Radial Basis Function Interpolation Research Articles

Optimising shapes of multiple pin fins in a microchannel using deep reinforcement learning and mesh deformation techniques

The utilisation of advanced pin fin designs in microchannels is useful for enhancing cooling efficiency. Advancements in machine learning and processing power have sparked interest in shape optimisation techniques. This research employs a novel framework that integrates Deep Artificial Neural Networks and Reinforcement Learning with a Computational Fluid Dynamics (CFD) solver to optimise multiple pin fin shapes within a microchannel. By incorporating Radial Basis Function interpolation and Proximal Policy Optimisation alongside FLUENT, acting as the CFD environment, the reinforcement learning agent adeptly explores the design space to enhance the thermohydraulic performance factor (TPF), aiming to maximise Nusselt number while minimising pressure loss. Unlike previous heat transfer optimisation studies, which typically required mesh regeneration at each step, the proposed framework could bypass the meshing step and alter the geometry directly by relying on the RBF interpolation technique to deform the mesh directly. Three distinct scenarios investigated in this study are uniform deformation of all pin fins, deformation of all pin fins arranged in two rows, and individual deformation of each pin fin. Extensive simulations, exceeding 90,000 different cases, demonstrate that although the optimisation process for the individual deformation of each pin fin requires more iterations compared to others, it surpasses them in terms of TPF improvement. Notably, significant improvements are achieved, such as a 49 % enhancement in Nusselt number and a 33 % reduction in pressure drop, culminating in an impressive 63 % increase in TPF compared to the initial geometry.

A comprehensive MRI-based computational model of blood flow in compliant aorta using radial basis function interpolation

BackgroundProperly understanding the origin and progression of the thoracic aortic aneurysm (TAA) can help prevent its growth and rupture. For a better understanding of this pathogenesis, the aortic blood flow has to be studied and interpreted in great detail. We can obtain detailed aortic blood flow information using magnetic resonance imaging (MRI) based computational fluid dynamics (CFD) with a prescribed motion of the aortic wall.MethodsWe performed two different types of simulations—static (rigid wall) and dynamic (moving wall) for healthy control and a patient with a TAA. For the latter, we have developed a novel morphing approach based on the radial basis function (RBF) interpolation of the segmented 4D-flow MRI geometries at different time instants. Additionally, we have applied reconstructed 4D-flow MRI velocity profiles at the inlet with an automatic registration protocol.ResultsThe simulated RBF-based movement of the aorta matched well with the original 4D-flow MRI geometries. The wall movement was most dominant in the ascending aorta, accompanied by the highest variation of the blood flow patterns. The resulting data indicated significant differences between the dynamic and static simulations, with a relative difference for the patient of 7.47±14.18% in time-averaged wall shear stress and 15.97±43.32% in the oscillatory shear index (for the whole domain).ConclusionsIn conclusion, the RBF-based morphing approach proved to be numerically accurate and computationally efficient in capturing complex kinematics of the aorta, as validated by 4D-flow MRI. We recommend this approach for future use in MRI-based CFD simulations in broad population studies. Performing these would bring a better understanding of the onset and growth of TAA.

Enhancing non-intrusive reduced-order models with space-dependent aggregation methods

AbstractIn this manuscript, we combine non-intrusive reduced-order models (ROMs) with space-dependent aggregation techniques to build a mixed-ROM, able to accurately capture the flow dynamics in different physical settings. The flow prediction obtained using the mixed formulation is derived from a convex combination of the predictions of several previously trained reduced-order models (ROMs), with each model assigned a space-dependent weight. The ROMs incorporated in the mixed model utilize different reduction methods, such as proper orthogonal decomposition and autoencoders, and various approximation techniques, including radial basis function interpolation (RBF), Gaussian process regression, and feed-forward artificial neural networks. Each model’s contribution is given higher weights in regions where it performs best and lower weights where its accuracy is lower compared to the other models. Additionally, a random forest regression technique is used to determine the weights for previously unseen conditions. The performance of the aggregated model is assessed through two test cases: the 2D flow past a NACA 4412 airfoil at a 5-degree angle of attack, with the Reynolds number ranging between $$1 \times 10^{5}$$ 1 × 10 5 and $$1 \times 10^{6}$$ 1 × 10 6 , and a transonic flow over a NACA 0012 airfoil, with the angle of attack as the varying parameter. In both scenarios, the mixed-ROM demonstrated improved accuracy compared to each individual ROM technique, while providing an estimate for the predictive uncertainty.

Accurately identifying the defects of bubbles and foreign objects under the protective films of electric vehicle batteries by using 3D point clouds

Abstract For the defects of bubbles and foreign objects under the protective film of electric vehicle batteries, it is difficult to accurately identify them over traditional 2D optical images. In this paper, we first propose a supervoxel-based region growing algorithm for pre-segmentation of point clouds. Secondly, we utilize radial basis function interpolation and threshold segmentation methods to accurately segment defect point clouds from the entire point cloud. Finally, we develop a feature descriptor and combine it with support vector machine to classify bubbles and foreign objects under the film. This paper achieves the identification of bubbles and foreign objects under the film through two steps: point cloud segmentation and point cloud classification. Experimental results demonstrate that the proposed point cloud segmentation method exhibits high robustness to noise and the intrinsic curvature of the workpiece. Additionally, in the classification scenario presented in this paper, the proposed feature descriptor outperforms classical feature descriptors. Compared to image-based deep learning methods, the defect recognition algorithm proposed in this paper has clear principles and superior performance, with precision and recall of 95.63% and 96.95%, and an intersection over union metric of 0.926.

3D spatial distribution of soil pollutants based on geo-shadowing anisotropic RBF-PCA

Research on soil contamination has become increasingly important, but there is limited information about where to sample for pollutants. Thus, the use of three-dimensional (3D) spatial interpolation techniques has been promoted in this area of study. However, the application of traditional interpolation methods is limited in geography, especially in the expression of anisotropy, and it is not associated with geographical properties. To address this issue, we used a test site (a factory in Nanjing) to develop a new research method based on the geographical shading radial basis function (RBF) interpolation method, which considers 3D anisotropy and geographical attribute expression. Drilling and uniform sampling were used to sample the contaminated area at this test site. This approach included two steps: i) An ellipsoid with anisotropic properties was constructed. Thus, the first step was to determine the shape of the ellipsoid using principal component analysis (PCA) to determine the main orientations and construct a rotational and stretched matrix. The second step was determining the ellipsoid size by computing the range using the variogram method for orientations. ii) During field measurement, the geospatial direction influences soil attribute values, so a shadowing calculation method was derived for quadratic weight determination. Then, the weight of the attribute value of known points can be assigned to meet the field conditions. Lastly, the model was evaluated using the root mean square error (RMSE). For the 2D space, the RMSE values of Kriging, RBF, and the proposed method are 6.09, 7.12, and 5.02, respectively. The R2 values of Kriging, RBF, and the proposed method are 0.871, 0.832, and 0.946, respectively. For the 3D space, the RMSE values of Kriging, RBF, and the proposed method are 2.65, 2.23, and 2.58, respectively. The R2 values of Kriging, RBF, and the proposed method are 0.934, 0.912, and 0.953, respectively. The resulting fitted model was relatively smooth and met experimental needs. Thus, we believe that the interpolation method can be applied as a new method to predict the distribution of soil pollutants.

Aerodynamic and Structural Assessment of Floating Wind Turbine Rotor Under Varying Tilt Angle

Predicting the aerodynamic performance of floating offshore wind turbines (FOWTs) proves challenging due to platform motion induced by waves. The effect of wind and waves results in a six-degree-of-freedom motion of the platform, directly influencing turbine performance. Understanding the impact of specific degrees of freedom (DOF) motions on aerodynamics and structural response is crucial for effective wind turbine design. This research examines the impact of rotor tilt on both aerodynamic performance and structural response. The investigation employs computational fluid dynamics (CFD) analysis and mapping aerodynamic loads onto the finite element (FE) mesh for structural analysis. The study employs a comprehensive 3D simulation, utilizing the moving reference frame (MRF) method for the NREL 5 MW reference wind turbine CFD simulations. It explores different rotor tilt angles (5°, 10°, 15°, and 20°) encountered by offshore structures during their operation and examines their impact on aerodynamic performance. Predicted aerodynamic loads were mapped onto the blade FE mesh using the radial basis function (RBF) interpolation technique and solved using the open-source FE solver CalculiX. The analysis shows that the turbine performance is relatively unaffected up to a tilt angle of 10°. However, further increase in rotor tilt angle adversely impacts turbine performance, leading to notable reductions in thrust and power output. The fluid-structure coupled analysis provided insights into the deformations and stresses experienced by the turbine blade, indicating a notable increase in flap-wise displacement for larger tilt angles, while edge-wise displacement is not as significantly affected. The maximum stress location on the blade generally correlates well with actual observations.

Enhancing OpenEP: atrial conduction velocity and conduction velocity heterogeneity quantification through EP Workbench

Abstract Background Alterations in atrial conduction velocity have been implicated in the pathogenesis of atrial arrhythmias, with conduction velocity thought to be slower and more heterogeneous in areas promoting atrial fibrillation. However, clinical studies of conduction velocity are challenging since there are no available platforms that provide researchers and clinicians with the ability to quantify conduction velocity and analyse conduction heterogeneity from electroanatomic mapping data. Purpose In this paper we sought to address these limitations by enhancing the OpenEP Python library by (1) automating calculations using previously-published conduction velocity estimation algorithms; (2) providing an interactive graphical representing of conduction velocity and conduction heterogeneity and (3) implementing a histogram analysis tool to quantify conduction velocity and enable the identification of slow conduction regions. Method Two well-known and previously published methods for calculating cardiac conduction velocity, Triangulation and polynomial surface fitting, were implemented. Conduction velocity estimation was improved further by excluding regions with wave collisions and focal discharges. These regions were identified using the divergence of conduction velocity vector fields, with positive divergence representing focal sources and negative divergence representing areas of collision. Finally, calculated conduction velocities using electrogram data were interpolated over the surface mesh via Radial Basis Function (RBF) interpolation method. All algorithms were implemented in the OpenEP Python library (openep-py), with visualisation tools provided in the complementary graphical interface, EP Workbench. Results An extensible "Analysis" feature was created in openep-py, to provide conduction velocity and divergence calculations which are fully customisable through the graphical interface based on user preferences. The EP Workbench desktop application displays resulting cardiac surface maps to depict calculated conduction velocity and divergence fields, together with conventional activation and voltage maps (Figure 1). The histogram analysis tool can be used to identify regions of slow conduction velocity from electroanatomic mapping data. Using a threshold of <0.3m/s, two slow conducting channels are visualised on the posterior wall of a test case (Figure 2). The tool is not specific to conduction velocity and may also be used to quantify other data types in EP Workbench. Finally, computed conduction velocity and divergence values and surface maps are stored in the standard OpenEP data structure, allowing further analysis following data export. Conclusion We have extended OpenEP to provide a comprehensive set of tools for conduction velocity calculation and analysis which will facilitate investigation of the structural and functional basis of conduction velocity alterations in disease states.

Quasi-Interpolation on Chebyshev Grids with Boundary Corrections

Quasi-interpolation is a powerful tool for approximating functions using radial basis functions (RBFs) such as Gaussian kernels. This avoids solving large systems of equations as in RBF interpolation. However, quasi-interpolation with Gaussian kernels on compact intervals can have significant errors near the boundaries. This paper proposes a quasi-interpolation method with Gaussian kernels using Chebyshev points and boundary corrections to improve the approximation near the boundaries. The boundary corrections use a linear approximation of the function beyond the interval to estimate the truncation error and add correction terms. Numerical studies on test functions show that the proposed method reduces errors significantly near boundaries compared to quasi-interpolation without corrections, for both equally spaced and Chebyshev points. The convergence and accuracy with the boundary corrections are generally better with Chebyshev points compared to equally spaced points. The proposed method provides an efficient way to perform quasi-interpolation on compact intervals while controlling the boundary errors. This study introduces a novel approach to quasi-interpolation modification, which significantly enhances convergence rates and minimizes errors at boundary points, thereby advancing the methods for boundary approximation.

Investigation of radial basis function dynamic mesh method with rotation correction based on adaptive background mesh

Strategies to improve the quality, robustness, and efficiency of dynamic mesh deformation are investigated. Radial basis function (RBF) interpolation is used for the background mesh, and volume-weighted interpolation is utilized for the computational mesh. An adaptive background mesh with an octree structure is automatically generated. It is refined according to the resolution near the boundaries and the wall's interior is also covered by mesh which is of sufficient resolution and prevents a negative volume due to large deformations of complex configurations. Secondly, the RBF method based on adaptive background mesh, the RBF interpolation angle method based on adaptive background mesh, and the damping function-RBF interpolation angle method based on adaptive background mesh have been developed. This method is applicable to complex shapes with multiple gaps, ensuring a high-quality mesh and high computational efficiency for large deformations. The numerical validation of the interpolation rotation angles for different RBFs indicates that the CTPS C0 function is suitable for large gradients in the boundary deformation. The RBF interpolation angle method ensures a smoother deformation for rigid and large deformations. The damping function-RBF interpolation angle method ensures mesh orthogonality near the boundary during elastic deformation, improving the boundary layer's quality and deformation robustness. Several examples show that the proposed method improves the mesh quality by about 30% and increases the time consumption by about 10%. In addition, the methods are applicable to rigid/elastic deformation of objects with multiple boundaries.

A new contact and road model for multi-body dynamic simulation of wheeled vehicles on soft-soil terrain

AbstractIn this paper, a new high-performance and memory-efficient contact and road model is developed. Specifically, the road is modeled as a rectangular structured grid of deformable springs in the vertical direction, thus enabling fast execution. The new road model stands out due to its ability to handle large road scenarios by allocating computer memory dynamically for each spring, resulting in efficient memory utilization. Furthermore, each spring represents a small road patch that entails various information, such as the soil elevation, the soil properties, and the soil compaction, allowing for complicated simulations incorporating spatially varying soil properties and phenomena related to the multi-pass effect. In addition, using the new contact model, complex terrain geometries are handled in a computationally efficient way by approximating locally the irregular road profile with a suitable equivalent plane. For this, two different strategies are proposed, namely the radial basis function (RBF) interpolation method and the 3D enveloping contact model. Finally, the proposed techniques are implemented in Altair MotionSolve, a comprehensive multi-body simulation software for complex mechanical systems. In particular, a single-wheel test bed is initially examined followed by a four-wheeled rover model and the next-generation NATO reference mobility model (NG-NRMM). In all cases, the proposed model is validated by using available experimental data. Lastly, a case involving both wheeled and tracked vehicles is also examined by using a shared road model.

Extending error bounds for radial basis function interpolation to measuring the error in higher order Sobolev norms

Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal rates for the whole RKHS are often known. Schaback added the doubling trick [Math. Comp. 68 (1999), pp. 201–216], which shows that functions having double the smoothness required by the RKHS (along with specific, albeit complicated boundary behavior) can be approximated with higher convergence rates than the optimal rates for the whole space. Other advances allowed interpolation of target functions which are less smooth, and different norms which measure interpolation error. The current state of the art of error analysis for RBF interpolation treats target functions having smoothness up to twice that of the native space, but error measured in norms which are weaker than that required for membership in the RKHS. Motivated by the fact that the kernels and the approximants they generate are smoother than required by the native space, this article extends the doubling trick to error which measures higher smoothness. This extension holds for a family of kernels satisfying easily checked hypotheses which we describe in this article, and includes many prominent RBFs. In the course of the proof, new convergence rates are obtained for the abstract operator considered by Devore and Ron in [Trans. Amer. Math. Soc. 362 (2010), pp. 6205–6229], and new Bernstein estimates are obtained relating high order smoothness norms to the native space norm.

A comparative study of different radial basis function interpolation algorithms in the reconstruction and path planning of γ radiation fields

Accurate reconstruction of radiation field and path planning are very important for the safety of operators in the process of dismantling nuclear facilities. Based on radial basis function (RBF) interpolation algorithm, this paper discussed the application of inverse multiquadric radial basis Function (IMRBF) interpolation method to the reconstruction of gamma radiation field, and proved the feasibility of reconstructing a radiation field with multiple γ sources. The average relative errors of IMRBF interpolation results were 4.28% and 8.76%, respectively, for the experimental scenarios with single and double gamma sources. After comparing the consistency between the simulated scene and the experimental scene, IMRBF method and Cubic Spline method were respectively used to reconstruct the gamma radiation field by Geant4 simulation data. The results showed that the interpolation accuracy of IMRBF method was superior to that of Cubic Spline method. Further, more RBF interpolation algorithms were used to reconstruct the multi-γ source radiation field, and then the Probabilistic Roadmap (PRM) algorithm was used to optimize the human walking path in the radiation field reconstructed by different interpolation methods. The optimal paths in radiation fields generated by multiple interpolation methods were compared. The results herein contribute to a comprehensive understanding of RBF interpolation methods in reconstructing γ radiation fields and their application in optimizing paths in radiation environments. The insights may provide valuable information for decision-making in radiation protection during the decommissioning of nuclear facilities.

Three-dimensional geological modelling of igneous intrusions in LoopStructural v1.5.10

Abstract. Over the last 2 decades, there have been significant advances in the 3D modelling of geological structures via the incorporation of geological knowledge into the model algorithms. These methods take advantage of different structural data types and do not require manual processing, making them robust and objective. Igneous intrusions have received little attention in 3D modelling workflows, and there is no current method that ensures the reproduction of intrusion shapes comparable to those mapped in the field or in geophysical imagery. Intrusions are usually partly or totally covered, making the generation of realistic 3D models challenging without the modeller's intervention. In this contribution, we present a method to model igneous intrusions in 3D considering geometric constraints consistent with emplacement mechanisms. Contact data and inflation and propagation direction are used to constrain the geometry of the intrusion. Conceptual models of the intrusion contact are fitted to the data, providing a characterisation of the intrusion thickness and width. The method is tested using synthetic and real-world case studies, and the results indicate that the method can reproduce expected geometries without manual processing and with restricted datasets. A comparison with radial basis function (RBF) interpolation shows that our method can better reproduce complex geometries, such as saucer-shaped sill complexes.

Multi‐Spacecraft Magnetic Field Reconstructions: A Cross‐Scale Comparison of Methods

AbstractSpace plasma studies frequently use in situ magnetic field measurements taken from many spacecraft simultaneously. A useful data product of these measurements is the reconstructed magnetic field in a volume near the spacecraft observatory. We compare a standard Linear method of computing the magnetic field at arbitrary spatial points to two novel approaches: a Radial Basis Function interpolation and a time‐dependent 2D inverse distance weighted interpolation scheme called Timesync. These three methods, which only require in situ measurements of the magnetic fields and bulk plasma velocities at a sparse set of spatial points, are implemented on synthetic data drawn from a time‐evolving numerical simulation of plasma turbulence. We compare both the topology of the reconstructed field to the ground truth of the simulation and the statistics of the fluctuations found in the reconstructed field to those from the simulated turbulence. We conclude that the Radial Basis Function and Timesync methods outperform the Linear method in both topological and statistical comparisons.

An adaptive harmonic polynomial cell method for three-dimensional fully nonlinear wave-structure interaction with immersed boundaries

To accurately simulate wave-structure interaction based on fully nonlinear potential flow theory, a three-dimensional (3 D) high-order immersed-boundary adaptive harmonic polynomial cell (IB-AHPC) method is proposed. Both the free surface and body surface are immersed in background octree cells that are adaptively refined near the boundaries of interest, thereby dramatically reducing computational costs without loss of accuracy. We also propose an easy-to-implement IB strategy to deal with possible instabilities in the time-domain solution arising from the intersection of Dirichlet–Neumann boundaries. For a linearized problem of wave-wall interaction, a matrix-based stability analysis is performed, providing mathematical support for the robustness of the proposed IB strategy. In contrast to the two-dimensional HPC method, compressed cells are found to offer superior stability compared to stretched cells in the vertical direction, while equal mesh aspect ratio in the horizontal plane is superior. Cubic octree cells are, however, still preferred in practice. The free surface is primarily described by a set of massless background wave markers; however, to address the challenges of IB methods in tracking the free surface evolution near the structure, additional body-fitted wave markers are introduced close to the waterline. The information exchange between these two sets of wave markers is realized by radial basis function (RBF) interpolation. While standard RBF schemes have grid-size-dependent filtering performance, we propose a normalized RBF scheme, which is then optimized in terms of the number of neighboring nodes, a smoothing coefficient and the basis functions. Excellent accuracy properties of the proposed 3 D IB-AHPC method are demonstrated by studying fully nonlinear wave propagation. The method is further applied to study relevant fully nonlinear wave-structure interaction problems, including sloshing in 3 D rectangular tanks and wave diffraction of a bottom-mounted cylinder in regular waves. Satisfactory agreement is demonstrated with existing experimental and numerical results, suggesting that the proposed 3 D IB-AHPC method is a promising potential-flow method in marine hydrodynamics.

On reduced-order modeling of gas–solid flows using deep learning

Reduced-order models (ROMs) have been extensively employed to understand complex systems efficiently and adequately. In this study, a novel parametric ROM framework is developed to produce Eulerian–Lagrangian simulations. This study employs two typical parametric strategies to reproduce the physical phenomena of a gas–solid flow by predicting the adequate dynamics of modal coefficients in the ROM: (i) based on the radial-basis function (RBF) interpolation, termed ROM-RBF and (ii) based on a long–short term memory (LSTM) neural network, termed ROM-LSTM. In the ROM, an advanced technique, namely, Lanczos-based proper orthogonal decomposition (LPOD), is employed to efficiently transform numerical snapshots into the modal coefficients. Validation tests are conducted in a typical gas–solid flow system such as a spouted bed. The coherent structures of the gas–solid flows are shown to be captured by the LPOD technique. Besides, in comparison with the high-fidelity simulations, our proposed ROMs are shown to simulate the gas–solid flows by significantly reducing the calculation time by several orders of magnitude and faithfully predicting the macroscopic properties. In particular, compared to the ROM-RBF, the ROM-LSTM can capture the flow fields more accurately within the gas–solid flows.

NON-NESTED MULTILEVEL ACCELERATION OF MESHLESS SOLUTION OF HEAT CONDUCTION EQUATION IN COMPLEX DOMAINS

In this paper, we developed and demonstrated a non-nested multilevel procedure for solving the heat conduction equation in complex domains using a meshless discretization method. Previous multilevel methods for solving elliptic partial differential equations in complex domains have mostly used one of the four approaches: nested refinement, agglomeration, additive corrections, and algebraic coarsening. Each of these techniques has some issues of generality, robustness, and speed of acceleration. In this paper, we developed a generally applicable multilevel algorithm for partial differential equations discretized on complex domains using unstructured finite volume, finite element, and meshless methods. We applied this multilevel method to accelerate convergence of the set of discrete equations obtained by a meshless technique. The heat conduction equation is discretized at scattered points using a polyharmonic spine (PHS) radial basis function (RBF) interpolation with appended polynomials to achieve exponential convergence of discretization errors. The RBF interpolations are performed over clouds of points, and the partial differential equation is collocated at the scattered points. The multilevel algorithm to solve the set of linear equations utilizes multiple independently generated coarser sets of points. Restriction of residuals and prolongation of the corrections are also performed using the RBF interpolations. The fast convergence of the algorithm is demonstrated for solution of the heat conduction equation in three model complex domains with manufactured solutions. A simple successive over-relaxation point solver is used as the relaxation scheme.

Study on the regional ASF prediction method based on the ordinary kriging interpolation

The eLoran system functions as a robust backup to the Global Navigation Satellite System (GNSS), providing substantial signal power, robust anti-jamming capabilities, and an easily maintainable ground system. Nonetheless, the propagation time delay of the system, primarily driven by the Additional Secondary Phase Factor (ASF), significantly influences the positioning accuracy. Compensating for ASF can effectively enhance the positioning performance. Traditional propagation delay theories frequently yield substantial discrepancies between the predicted and measured values in regions characterized by extended propagation distances and varying topographic features. To address this issue, we conducted ASF measurements within a selected test area using a limited number of measurement points. We employed the ordinary Kriging interpolation method to predict ASF values across the entire test area, and used cross-validation to validate our predictions. The results confirmed the accuracy and effectiveness of the Kriging interpolation algorithm in predicting ASF values within specific regions. The cross-validation demonstrated that the errors remained within acceptable ranges. Furthermore, we applied Ordinary Kriging, Inverse Distance Weighting, and Radial Basis Function Interpolation methods to evaluate the positioning accuracy of the test area before and after ASF correction. Compared to other methods, using the Ordinary Kriging interpolation algorithm for predicting ASF values resulted in a corrected positioning accuracy of up to 68.8 m at various test locations. This approach effectively resolves the challenge of low accuracy in theoretical calculations in complex environments. By utilizing ordinary Kriging interpolation, we required measurements of only a few ASF values within a specific region to create an ASF correction map, addressing the challenges related to inaccurate theoretical calculations in complex pathways and avoiding time-consuming and labor-intensive large-scale measurements. The results of this study offer valuable theoretical support for improving the accuracy of land-based navigation systems.

Application of Trigonometric Polynomial Fitting Method in Simulating the Spatial Distribution of PM2.5 Concentration in South-Central China

Near-surface PM2.5 estimates remain a global scientific research challenge due to their effect on human fitness and atmospheric environmental quality. However, practical near-surface PM2.5 estimates are impeded by the incomplete monitoring data. In this study, we propose the trigonometric polynomial fitting (TPF) method to estimate near-surface PM2.5 concentrations in south-central China during 2015. We employ 10-fold cross-validation (CV) to assess the reliability of TPF in estimating practical PM2.5 values. When compared to alternative methods such as the orthogonal polynomial fitting (OBF) method based on Chebyshev basis functions, Kriging interpolation, and radial basis function (RBF) interpolation, our results show that utilizing TPF31, with a maximum order of 3 in the x direction and a maximum order of 1 in the y direction, leads to superior efficiency through error minimization. TPF31 reduces MAE and RMSE by 1.93%, 24%, 6.96% and 3.6%, 23.07%, 10.43%, respectively, compared to the other three methods. In addition, the TPF31 method effectively reconstructs the spatial distribution of PM2.5 concentrations in the unevenly distributed observation stations of Inner Mongolia and the marginal regions of the study area. The reconstructed spatial distribution is remarkably smooth. Despite the non-uniform distribution of observation stations and the presence of missing data, the TPF31 method demonstrates exceptional effectiveness in accurately capturing the inherent physical attributes of spatial distribution. The theoretical and experimental results emphasize that the TPF method holds significant potential for accurately reconstructing the spatial distribution of PM2.5 in China.

Real-space tight-binding model for twisted bilayer graphene based on mapped Wannier functions

Twisted multi-layer heterostructures have been considered a platform for studying highly correlated many-particle systems, hosting the emerging phenomena of correlation physics. Electronic structure calculations of such materials which mainly focused on Twisted Bilayer Graphene (TBG) in the framework of the independent-electron approximation, despite the complexity, accurately match the experimental results around the Fermi level. Here, we present a convenient real space π-bands tight-binding model to calculate the band structure of TBG based on the Wannier interlayer hopping parameters obtained from non-twisted bilayer graphene. Our approach is based on mapping hopping parameters onto the real space TBG interlayer couplings, using Radial Basis Function (RBF) interpolation method. This accurate mapping, naturally, transfers all the symmetries and orbital shape features of the bilayer graphene lattice to the TBG lattice and preserves coupling orientation dependencies in addition to distance dependencies. The importance of this fact is explained by showing the effect of the threefold rotation symmetry of AB′ interlayer coupling on the flat band of the TBG band structure. In addition, due to real space study, the model gives us a comprehensive and intuitive view of the role of each interlayer orbital coupling in the TBG band structure and the structural variation relative to twisting.