) method to transform the original problem into a series of linear quadratic (LQ) problems, addressing nonlinearity. Then, leveraging time-scale transformations and parametric variational principles, we derive the first-order necessary conditions (FONCs) for these convexified problems, including a Hamiltonian two-point boundary value problem (HTBVP) with stretching terms and a linear complementarity problem. To handle pantograph time delays effectively, we employ a local Legendre-Gauss-Lobatto (LGL) pseudospectral method with proportional grid discretization. Finally, employing a symplectic indirect algorithm, we develop a symplectic pseudospectral method (SPM) for solving the transformed LQ problems, converting them into a system of sparse linear algebraic equations and a linear complementarity problem. Validation of the framework is performed using diverse illustrative examples, demonstrating strong agreement between SCvx and SPM in computational efficiency. Numerical experiments confirm the sparsity of the core matrix and robustness to initial guesses.

Elastic reverse time migration using an efficient and accurate adaptive variable grid discretization method

Abstract The conventional reverse time migration utilizes regularly sampled computational grids to simulate wave propagation. Selecting the appropriate grid sampling is important for computational accuracy and efficiency. In general, the uniform-size grid cannot represent the complexity of the geology well. The grid may appear sparse in the low-velocity zone, especially in shallow depths where dispersion may occur. Conversely, it may appear excessively dense in the high-velocity zone, such as at greater depths or within a salt body, which results in higher computational memory and time consumption. To overcome these issues, we developed an efficient and accurate adaptive variable grid discretization method that automatically selects the vertical grid size based on the velocity, depth, and dominant frequency of the wavelet in elastic medium. Then we reformulated the elastic equations based on the adaptive variable grid by introducing a mapping relationship. To test the effectiveness, accuracy, and efficiency of the equation, we implemented it to both the forward propagation and migration of elastic wavefield. Synthetic numerical examples demonstrate that our proposed method can achieve elastic wavefield separation and no significant dispersion phenomenon. The multi-component imaging accuracy of reverse time migration is nearly equivalent to the traditional method, while significantly improving computational efficiency and saving storage space.

Near‐surface wind profiles from numerical model predictions. Part I: Algorithms and comparisons with wind profile based on Monin–Obukhov similarity theory

AbstractWinds are predicted on the discrete grid of numerical weather and climate models. Winds distribute nonlinearly on the height in the near‐surface layer, and a 10 m wind prediction within the layer is often diagnosed upon the Monin–Obukhov similarity theory flux–profile relationship determined from winds at the lowest grid level, the near‐surface atmospheric stability, and surface properties, which leads to concerns that systemic biases may be introduced to the diagnosed wind. Algorithms are proposed to derive near‐surface wind profiles from the grid‐based numerical model forecasts at multiple model levels under the framework of momentum conservations with an implicit solution, associated with simple logarithmic plus linear interpolation in exceptional exemptional conditions. The diagnosed wind profile coheres to the model prediction at the grid level and exhibits differences from the profile using the conventional scheme in the quasi‐steady thermal stratification and non‐steady transitional conditions, retreating to the same logarithmic profile in the neutral condition.

Enhancing the Resolution of Satellite Ocean Data Using Discretized Satellite Gridding Neural Networks

Ocean satellite data are often impeded by intrinsic limitations in resolution and accuracy. However, conventional data reconstruction approaches encounter substantial challenges when facing the nonlinear oceanic system and high-resolution fusion of variables. This research presents a Discrete Satellite Gridding Neural Network (DSGNN), a new machine learning method that processes satellite data within a discrete grid framework. By transforming the positional information of grid elements into a standardized vector format, the DSGNN significantly elevates the accuracy and resolution of data fusion through a neural network model. This method’s innovative aspect lies in its discretization and fusion technique, which not only enhances the spatial resolution of oceanic data but also, through the integration of multi-element datasets, better reflects the true physical state of the ocean. A comprehensive analysis of the reconstructed datasets indicates the DSGNN’s consistency and reliability across different seasons and oceanic regions, especially in its adept handling of complex nonlinear interactions and small-scale oceanic features. The DSGNN method has demonstrated exceptional competence in reconstructing global ocean datasets, maintaining small error variance, and achieving high congruence with in situ observations, which is almost equivalent to 1/12° hybrid coordinate ocean model (HYCOM) data. This study offers a novel and potent strategy for the high-resolution reconstruction and fusion of ocean satellite datasets.

A full-field approach for precipitation in metallic alloys. Comparison with a mean-field model

Modeling precipitation in metallic alloys is a topic of great importance in physical metallurgy as the resulting strengthening strongly depends on the precipitate microstructure. We propose here a numerical full-field model for precipitation that describes precipitates with shape functions, thereby allowing to bridge scales between phase-field approaches - that accurately describe the precipitate evolution but require a fine discretization grid - and mean-field approaches - that are computationally very efficient but rely on strong assumptions. Our results demonstrate the capability of the full-field approach to model the different stages of precipitation during isothermal treatments. The comparison with mean-field results allow to discuss the influence of solutal impingement and precipitate coagulations on the evolution of the precipitate microstructure.

Discontinuous Galerkin approximations of the heterodimer model for protein–protein interaction

Mathematical models of protein–protein dynamics, such as the heterodimer model, play a crucial role in understanding many physical phenomena, e.g., the progression of some neurodegenerative diseases. This model is a system of two semilinear parabolic partial differential equations describing the evolution and mutual interaction of biological species. This article presents and analyzes a high-order discretization method for the numerical approximation of the heterodimer model capable of handling complex geometries. In particular, the proposed numerical scheme couples a Discontinuous Galerkin method on polygonal/polyhedral grids for space discretization, with a θ-method for time integration. This work presents novelties and progress with respect to the mathematical literature, as stability and a-priori error analysis for the heterodimer model are carried out for the first time. Several numerical tests are performed, which demonstrate the theoretical convergence rates, and show good performances of the method in approximating traveling wave solutions as well as its flexibility in handling complex geometries. Finally, the proposed scheme is tested in a practical test case stemming from neuroscience applications, namely the simulation of the spread of α-synuclein in a realistic test case of Parkinson’s disease in a two-dimensional sagittal brain section geometry reconstructed from medical images.

Streaming Algorithms for Geometric Steiner Forest

We consider a generalization of the Steiner tree problem, the Steiner forest problem , in the Euclidean plane: the input is a multiset \(X\subseteq{\mathbb{R}}^{2}\) , partitioned into \(k\) color classes \(C_{1},\ldots,C_{k}\subseteq X\) . The goal is to find a minimum-cost Euclidean graph \(G\) such that every color class \(C_{i}\) is connected in \(G\) . We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to \(X\) . Each input point \(x {\in} X\) arrives with its color \(\mathsf{color}(x) {\in} [k]\) , and as usual for dynamic geometric streams, the input is restricted to the discrete grid \(\{1,\ldots,\Delta\}^{2}\) . We design a single-pass streaming algorithm that uses \(\operatorname{poly}(k\cdot\log\Delta)\) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio \(\alpha_{2}\) (currently \(1.1547\leq\alpha_{2}\leq 1.214\) ). This approximation guarantee matches the state-of-the-art bound for streaming Steiner tree, i.e., when \(k=1\) , and it is a major open question to improve the ratio to \(1+\varepsilon\) even for this special case. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and so far has not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite multiplicative approximation requires \(\Omega(k)\) bits of space.

Meshing theory and contact analysis of double enveloping hourglass worm drive with planar generatrix

The meshing and limit equations of worm drive usually have strong nonlinearities such as multiple solutions, solution nonexistence and equation singularity. Meanwhile, the tooth surfaces of worm drive are in nonconforming line contact, which often requires mesh refinement of contact region for the loaded contact analysis. These two challenges cause that the modeling of worm drive heavily relies on manual adjustment and the loaded contact analysis of worm drive is still rare especially when edge contact and assembly error are concerned. Focusing on the double enveloping hourglass worm (DEHW) drive with planar generatrix, this work presents procedures to solve meshing and limit equations with global convergence. The instantaneous contact line, meshing limit line, curvature interference limit line and tooth surface grid discretization are adaptively generated, without manual trial or adjustment. On the basis of adaptive mesh refinement of tooth surface, the mortar virtual element method is adopted for loaded contact analysis of DEHW drive with edge contact and center distance error. Under sliding friction, the discrete system of governing equalities and inequalities is solved by semi-smooth Newton algorithm after constraint condensation. Numerical results for meshing theory and loaded contact analysis of DEHW drive with planar generatrix are discussed.