In this study, an innovative strategy integrating the moving least squares (MLS) method with the Genocchi-collocation technique is advanced to approximate the solution of fractal-fractional integro-differential equations. An essential advantage of the proposed technique is that it does not apply meshing and does not depend on the geometry of the computational domain, hence, this method can be considered as a meshless method. Also, accurate results can be achieved with a small number of points and basis functions, thereby significantly reducing computational complexity. By employing the MLS method, Genocchi polynomials, the Gauss-Legendre quadrature rule, and the collocation method, the problem under investigation is transformed into a system of algebraic equations. The convergence analysis of the obtained approximation is established by proving theorems. Several illustrative examples are provided to demonstrate the applicability and efficacy of the proposed strategy.

In this paper, meshfree collocation with fast-moving least-squares reproducing kernel was implemented to write down and execute numerical solutions for some applications in elastostatics and elastodynamics. The spatial discretization using meshfree collocation method was carried out on the equilibrium differential equations of elastostatics and elastodynamics and the corresponding boundary conditions. The resulting discrete forms were solved for benchmark problems in the one- and two-dimensional cases. In each case, a convergence study was conducted to ascertain the utility and efficacy of the developed solutions. For elastodynamics, the time domain, however, was discretized using the Newmark beta time-integration scheme. The latter combination was implemented to solve suitable benchmark problems in the one-dimensional and two-dimensional cases. In each case, a stability study was conducted to demonstrate, again, the method’s efficacy in handling elastodynamic problems.

Abstract We propose LeafFit, a pipeline that converts 3D Gaussian Splatting (3DGS) of individual plants into editable, instanced mesh assets. While 3DGS faithfully captures complex foliage, its high memory footprint and lack of mesh topology make it incompatible with traditional game production workflows. We address this by leveraging the repetition of leaf shapes; our method segments leaves from the unstructured 3DGS, with optional user interaction included as a fallback. A representative leaf group is selected and converted into a thin, sharp mesh to serve as a template; this template is then fitted to all other leaves via differentiable Moving Least Squares (MLS) deformation. At runtime, the deformation is evaluated efficiently on‐the‐fly using a vertex shader to minimize storage requirements. Experiments demonstrate that LeafFit achieves higher segmentation quality and deformation accuracy than recent baselines while significantly reducing data size and enabling parameter‐level editing. Our source code is publicly available at https://github.com/netbeifeng/leaf_fit .

This study presents and evaluates four distinct case studies concerning the transient (unsteady) heat conduction equation. The Meshless Local Petrov-Galerkin (MLPG) method is employed, and a direct comparison is performed between the distinct approximation functions: the Moving Least-Squares (MLS), the Radial Point Interpolation Method (RPIM) and the Interpolating Moving Least-Squares(IMLS). The first numerical example investigates a more complex two-dimensional geometry, specifically a quarter of a circular plate, to which a Dirichlet boundary condition is applied. The second example involves a square domain, utilizing both a Neumann boundary condition and a Dirichlet boundary condition that varies over time, thus constituting a time-dependent problem. The third example analyzes a plate with a circular arc at its bottom edge. The four example assess a three dimensional case and the response of the MLPG Method. Higher order MLS and IMLS bases are evaluated, such as quadratic and cubic bases. A regular and irregular node distribution for each study case investigates how this parameter affect the solution and convergence.The results derived from the MLPG implementation are compared against a reference solution obtained from the commercial software ABAQUS, which utilizes the consolidated Finite Element Method (FEM). The primary objective is to assess the performance and efficacy of the MLPG method when applied to the transient heat conduction equation.

This paper focuses on the challenging problem of detecting the surface delamination damage during the drilling of carbon fiber reinforced polymer (CFRP). Considering the issues such as the low efficiency of traditional manual inspection, the limitations of non-destructive testing technologies, and the deficiencies of machine vision inspection algorithms, the point cloud processing technology is innovatively introduced. By analyzing the delamination damage mechanism and evaluation criteria, a drilling experiment is designed, and the point cloud data are collected. Voxel grid filtering and statistical filtering are applied to preprocess the point cloud. The moving least squares (MLS) method is adopted to smooth the point cloud. Based on the region growing algorithm and combined with the dual constraints of curvature and normal vector, the precise segmentation of the point cloud is realized. The experimental results demonstrate that the algorithm proposed in this paper achieves an accuracy rate of 91% for the detection of the surface delamination area with a repeatability accuracy of 1 mm 2 , and an accuracy rate of 93% for the depth detection with a repeatability accuracy of 40 μm. This provides a novel solution for the detection of surface delamination damage during the drilling of CFRP.

A hybrid numerical method for multi-domain wave propagation problems

Abstract Accurate simulation of acoustic-wave propagation is important in many engineering and biomedical applications, such as nondestructive testing, noise control, and focused ultrasound therapies. Although the finite-element method (FEM) is commonly used for these analyses, it often suffers from numerical dispersion—especially when the wavelength is small—which can reduce the reliability of the results. Meshfree methods offer an alternative by using scattered nodes instead of fixed element meshes; this flexibility helps reduce dispersion and eliminates the need for complex mesh generation. This study introduces the Moving Kriging Interpolation Method (MKIM), which is based on the Moving Least-Squares (MLS) approach. The combination yields a continuously differentiable and more stable solution space. The problem is formulated for a two-dimensional chamber with Dirichlet boundary conditions and for a two-dimensional submarine-like shape with Neumann boundary conditions. The method is implemented in MATLAB, employing Gauss points for domain integration to solve the weak form of the Helmholtz wave equation. To evaluate its performance, the numerical results are validated against both closed-form analytical solutions and the other numerical schemes discussed above. The outcomes demonstrate higher accuracy and better control of numerical dispersion than traditional methods. These findings highlight the potential of meshfree techniques for tackling increasingly complex acoustic problems in the future.

Immersed boundary methods (IBMs) provide a convenient and efficient approach for fluid-structure interaction (FSI) problems, as they allow for flexible handling of moving boundaries without conforming mesh requirements. However, computing the hydrodynamic loads exerted by the fluid on the immersed body is a well-known issue, as fluid grid points are not directly available on the wet surface. In the literature, several procedures have been proposed, but they typically yield non-smooth stress distributions or have low accuracy. On the other hand, higher-order methods can only be applied to rigid bodies in the limit of boundary layer flow. In this paper, we propose a novel procedure capable of accurately evaluating the stresses acting on solid surfaces, which is based on a truncated Taylor series of the whole fluid stress tensor, without relying on any assumption on the underlying flow or body dynamics. The terms of the Taylor series are evaluated directly from the flow solution using a versatile moving-least-squares (MLS) interpolation with a small computational overhead. We also propose a variation of the procedure for computing hydrodynamic loads within thin lubrication layers, which can arise during the interaction of multiple deformable bodies mediated by a fluid. The method is validated in a series of numerical experiments encompassing analytical flow solutions, separated flows over rigid bodies, and FSI over rigid and deformable bodies. Although the method is here applied to a second-order IBM code, it can be applied to any-order fluid solver, including body-fitted ones.

ABSTRACT This paper presents a time spectral generalized finite difference method (TS‐GFDM) for three‐dimensional (3D) transient heat conduction in functionally graded materials (FGMs) with space–time dependent coefficients. The time derivative of temperature in the governing equation is approximated as a linear combination of temperatures at Gaussian points within each time step, achieved via the inverse transform of spectral integration. Space derivatives of temperature are evaluated as linear combinations of nodal temperatures, constructed using Taylor series expansion in conjunction with the moving least squares (MLS) approximation. The proposed method allows for large time steps in the temporal direction while ensuring stability over long‐time simulations. In the spatial domain, it eliminates the need for mesh generation, making it particularly well suited for heat conduction analysis in complex structures. The numerical results obtained using the TS‐GFDM are compared with those from existing methods and the analytical solution, demonstrating the higher computational efficiency of the proposed approach.

ABSTRACT In this study, we present a novel mesh‐free approach for solving incompressible fluid flow problems, which is introduced here for the first time. Our approach solves the steady‐state Navier–Stokes equations without requiring traditional mesh generation. For this purpose, we adopt a discrete framework in which variables are defined at specific points within the domain, thereby eliminating the need for numerical integration. The proposed approach combines a weighted least squares (WLS) approximation with a high‐order continuation method (HOCM). This approach significantly enhances the accuracy of steady‐state incompressible flow simulations, offering both improved precision and reduced computation time compared to classical methods. Our results indicate that this approach holds substantial potential for expanding practical applications across various engineering fields. In contrast to the coupling of the moving least squares (MLS) method with the HOCM, our approach avoids computing derivatives of the weight function within the influence domain, which reduces the computational cost and enhances accuracy. This original combination highlights the novelty of our work compared to research conducted in recent years. A comparison is presented between the results obtained using the HOCM with MLS approximation and those reported in the literature.

Abstract With the development of three-dimensional laser scanning technology, high-density point cloud data provides a reliable database, and there is also a large amount of redundant information, which increases the storage and calculation burden of data processing. A point cloud simplification method that maintains the integrity of the geometric structure while compressing data is urgently needed. In this paper, we propose a density-aware sampling strategy following the construction of a grid structure, and the number of local sampling points is dynamically adjusted according to local density variations to enhance the global consistency of the simplified point cloud. During the feature point sampling stage, the sampling starting point is optimized based on the farthest point sampling (FPS) algorithm, and curvature weights are incorporated into the iterative selection strategy. This approach allows more points along the sampling path to be chosen to represent geometric features, thereby ensuring that the simplified point cloud more accurately preserves the features of the original model. However, feature-based sampling tends to sparsely select points in flat regions of the point cloud surface. In this study, surface fitting is performed using the moving least squares method, and auxiliary feature points are subsequently selected via uniform sampling. This combination effectively improves the geometric uniformity of the simplified results. By integrating both sampling strategies, a balance is achieved that maintains high geometric fidelity while ensuring uniformity, thereby supporting subsequent point cloud processing tasks with higher accuracy. The experimental results show that compared with existing methods such as AIVS, GF-Sim, FPS, and curvature-based, the proposed method has better fidelity and robustness in terms of running time, information entropy, geometric error spacing and error, and reconstruction quality on multiple public point cloud data sets.

Numerical simulation of ternary flow is still a challenging problem due to the presence of complex interfacial dynamics and irregular geometric domains. Conventional lattice Boltzmann methods (LBM) are limited by their dependence on structured grids and the strict coupling between spatial and temporal discretizations, which reduces their flexibility in handling such complex systems. This paper presents a novel generalized semi-Lagrangian meshfree lattice Boltzmann method (SL-M-LBM) for ternary fluid flows by using a two-component phase-field model. Our approach fundamentally decouples the discretization by integrating a semi-Lagrangian streaming algorithm with moving least squares (MLS) reconstruction. This enables entirely mesh-free simulations with flexible, non-uniform node distributions and independent control of time stepping, thereby enhancing numerical stability and allowing for local refinement in complex domains. To solve the hydrodynamic equations for incompressible flows and numerically capture interface evolution, our approach creates a Lattice Boltzmann (LB) model with two LB equations (LBEs) for phase-field evolution and a single LB equation (LBE) for hydrodynamics. A wide range of benchmark tests are conducted to evaluate the model, such as Zalesak's rotating disk, diagonal translation, two circular interfaces deformation under shear flow, binary and ternary Rayleigh–Taylor instability, compound droplet passing through a capillary throat under pressure-driven, interaction between a droplet and a rising bubble under gravity, and finally liquid lens spreading. The simulation results indicate that the SL-M-LBM performs effectively for many flow conditions with accurate interface tracking and good mass conservation ability even with non-uniform node distributions. Furthermore, the model can handle irregular geometric domains, high density ratios, and highlights its potential for versatile and efficient simulation of complex multiphase systems containing three immiscible fluids.

Abstract Point cloud data acquired from complex surface measurements often contain noise and incompleteness due to equipment and registration errors. To address these challenges, this study proposes a mathematical model-driven inverse precision measurement and error evaluation method aimed at improving the accuracy and robustness of surface reconstruction. The approach enhances normal vector consistency through anisotropic normal vector smoothing, removes small-scale noise using bilateral filtering while preserving feature edges, and generates high-quality manifold point clouds using moving least squares (MLS) smoothing. Surface reconstruction is then performed using greedy projection triangulation with Delaunay triangulation to ensure mesh regularity. Experimental results show that the proposed method achieves RMSE values of 0.025 mm, 0.029 mm, and 0.022 mm on the Bunny, Dragon, and Blade models, respectively, outperforming Poisson reconstruction (0.049 mm, 0.057 mm, and 0.053 mm). In noise robustness testing, the method achieves an RMSE of 0.049 mm at 0.1 mm noise level, surpassing comparable techniques. Ablation analysis reveals that removing MLS smoothing increases the reconstruction error by 72 %, confirming its significance. The method also achieved an engineer subjective score of 90.5 points, and scalability experiments on the Lucy model maintained high accuracy (RMSE 0.062 mm). Overall, the proposed method provides a highly accurate, robust, and practical mathematical model-driven solution for reverse engineering of complex surfaces, significantly improving measurement reliability in real-world applications.

In this study, we combine orthonormal basis functions with the traditional moving least-squares (MLS) approximation to establish a new approximation function via the improved moving least-squares (IMLS) approximation, and the corresponding formula derivation is provided in Section 2. Afterwards, the equilibrium, geometrical, and physical equations for bending problems of plates on elastic foundations are presented respectively, and the equivalent functional of such problems is established by imposing the essential boundary conditions via the penalty method. Subsequently, the calculation formulas for the numerical solution are derived using the improved element-free Galerkin (IEFG) method based on the IMLS approximation. In the numerical examples, we verify the convergence of the IEFG method by increasing the number of nodes. Compared with the EFG method, the IEFG method exhibits faster convergence. Furthermore, by adopting the IEFG method for solving three numerical examples, smaller errors and higher computational speed are achieved.

High-dimensional integral equations appear in many fields of science and engineering, but solving them on non rectangular domains is challenging and deserves particular attention in scientific computing. A computational approach constructed using the regularized moving least squares approximation (RMLS) integrated with Gauss-Legendre quadrature rule is expanded to solve three-dimensional Fredholm integral equations on 3D irregular domain. The proposed method is meshless because they do not require any background mesh or cell structures and so they are independent of the geometry of the domain. This method is introduced to address the issue of a singular moment matrix within meshfree frameworks that utilize the moving least squares (MLS) approximation. One of the key benefits of this technique lies in its ability to achieve faster convergence to the analytical solution, even when using a relatively small support domain. It also offers enhanced adaptability by allowing easy adjustments to the distribution of nodes. The algorithm's computational cost is analyzed to assess the execution time. Additionally, the convergence behaviour of the new tool is examined. Numerical examples are provided and compared with the outcomes from the conventional MLS method to demonstrate the proposed scheme's effectiveness and precision.

In this paper, a moving-least square (MLS) approximation meshless method combined with Timoshenko beam theory is proposed to first investigate the free vibration and forced vibration of functionally graded graphene origami-enabled auxetic metamaterials (FG-GOEAMs) curved beams with variable curvature and thickness. The FG-GOEAMs curved beams are composed of multiple GOMAM layers with uniform or nonuniform functional gradient layers in the thickness direction, and the determination of material characteristics is accomplished through a synergistic approach combining micromechanical modeling enhanced by genetic programming (GP) with the rule of mixture. The meshless model of FG-GOEAMs curved beams is established by a series of discrete points, and then the meshless governing equations of free and forced vibration for the FG-GOEAMs curved beams are derived by Hamilton’s principle and the principle of minimum potential energy, respectively. The enforcement of essential boundary conditions is systematically achieved by the complete transformation method. The eigenvalue method and Newmark-[Formula: see text] time integration scheme are employed to resolve the governing equations, deriving the system’s natural frequency and time–history curve, respectively. The convergence behavior and numerical precision of the present method are rigorously validated through comparative analysis against both benchmark simulations and established theoretical data documented in the prior studies. At the end of the paper, the influence of GOri’s distribution patterns, content, folding degree, damping ratios, boundary conditions and other key parameters on the natural frequency and dynamic transient response of FG-GOEAMs curved beams with variable curvature and thickness are discussed.

The recently developed wind gust generator, the adaptive nozzle (Wood and Breuer, 2025) , features a nozzle with a fully rotatable upper contour, enabling a smooth gust generation with low unwanted flow disturbances. While preserving the underlying gust-generation principle of its predecessor, the new design significantly reduces pressure losses caused by flow blockage and preserves the original horizontal trajectory of the flow along the streamwise direction. While experiments have validated the improved design, a comprehensive numerical analysis is crucial to resolve the three-dimensional flow fields across the entire computational domain. This shall also facilitate capturing the resulting transient aerodynamic loads on a wind tunnel specimen — quantities difficult to measure experimentally. To accurately capture the complex flow dynamics, high-fidelity large-eddy simulations are conducted, modeling the nozzle’s upper contour as a dynamic immersed boundary (IB). A curvilinear Eulerian grid is employed to ensure both efficient and precise spatial resolution of the problem. The moving least-squares (MLS) version of the direct forcing IB approach (Vanella and Balaras, 2009) is used to construct an IB kernel for each Lagrangian marker. Additionally, the MLS approach is also applied to construct one-sided kernel functions for Lagrangian points near the boundaries of the computational domain. Challenges related to the efficient IB simulation on curvilinear grids are addressed, and a solution is proposed within the MLS framework. The predicted results are analyzed in detail and validated against the experimental data by (Wood and Breuer, 2025) , providing insights into the effectiveness of the new design in generating controlled wind gusts. • Dynamic immersed boundary (IB) for curvilinear Eulerian grids. • Moving least-squares (MLS)-based direct forcing IB approach. • One-sided interpolation kernels for Lagrangian points near domain boundaries. • Detailed validation against experimental measurements and numerical results. • Simulation of an artificial wind gust generator based on an adaptive nozzle.

This study investigates steady-state heat conduction in materials with stepwise discontinuities in thermal conductivity, a phenomenon frequently encountered in layered composites, thermal barrier coatings, and electronic packaging. The problem is formulated for a 2D two-domain region, where each subdomain has a distinct constant conductivity. Both the direct problem—determining the temperature field from known conductivities—and the inverse problem—identifying conductivities and the internal heat source from limited temperature measurements—are addressed. To this end, three deterministic finite-difference-type models are developed: two for the standard formulation and one for a meshless formulation based on Moving Least Squares (MLS), all derived within a local framework that efficiently enforces interface conditions. In addition, two Monte Carlo models are proposed—one for the standard and one for the meshless setting—providing pointwise estimates of the solution without requiring computation over the entire domain. Finally, an algorithm for solving inverse problems is introduced, enabling the reconstruction of material parameters and internal sources. The performance of the proposed approaches is assessed through 2D benchmark problems of varying geometric complexity, including both structured grids and irregular node clouds. The numerical experiments cover convergence studies, sensitivity of inverse reconstructions to measurement noise and input parameters, and evaluations of robustness across different conductivity contrasts. The results confirm that the hybrid difference-meshless Monte Carlo framework delivers accurate temperature predictions and reliable inverse identification, highlighting its potential for engineering applications in thermal design optimization, material characterization, and failure analysis.

This study investigates the primary mechanisms contributing to damage accumulation in additively manufactured Scalmalloy (Al-Mg-Sc) subjected to tensile tests at low and high strain rates. High-speed X-ray Phase Contrast Imaging ( XPCI ) performed at beamline ID19 of the European Synchrotron Radiation Facility ( ESRF ) provide real-time, high-resolution through-volume visualization of internal void evolution processes, including void nucleation, growth, and coalescence. This advanced experimental approach facilitates the precise calibration of established physics-based fracture models, such as the Gurson-Tvergaard-Needleman ( GTN ) model, which otherwise relies on assumptions that are very difficult to verify experimentally. The 2D images captured during in-situ testing were segmented to identify and track individual voids using advanced computational techniques. The strain fields within the material were calculated via the Moving Least Squares ( MLS ) method, enabling accurate local strain estimation in materials with complex and evolving microstructures. The results show significant strain rate effects on the void evolution in Scalmalloy. At low strain rates, the void fraction increased steadily as a result of isolated void growth. In contrast, high strain rates demonstrated complex deformation behaviors, with slow initial void growth transitioning to rapid coalescence beyond a critical strain threshold, ultimately resulting in extensive internal damage. Moreover, the analysis of the time derivative of the apparent void fraction and its relationship with the local strain rate reveals proportional damage evolution at low strain rates, indicating progressive void growth. At high strain rates, the strong linear relationships observed between the rate of change of the apparent void fraction and the local strain rate, validate the applicability of the GTN model and demonstrate its ability to predict rapid void coalescence and ductile fracture under dynamic loading conditions. • High-speed XPCI used in in-situ tensile tests at low and high strain rates. • Void tracking via Point-In-Polygon test with Euclidean Distance approach. • Local strain measured using the Moving Least Squares (MLS) technique. • Analysis of void nucleation, growth, and coalescence at both strain rates.

An improved MLS-based numerical manifold method for saturated-unsaturated seepage in porous media