Local Interpolation Research Articles

A comparative study of interpolation methods for the development of ore distribution maps

Mining projects require precise knowledge about tonnage and quality of ore reserves for planning and decision-making. This is hard to establish as exploration operations, which are costly, time-consuming and require an accurate description of the deposit. Geologists use deterministic and geostatistical methods as interpolation tools to generate a continuous (or prediction) ore distribution map from known sampled data. A comparative study between four different geostatistical interpolation techniques: Ordinary Kriging (OK), Simple Kriging (SK), Universal Kriging (UK) and Empirical Bayesian Kriging (EBK); and four deterministic techniques: Inverse Distance Weighting (IDW), Global Polynomial Interpolation (GPI), Radial Basis Function (RBF), and Local Polynomial Interpolation (LPI) were applied. The 95% confidence level was measured and the effectiveness of each interpolation method was assessed through cross-validation for the construction of the corresponding maps displaying the distribution of ore on the surface with the use of GIS. The output methods are ranked to perform a comprehensive analysis using the error statistics methods: Mean Error (ME), Root Mean Square Error (RMSE), Mean Standardized Error (MSE) and Root Mean Square Standardized Error (RMSSE). The results show that GPI, EBK and Kriging methods perform the best results while IDW has good statistical analysis factors but it came in the tail of the rank. Therefore, there is no appropriate interpolation method accurate for all cases, each method must be statistically evaluated before each application and essentially based on real data.

Dynamic response of a MEMS electrothremal actuator by the local radial point interpolation method

Dynamic response of a MEMS electrothremal actuator by the local radial point interpolation method

THE CHARACTERIZATION OF HIGH-ALTITUDE GRASSLANDS IN THE ALMAJ MOUNTAINS

Grasslands are essential for providing animal feed and ecosystem services, having important roles in regulating the climate and maintaining biodiversity and natural cycles. This study analyzed the distribution and characteristics of the grasslands in the Almăj Mountains, highlighting the influence of topographic and climatic factors on them. The research was carried out in the Almăj Mountains, a mountainous area in southwestern Romania, using geospatial techniques in ArcGIS 10.4. The grasslands were delimited and analyzed based on a Digital Elevation Model (DEM), slopes, slope exposure, temperature and precipitation. Climate data were obtained from local weather stations and spatial interpolation provided a detailed analysis of environmental variables. The grasslands of the Almăj Mountains are concentrated between altitudes of 500 - 800 m (76% of the surface). Lands with slight and moderate slopes (5.1 - 15°) host 77% of them, and the dominant exposure is south and southeast, an orientation that favours grassland vegetation. Multiannual average temperatures are high, as an effect of global warming, which also affects mountainous areas with low altitudes. The distribution of grasslands in the Almăj Mountains is significantly influenced by altitude, slope, exposure and climatic conditions. Grasslands are used agro-pastorally, having a crucial importance for local communities. The geospatial methodology facilitated a detailed analysis, contributing to the development of sustainable management practices for these ecosystems.

Interpolation methods for spatial distribution of groundwater mapping electrical conductivity

This study was carried out to develop a conceptual framework for determining the best interpolation method which mainly is employed to calculate the variability maps of electrical conductivity (EC) in neighboring regions. The considered case study is parts of the Khorasan Razavi province, Iran (including five aquifers Kashmar, Fariman, Doruneh, Sarakhs and Joveyn). In the first step, the empirical variogram (semi-variogram) was computed for the study area. The methods of the variability of a variable with spatial or temporal distance were considered to measure the semi-variogram function. In the next step, the best variogram model (e.g. spherical, exponential or Gaussian) was considered in the Geographic Information System (GIS) environment and f for the Environmental Sciences (GS+) software. By plotting the semi-variogram in GS+ program based on different method as Global Polynomial Interpolation (GPI), Inverse distance weighing (IDW), Radial basis function (RBF), Kriging method, Global Polynomial Interpolation (GPI), Local Polynomial Interpolation (LPI), the best variogram model fitted to spatial structure of the EC. Finally, by considering the acceptable range for different parameters which impact on EC and evaluating their impacts by scaling, the best interpolation method has been selected for that area for employing their neighborhood basin. Result indicated that the precipitation located within the range of 140 to 180 mm, RBI has the priority. This process is continued for all 14 parameters and eventually one method gets the most points.

Fluid Implicit Particles on Coadjoint Orbits

We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We start with a Hamiltonian formulation of the incompressible Euler Equations, and then, using a local, explicit, and high order divergence free interpolation, construct a modified Hamiltonian system that governs our discrete Euler flow. The resulting discretization, when paired with a geometric time integration scheme, is energy and circulation preserving (formally the flow evolves on a coadjoint orbit) and is similar to the Fluid Implicit Particle (FLIP) method. CO-FLIP enjoys multiple additional properties including that the pressure projection is exact in the weak sense, and the particle-to-grid transfer is an exact inverse of the grid-to-particle interpolation. The method is demonstrated numerically with outstanding stability, energy, and Casimir preservation. We show that the method produces benchmarks and turbulent visual effects even at low grid resolutions.

Bending analysis of viscoelastic plates according to the Reissner’s theory using meshless local Petrov–Galerkin method and hereditary integral formulation

Purpose The purpose of this study is to apply the Meshless Local Petrov–Galerkin (MLPG) method to solve the bending problems of linear viscoelastic plates, considering Reissner’s theory.Design/methodology/approachThe weak formulation for the set of equations that govern Reissner’s plate theory is implemented in conjunction with the integral formulation applied to viscoelastic constitutive expressions. A meshless method based on the Moving Least Squares (MLS) approximation is considered in the numerical implementation. The final equation system is assembled by adopting simple and efficient schemes for numerical integration, considering a simplified formulation through centralization of the local interpolation domains and Gaussian quadrature at the same field point. The results obtained are compared with available solutions to demonstrate the efficiency of the proposed formulation.FindingsThe hereditary integral approach proved to be the most general way to analyze the viscoelastic problem, especially when applied together with the modified scheme for numerical integration. In addition, the variable changing technique is demonstrated to be an efficient formulation for solving shear-locking effects in thin plate problems.Originality/valueThe differential of the present study is related to the manner in which the properties of linear viscoelastic materials are considered in the formulation. Although most authors consider this point through the application of the correspondence principle, the present study works with a hereditary integral formulation. In addition, the variable changing technique is applied to solve the shear-locking effects, and an alternative approximation technique is considered to speed up the numerical integration process.

Modelling of the trade-off between the deep soil moisture and vegetation restoration in the hilly area of the Loess Plateau, China

Excessive depletion of soil moisture by artificial forests in the vegetation restoration areas of the Loess Plateau has attracted widespread attention. To assess potential risks of soil moisture deficit, we needed on-site vegetation and soil sampling data, as well as UAV images from the Chaigou Watershed for three-dimensional analysis, combining both sampling and raster data. Three-dimensional surfaces for assessment of trade-off were established innovatively by the local regression and interpolation methods. The results indicated that soil moisture benefits at 20–40 cm depth are lower than at 0–20 cm due to infiltration and surface disturbance. In some areas of the Chaigou Watershed, grass and shrub-grass vegetation are facing risks of soil moisture deficit based on trade-off values (RSMD) and multiparameter evaluations. Analysis of deep soil water content variability revealed the moisture decreases significantly with the deepening of the soil layer in some plots. R (Richness), H (Shannon’s Diversity), Margalef, COHESION, and CONTAG were applicable in interpolation and fitted with the local regression model (R2 > 0.6) corresponding to the trade-off, but SPLIT was proven to be inapplicable in this study area. The zero trade-off inflection points were 6 %–8 %, while the trend inflection points were 7.5 %–9 % for soil moisture and vegetation indices of vegetation and landscape in typical sampling sites of the Chaigou Watershed. Three-dimensional fitting model is more comprehensive and effective in assessing deep soil moisture conditions and grass plots on shady slopes generally had the best trade-off status in this region.

Research on a coal seam modeling construction method based on improved kriging interpolation

To address the issues of large anomalous triangulations, invalid interpolations, and uneven boundary interpolations in kriging interpolation, we propose research on a coal seam modeling construction method based on improved kriging interpolation. The work methodology assumes that by introducing kriging interpolation and analyzing its problems, we improve the interpolation method via a local interpolation algorithm for large anomalous triangulations, an optimization algorithm for locally redundant interpolation points, and a nonuniform boundary adaptive local interpolation algorithm. These improvements allow the interpolation method to better reflect the variability and realistic nature of coal seams. The research results indicate that applying this method to the construction of the Dananhu No. 2 open-pit mine coal seam model has improved the issue of coal seam transition stiffness, such as abnormal large-area triangulation in areas with significant elevation differences. This approach appropriately reduces the memory space usage without altering the coal seam morphology (which saves approximately 27,000 KB of memory, equivalent to the space occupied by 4 out of 21 coal seams). It has also prevented inaccuracies in boundary line positioning and transitions caused by too low a density of points on the coal seam reserve boundary line, resulting in smoother model transitions at the boundaries that better align with the actual coal seam change trends, the error rate in coal quality estimation decreased by 62.69%. This study provides data support for mining planning and reduces costs. This method can be extended to the construction of all mine models.

NUMERICAL COMPARISON OF THREE LOCAL MULTIVARIATE GAUSSIAN-BASED INTERPOLATION SCHEMES

This study introduces the Local Explicit Radial Basis Function scheme (LE-RBF), alongside an investigation of its effectiveness compared with two other Gaussian RBF-based methods: the Modified Radial Basis Function Shepard's method (MRBFS) and the Local Radial Basis Function Karel's algorithm (KAREL), for interpolation problems. The research focuses on three challenging applications: Franke’s function in two and three dimensions, and grayscale image reconstruction. All three schemes, operating in a local interpolation manner, are assessed across multiple criteria, including accuracy, CPU time and storage, and sensitivity to parameters and node sizes. The numerical experiments conducted reveal that while all three schemes yield reasonably good results in most scenarios, the proposed LE-RBF method stands out for its higher accuracy, reduced sensitivity to nodes and shape choices, and lower CPU time and storage requirements. Notably, LE-RBF demonstrates superior performance in various node densities and shape parameter-selecting strategies, especially when combined with specific strategies like the Carlson shape at 142×142 nodes. Its efficiency in processing time further highlights its practicality for complex applications. The study concludes that LE-RBF, with its robust performance and flexibility, presents a promising avenue for future application in diverse scientific and engineering tasks.

Deformation monitoring for fixed-wing UAS through Inverse Mesh-free Method

This study proposes the Inverse Local Radial Point Interpolation Method (iLRPIM) for reconstructing the deformation of irregular plate structures. The proposed methodology eliminates the drawbacks of the existing inverse finite-element method (iFEM), where the Jacobian matrix results in reduced computational accuracy for irregular plate-shape sensing. To this end, this paper released from the element and used the Mesh-Free method to describe the calculated strains. Subsequently, the system equation was formulated by minimizing the error between the measured and calculated strains. The deformation parameters were then computed by solving this system equation. Additionally, compared to iFEM, iLRPIM allows for easier adjustment of the interpolation function, making it more suitable for real-world engineering applications. The effectiveness of iLRPIM has been validated through simulations and practical experiments using four typical models. In simulations of the UAS wing box equivalent model, peak errors were below 0.8 mm for a deformation range of 25 mm. In practical scenarios, peak errors were below 3.1 mm for a deformation range of 85 mm. These results underscore the system’s capability to accurately sense the shapes of irregular plate structures using iLRPIM.

A novel peak positioning method for nanometer displacement measurement by optical linear encoder

In the field of precision measurement, subdividing displacement within the grating period to achieve high precision and high resolution remains a significant challenge. For subdivision, traditional optical linear encoders need complex coding and complicated optical systems, which struggle to meet the demands of nanometer measurement. In this paper, based on the fundamental principles of grating displacement measurement, a high-precision peak positioning method from grating images is proposed for linear displacement measurements with nanometer resolution and submicron accuracy. By utilizing a local gradient interpolation algorithm to optimize curves, it could further improve the accuracy of displacement measurement. Experiments demonstrate that this method achieves a resolution of 2 nm and an accuracy of ± 0.1 μm in a range of 50 mm using a grating with a 20 μm period. This sets the groundwork for subdividing optical linear encoders further, the resolution and accuracy could be further improved by optimizing the subdivision method.

Empirical Bayesian Kriging, a Robust Method for Spatial Data Interpolation of a Large Groundwater Quality Dataset from the Western Netherlands

No single spatial interpolation method reigns supreme for modelling the precise spatial distribution of groundwater quality data. This study addresses this challenge by evaluating and comparing several commonly used geostatistical methods: Local Polynomial Interpolation (LPI), Ordinary Kriging (OK), Simple Kriging (SK), Universal Kriging (UK), and Empirical Bayesian Kriging (EBK). We applied these methods to a vast dataset of 3033 groundwater records encompassing a substantial area (11,100 km2) in the coastal lowlands of the western Netherlands. To our knowledge, no prior research has investigated these interpolation methods in this specific hydrogeological setting, exhibiting a range of groundwater qualities, from fresh to saline, often anoxic, with high natural concentrations of PO4 and NH4. The prediction performance of the interpolation methods was assessed through statistical indicators such as root means square error. The findings indicated that EBK outperforms the other geostatistical methods in forecasting groundwater quality for the five variables considered: Cl, SO4, Fe, PO4, and NH4. In contrast, SK performed worst for the species except for SO4. We recommend not using SK to interpolate groundwater quality species unless the data exhibit low spatial variation, high sample density, or evenly distributed sampling.

The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system

In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time.

Manifold learning-based reduced-order model for full speed flow field

Reduced-order models (ROMs) can effectively balance the accuracy and efficiency of computational fluid dynamics (CFD). The nonlinear flow field characteristics cannot be captured accurately by traditional ROMs, such as proper orthogonal decomposition (POD). Combining isometric mapping (ISOMAP) and local linear embedding (LLE), a novel manifold learning method (ISOMAP-Local LLE) is proposed, performing global and accurate reconstruction of the nonlinear flow field. First, the nonlinear dimensionality reduction is derived by considering the global isometric characteristic and the local reconstruction relationship simultaneously. Then, the local interpolation model is constructed to improve the interpolation accuracy introduced by the global interpolation model. Finally, the flow field reconstruction is accomplished based on the inverse mapping. Furthermore, the criteria of hyperparameters have been established to achieve high-precision prediction. Several examples covering full speed flow field are carried out to demonstrate the accuracy and efficiency of ISOMAP-Local LLE. The proposed manifold learning-based ROM achieves prediction accuracy that surpasses that of other ROMs (such as POD, ISOMAP, LLE, etc.), resulting in significant time cost savings for CFD simulations.

Eighth-Order Numerov-Type Methods Using Varying Step Length

This work explores a well-established eighth-algebraic-order numerical method belonging to the explicit Numerov-type family. To enhance its efficiency, we integrated a cost-effective algorithm for adjusting the step size. After each step, the algorithm either maintains the current step length, halves it, or doubles it. Any off-step points required by this technique are calculated using a local interpolation function. Numerical tests involving diverse problems demonstrate the significant efficiency improvements achieved through this approach. The method is particularly effective for solving differential equations with oscillatory behavior, showcasing its ability to maintain high accuracy with fewer function evaluations. This advancement is crucial for applications requiring precise solutions over long intervals, such as in physics and engineering. Additionally, the paper provides a comprehensive MATLAB-R2018a implementation, facilitating ease of use and further research in the field. By addressing both computational efficiency and accuracy, this study contributes a valuable tool for the numerical analysis community.

Local Polynomial Interpolation Method Optimization in the Process of Digital Terrain Model Creation Based on Data Collected From a Multibeam Echosounder

Local Polynomial Interpolation Method Optimization in the Process of Digital Terrain Model Creation Based on Data Collected From a Multibeam Echosounder

LPV interpolation modeling and modal-based pole placement control for ball screw drive with dynamic variations

This paper presents a linear parameter varying (LPV) interpolation modeling method and modal-based pole placement (PP) control strategy for the ball screw drive (BSD) with varying dynamics. The BSD is modeled as a global LPV model with position-load dependence by selecting position and load as scheduling variables. The global LPV model is obtained from local subspace closed-loop identification and LPV interpolation modeling. A modal-based global LPV model is obtained through the similarity transformation. Based on this model, a modal-based LPV PP control strategy is proposed to achieve various modal control. Specifically, a state feedback control structure with an LPV state observer is designed to realize online state estimation and real-time state feedback control of modal state variables which cannot be measured directly. The steady-state error is minimized by introducing an error state space (SS) model with the integral effects. Moreover, the stability of the closed-loop system is analyzed according to the controllable decomposition and principle of separation. It is experimentally demonstrated that the proposed modal-based LPV PP control strategy can effectively achieve precise tracking and outstanding robustness meantime.

Fractional Boundary Element Solution for Nonlinear Nonlocal Thermoelastic Problems of Anisotropic Fibrous Polymer Nanomaterials

This paper provides a new fractional boundary element method (BEM) solution for nonlinear nonlocal thermoelastic problems with anisotropic fibrous polymer nanoparticles. This comprehensive BEM solution comprises two solutions: the anisotropic fibrous polymer nanoparticles problem solution and the nonlinear nonlocal thermoelasticity problem. The nonlinear nonlocal thermoelasticity problem solution separates the displacement field into complimentary and specific components. The overall displacement is obtained using the boundary element methodology, which solves a Navier-type problem, and the specific displacement is derived using the local radial point interpolation method (LRPIM). The new modified shift-splitting (NMSS) technique, which minimizes memory and processing time requirements, was utilized to solve BEM-created linear systems. The performance of NMSS was evaluated. The numerical results show how fractional and graded parameters influence the thermal stresses of nonlinear nonlocal thermoelastic issues involving anisotropic fibrous polymer nanoparticles. The numerical findings further reveal that the BEM results correlate very well with the finite element method (FEM) and analytical results, demonstrating the validity and correctness of the proposed methodology.

Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation

This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented.

Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations

We investigate the construction of C2 cubic spline quasi-interpolants on a given arbitrary triangulation T to approximate a sufficiently smooth function f. The proposed quasi-interpolants are locally represented in terms of a simplex spline basis defined on the cubic Wang–Shi refinement of the triangulation. This basis behaves like a B-spline basis within each triangle of T and like a Bernstein basis for imposing smoothness across the edges of T. Any element of the cubic Wang–Shi spline space can be uniquely identified by considering a local Hermite interpolation problem on every triangle of T. Different C2 cubic spline quasi-interpolants are then obtained by feeding different sets of Hermite data to this Hermite interpolation problem, possibly reconstructed via local polynomial approximation. All the proposed quasi-interpolants reproduce cubic polynomials and their performance is illustrated with various numerical examples.