Moving Least Squares Research Articles

An optimal ansatz space for moving least squares approximation on spheres

We revisit the moving least squares (MLS) approximation scheme on the sphere Sd-1⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb S^{d-1} \\subset {\\mathbb R}^d$$\\end{document}, where d>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d>1$$\\end{document}. It is well known that using the spherical harmonics up to degree L∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L \\in {\\mathbb N}$$\\end{document} as ansatz space yields for functions in CL+1(Sd-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}^{L+1}(\\mathbb S^{d-1})$$\\end{document} the approximation order OhL+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( h^{L+1} \\right) $$\\end{document}, where h denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to L, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as h→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\rightarrow 0$$\\end{document}. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space.

Meshless Technique Based on Moving Least Squares Approximation for Numerical Solutions of Linear and Nonlinear Third-Kind VIEs

The principal objective of this paper is to propose a numerical scheme to solve linear and nonlinear third-kind Volterra integral equations (VIEs). The method approximates the solution using the collocation method based on moving least squares (MLS) approximation. The suggested scheme is meshless, since it needs no background mesh or cell structures, and is thus independent of the geometry of the domain. The approach reduces the solution of third-kind VIEs to the solution of systems of algebraic equations. All integrals appearing in the approach are calculated approximately by the Gauss–Legendre integration formula. The convergence analysis of the proposed scheme has been conducted, and its performance has been tested through a series of numerical examples, demonstrating its effectiveness and applicability in comparison to the existing methods.

FRACTURE MECHANISM ANALYSIS OF HIGH-DENSITY FIBREBOARD BASED ON DIGITAL IMAGE CORRELATION TECHNOLOGY

This paper analyses the scattering images of the bending deformation of high-density fibreboards based on the digital image correlation (DIC) technique, so as to study its mechanical deformation law. Three-point bending tests were carried out on fibreboards using a mechanical testing machine with a non-contact measuring system. The measured values of the displacements of the grid nodes in the region of interest (ROI) were combined with the Moving least squares (MLS) method to construct the strains of the high-density fibreboards at different loading forces, thus deriving the strain values of the fibreboards during the bending deformation process. To further analyze its force deformation mechanism, this paper used a portable electron microscope and scanning electron microscope to analyze the damage situation at the fracture damage, and at the same time, it verified that the constructed strain field model was accurate.

Automated Measurement of Cattle Dimensions Using Improved Keypoint Detection Combined with Unilateral Depth Imaging.

Traditional measurement methods often rely on manual operations, which are not only inefficient but also cause stress to cattle, affecting animal welfare. Currently, non-contact cattle dimension measurement usually involves the use of multi-view images combined with point cloud or 3D reconstruction technologies, which are costly and less flexible in actual farming environments. To address this, this study proposes an automated cattle dimension measurement method based on an improved keypoint detection model combined with unilateral depth imaging. Firstly, YOLOv8-Pose is selected as the keypoint detection model and SimSPPF replaces the original SPPF to optimize spatial pyramid pooling, reducing computational complexity. The CARAFE architecture, which enhances upsampling content-aware capabilities, is introduced at the neck. The improved YOLOv8-pose achieves a mAP of 94.4%, a 2% increase over the baseline model. Then, cattle keypoints are captured on RGB images and mapped to depth images, where keypoints are optimized using conditional filtering on the depth image. Finally, cattle dimension parameters are calculated using the cattle keypoints combined with Euclidean distance, the Moving Least Squares (MLS) method, Radial Basis Functions (RBFs), and Cubic B-Spline Interpolation (CB-SI). The average relative errors for the body height, lumbar height, body length, and chest girth of the 23 measured beef cattle were 1.28%, 3.02%, 6.47%, and 4.43%, respectively. The results show that the method proposed in this study has high accuracy and can provide a new approach to non-contact beef cattle dimension measurement.

Deep Learning-Meshless Method for Inverse Potential Problems

This paper combines the deep learning method with the meshless method to propose a new numerical method, which is the deep learning-improved element-free Galerkin (DL-IEFG) method, for solving inverse potential problems. In this method, the unknown term in the governing equation of the inverse potential problem is represented by the feedforward neural network (FNN). By employing the improved element-free Galerkin (IEFG) method to solve the inverse potential problem, the solution equations are established to obtain numerical solutions. The training set is constructed on the valid values obtained from discretized observation spatial points. The predicted values at the training sample points are calculated by combining the numerical solutions with the approximation function built by the improved moving least-squares (MLS) approximation. Then, the FNN representing the unknown term is iterated using the loss function. The effectiveness of the DL-IEFG method for solving potential inverse problems is validated through numerical examples. In addition, the factors impacting the calculation accuracy and efficiency of the DL-IEFG method are investigated.

Neural-IMLS: Self-Supervised Implicit Moving Least-Squares Network for Surface Reconstruction.

Surface reconstruction is a challenging task when input point clouds, especially real scans, are noisy and lack normals. Observing that the Multilayer Perceptron (MLP) and the implicit moving least-square function (IMLS) provide a dual representation of the underlying surface, we introduce Neural-IMLS, a novel approach that directly learns a noise-resistant signed distance function (SDF) from unoriented raw point clouds in a self-supervised manner. In particular, IMLS regularizes MLP by providing estimated SDFs near the surface and helps enhance its ability to represent geometric details and sharp features, while MLP regularizes IMLS by providing estimated normals. We prove that at convergence, our neural network produces a faithful SDF whose zero-level set approximates the underlying surface due to the mutual learning mechanism between the MLP and the IMLS. Extensive experiments on various benchmarks, including synthetic and real scans, show that Neural-IMLS can reconstruct faithful shapes even with noise and missing parts. The source code can be found at https://github.com/bearprin/Neural-IMLS.

Numerical approximation of a generalized time fractional partial integro-differential equation of Volterra type based on a meshless method

The moving least squares (MLS) approximation is used in this study to solve time-fractional partial integro-differential equations (TFPIDE). In our approach, we approximate the time Caputo fractional derivative term and the integral term by using a finite difference scheme and trapezoidal method, respectively, which yields an order of accuracy of O(τ+τ2−α), where 0<α<1. We thoroughly investigate the applicability and validity of the proposed method and provide an error estimate for its performance. Additionally, we solve various numerical problems to demonstrate the accuracy and computational efficiency of our approach, confirming the theoretical findings.

A robust reconstruction method based on local Bayesian estimation combined with CURE clustering

Due to their good approximation accuracy and local fitting characteristics, the moving least squares (MLS) and moving total least squares (MTLS) methods are widely used in various engineering fields. However, neither of these two methods is robust and they cannot effectively deal with outliers in measurement data. To eliminate the negative influence of outliers and achieve robust reconstruction, a novel MTLS method is proposed in this paper, which introduces local Bayesian estimation combined with clustering using representatives (CURE) algorithm. In the support domain, this method adopts a two-step process to remove the abnormal points and adjust the weights of discrete points through compound weighting. Bayesian estimation is first performed on discrete points to derive the reference model, and the residuals are calculated as the input of CURE clustering. The points with large residuals are classified into one cluster and removed. The remaining points undergo repeated processing until the iteration concludes. A gradient weight function based on the residuals and a compact support weight function are combined to determine the final estimated value using weighted Bayesian estimation. The simulations and experiments demonstrate that the proposed reconstruction method achieves excellent accuracy and robustness, surpassing several existing methods when handling highly contaminated datasets.

Determination of Accuracy and Usability of a SLAM Scanner GeoSLAM Zeb Horizon: A Bridge Structure Case Study

The presented paper focuses on testing the performance of a SLAM scanner Zeb Horizon by GeoSLAM for the creation of a digital model of a bridge construction. A cloud acquired using a static scanner Leica ScanStation P40 served as a reference. Clouds from both scanners were registered into the same coordinate system using a Trimble S9 HP total station. SLAM scanner acquisition was performed independently in two passes. The data acquired using the SLAM scanner suffered from relatively high noise. Denoising using the MLS (Moving Least Squares) method was performed to reduce noise. An overall comparison of the point clouds was performed on both the original and MLS-smoothed data. In addition, the ICP (Iterative Closest Point) algorithm was also used to evaluate local accuracy. The RMSDs of MLS-denoised data were approximately 0.02 m for both GeoSLAM passes. Subsequently, a more detailed analysis was performed, calculating RMSDs for several profiles of the construction. This analysis revealed that the deviations of SLAM data from the reference data did not exceed 0.03 m in any direction (longitudinal, transverse, elevation) which is, considering the length of the bridge of 133 m, a very good result. These results demonstrate a high applicability of the tested scanner for many applications, such as the creation of digital twins.

Radial basis point interpolation for strain field calculation in digital image correlation

In order to extract smooth and accurate strain fields from the noisy displacement fields obtained by digital image correlation (DIC), a point interpolation meshless (PIM) method with a radial basis function (RBF) is introduced for full-field strain calculation, which overcomes the problems of slow calculation speed and unstable matrix inverse calculation of the element-free Galerkin method (EFG). The radial basis point interpolation method (RPIM) with three different radial basis functions and the moving least squares (MLS) and pointwise least squares (PLS) methods are compared by analyzing and validating the strain fields with high-strain gradients in simulation experiments. The results indicate that the RPIM is nearly 80% more computationally efficient than the MLS method when a larger support domain is used, and the efficiency of the RPIM is nearly 26% higher than that of the MLS method when a smaller support domain is used; the strain calculation accuracy is slightly lower than that of the MLS method by 0.3–0.5%, but the stability of the calculation is significantly improved. In contrast with the PLS method, which is easily affected by the noise and the size of the strain calculation window, the RPIM is insensitive to the displacement noise and the size of the support domain and can obtain a similar calculation accuracy. The RPIM with multiquadric (MQ) radial basis functions performs well in balancing the computational accuracy and efficiency and is insensitive to shape parameters. The application cases show that the method can effectively compute the strain field at the crack tip, validating its applicability to the study of the plastic region at the crack tip. In conclusion, the proposed RPIM-based method provides an accurate, practical, and robust approach for full-field strain measurements.

An upwind moving least squares approximation to solve convection-dominated problems: An application in mixed discrete least squares meshfree method

Moving Least Squares (MLS), as a series representation type of approximation, is broadly used in a wide array of meshfree methods. However, using the existing standard form of the MLS causes an unphysical oscillation for the meshfree methods encountering the convection-dominated partial differential equations (PDEs). In this study, several approaches are investigated to enhance the MLS approximation for solving convection-dominated problems. A novel upwind version of MLS approximation called shifted upward MLS (SU-MLS), is presented, which is based on strengthening the effect of upwind nodal points in approximation by wisely adjusting the weight function. Regarding the presented theoretical/numerical investigation, the proposed SU-MLS approximation can yield monotone solutions encountering the convection-dominated PDEs, unlike the existing standard form of MLS. The suggested SU-MLS approximation is, then, utilized in the mixed discrete least squares meshfree (MDLSM) method, rather than the standard MLS which is conventionally used in existing MDLSM. The novel method, which uses SU-MLS, is named upwind MDLSM (UMDLSM). Several numerical examples are investigated, and the results are compared to existing MDLSM. The obtained results indicate that the suggested UMDLSM is remarkably more accurate than the existing MDLSM in convection-dominated PDEs. Furthermore, while the existing MDLSM dramatically suffers from spurious oscillations (wiggling) when the Peclet number is high, the presented UMDLSM can yield monotone and accurate solutions.

Moving least‐squares aided finite element method: A powerful means to predict flow fields in the presence of a solid part

AbstractWith the assistance of the moving least‐squares (MLS) interpolation functions, a two‐dimensional finite element code is developed to consider the effects of a stationary or moving solid body in a flow domain. At the same time, the mesh or grid is independent of the shape of the solid body. We achieve this goal in two steps. In the first step, we use MLS interpolants to enhance the pressure (P) and velocity (V) shape functions. By this means, we capture different discontinuities in a flow domain. In our previous publications, we have named this technique the PVMLS method (pressure and velocity shape functions enhanced by the MLS interpolants) and described it thoroughly. In the second step, we modify the PVMLS method (the M‐PVMLS method) to consider the effect of a solid part(s) in a flow domain. To evaluate the new method's performance, we compare the results of the M‐PVMLS method with a finite element code that uses boundary‐fitted meshes.

A study on the effect of defects on the buckling of double-walled carbon nanotubes under compression based on a new atomic-continuum coupling method

An atomic-continuum coupling (ACC) method is developed for the nonlinear mechanical analysis of defective double-walled carbon nanotubes (DWCNTs). The moving least squares (MLS) approximation is resorted to bridge the fully atomic discrete structures of defective DWCNTs and the corresponding virtual continuum solids. The intrinsic mechanic laws implied in nanostructures can be accurately mapped into the mechanical governing equations of the continuum models. Based on ACC method, a numerical computational scheme is developed for predicting the buckling and contact behaviors of defective DWCNTs, which do not need any numerical integration method to calculate potential functional and its derivatives. The numerical tests show that this method can furnish good predictions even with a small number of nodes. It is found that Stone–Wales (SW) defects can lead to greatly decrease in the buckling properties of DWCNTs. In contrary, the complex interlayer van der Waals (vdW) interactions can enhance the buckling resistance of DWCNTs.

Generalized finite difference method on unknown manifolds

In this paper, we extend the Generalized Finite Difference Method (GFDM) on unknown compact submanifolds of the Euclidean domain, identified by randomly sampled data that (almost surely) lie on the interior of the manifolds. Theoretically, we formalize GFDM by exploiting a representation of smooth functions on the manifolds with Taylor's expansions of polynomials defined on the tangent bundles. We illustrate the approach by approximating the Laplace-Beltrami operator, where a stable approximation is achieved by a combination of Generalized Moving Least-Squares algorithm and novel linear programming that relaxes the diagonal-dominant constraint for the estimator to allow for a feasible solution even when higher-order polynomials are employed. We establish the theoretical convergence of GFDM in solving Poisson PDEs and numerically demonstrate the accuracy on simple smooth manifolds of low and moderate high co-dimensions as well as unknown 2D surfaces. For the Dirichlet Poisson problem where no data points on the boundaries are available, we employ GFDM with the volume-constraint approach that imposes the boundary conditions on data points close to the boundary. When the location of the boundary is unknown, we introduce a novel technique to detect points close to the boundary without needing to estimate the distance of the sampled data points to the boundary. We demonstrate the effectiveness of the volume-constraint employed by imposing the boundary conditions on the data points detected by this new technique compared to imposing the boundary conditions on all points within a certain distance from the boundary, where the latter is sensitive to the choice of truncation distance and requires the knowledge of the boundary location.

Element‐free Galerkin method for a fractional‐order boundary value problem

AbstractIn this article, we develop a meshfree numerical solver for fractional‐order governing differential equations. More specifically, we develop a mesh‐free interpolation‐based element‐free Galerkin numerical model for the fractional‐order governing differential equations. The proposed fractional element‐free Garlekin (f‐EFG) numerical model is a lighter and more accurate alternative to existing mesh‐based finite element solvers for the fractional‐order governing differential equations. We demonstrate here that the f‐EFG with moving least squares (MLS) interpolants are naturally suitable for the approximation of fractional‐order derivatives in terms of the corresponding nodal values, thereby alleviating several issues with FE solvers for such integro‐differential governing equations. We demonstrate the efficacy of the proposed numerical model for numerical solutions with benchmark problems on the linear and nonlinear elastic response of nonlocal elastic solid modeled via fractional‐order governing differential equations. However, it must be noted that the proposed f‐EFG algorithm can be extended to fractional‐order governing differential equations in diverse applications, including multiscale and multiphysics studies.

Mass lumping schemes fitted to MLS-based numerical manifold method in vibration of plates with cutouts using CPT and FSDT

It is critical to accurately extract the natural frequencies in the transverse vibration of plates using the classical plate theory (CPT) and the first-order shear deformation plate theory (FSDT). The moving least squares (MLS) based numerical manifold method (MLS-based NMM) in conjunction with a suitable diagonally lumped mass matrix is a reasonable choice because it can naturally treat plates with cutouts and easily formulate an H2 regular approximation based on MLS interpolation, as well as mitigate shear locking issues for vibration analysis of thin plates based on FSDT. Moreover, the mass lumping techniques involving the row-sum method, the diagonal scaling method, and the mathematically rigorous manifold-based method are extended and derived in the unified framework of MLS-based NMM for transverse vibration analysis of plates. Furthermore, the positive-definiteness of the lumped mass matrix (LMM) generated by the manifold-based mass method is explained in MLS settings, and the row-sum method is proven to be a special case of the manifold-based method. A comprehensive comparative study of different LMMs is performed in accordance with the consistent mass matrix (CMM) on extensive numerical benchmarks. Numerical results demonstrate the convergence properties of LMMs compared with CMM in MLS-based NMM for plate vibration based on CPT and FSDT. It can be observed that LMMs yield comparable performance compared with CMM. The row-sum method obtains accurate results in most cases but can not guarantee the positivity. In contrast, the manifold method can guarantee the positive-definiteness of LMM and is more accurate than the diagonal scaling method in most cases.

A generalized finite difference method for 2D dynamic crack analysis

This paper presents a new framework for efficient and accurate analysis of transient elastodynamic cracks by using the generalized finite difference method (GFDM). The method first discretizes the solution domain into a set of overlapping small subdomains, and then in each of the subdomains, the unknown functions and their derivatives are approximated by using the local Taylor series expansions and moving-least square approximation. The degree of the Taylor series used in the local subdomain is increased automatically in the regions near the crack-tips, in order to appropriately describe the local asymptotic behavior of near-tip displacement and stress fields. The path-independent J-integral and sub-domain technique are adopted to compute the dynamic stress intensity factors (SIFs) of the cracked bodies. Preliminary numerical experiments for dynamic SIFs with both uniform and variable loading conditions are given to show the efficient and accuracy of the present method for transient elastodynamic crack analysis.

3D reconstruction of the scapula from biplanar X-rays for pose estimation and morphological analysis.

Patient-specific scapular shape in functional posture can be highly relevant to clinical research. Biplanar radiography is a relevant modality for that purpose with already two existing assessment methods. However, they are either time-consuming or lack accuracy. The aim of this study was to propose a new, more user-friendly and accurate method to determine scapular shape. The proposed method relied on simplified manual inputs and an upgraded version of the first 3D estimate based on statistical inferences and Moving-Least Square (MLS) deformation of a template. Then, manual adjustments, with real-time MLS algorithm and contour matching adjustments with an adapted minimal path method, were added to improve the match between the projected 3D model and the radiographic contours. The accuracy and reproducibility of the method were assessed (with 6 and 12 subjects, respectively). The shape accuracy was in average under 2mm (1.3mm in the glenoid region). The reproducibility study on the clinical parameters found intra-observer 95% confidence intervals under 3mm or 3° for all parameters, except for glenoid inclination and Critical Shoulder Angle, ranging between 3° and 6°. This method is a first step towards an accurate reconstruction of the scapula to assess clinical parameters in a functional posture. This can already be used in clinical research on non-pathologic bones to investigate the scapulothoracic joint in functional position.

Construction of G2 continuous curve on point-cloud surface

PurposeThe curve construction on surfaces is becoming more and more important in computer-aided design (CAD), computer graphics (CG) and the other related fields. This problem is often encountered in NC machining, tool path generation, automated fiber placement and so on. However, designing curves on curved surfaces is quite different from constructing a curve in Euclidean space. Therefore, the traditional methods of curve design are not suitable for constructing a continuous curve on surface. The authors need to perform interpolation directly on surface so that the final target curve is embedded into the given surface and also meets the continuous conditions.Design/methodology/approachFirstly, adopting a series of Hermite blending functions, the authors design a space curve passing the given knots on the point-cloud surface. Then, the authors construct a class of directrixes that are adopted to determine vector fields for projection. Finally, a complete G2 continuous curve embedded in point-cloud surfaces is constructed by solving the first-order ordinary differential equations (ODEs).Findings The authors’ main contribution is to overcome the problem of constructing G1 and G2 continuous curves on point-cloud surfaces and the authors’ schemes are based on the projection moving least square (MLS) surfaces and traditional differential geometric.Originality/valueBased on the framework of projection MLS surfaces, a novel method to overcome the problem of constructing G2 continuous curves on point-cloud surfaces is proposed.

A comparison of linear consistent correction methods for first-order SPH derivatives

A well-known issue with the widely used Smoothed Particle Hydrodynamics (SPH) method is the neighborhood deficiency. Near the surface, the SPH interpolant fails to accurately capture the underlying fields due to a lack of neighboring particles. These errors may introduce ghost forces or other visual artifacts into the simulation. In this work we investigate three different popular methods to correct the first-order spatial derivative SPH operators up to linear accuracy, namely the Kernel Gradient Correction (KGC), Moving Least Squares (MLS) and Reproducing Kernel Particle Method (RKPM). We provide a thorough, theoretical comparison in which we remark strong resemblance between the aforementioned methods. We support this by an analysis using synthetic test scenarios. Additionally, we apply the correction methods in simulations with boundary handling, viscosity, surface tension, vorticity and elastic solids to showcase the reduction or elimination of common numerical artifacts like ghost forces. Lastly, we show that incorporating the correction algorithms in a state-of-the-art SPH solver only incurs a negligible reduction in computational performance.