Moving Least Squares Research Articles

Proportional topology optimisation with maximum entropy-based meshless method for minimum compliance and stress constrained problems

In this paper, proportional topology optimisation (PTO) with maximum entropy (maxent)-based meshless method is presented for two-dimensional linear elastic structures for both minimum compliance (PTOc) and stress constraint (PTOs) problems. The computation of maxent basis functions is efficient as compared to the standard moving least square (MLS) and possesses a weak Kronecker delta property leading to straightforward imposition of Dirichlet boundary conditions. The PTO is a simple, non-gradient, accurate, and efficient method compared to the standard topology optimisation methods. A detailed and efficient implementation of the computational algorithms for both PTOc and PTOs is presented. The maxent basis functions are calculated only once at the start of simulation and used in each optimisation iteration. Young’s modulus for each background cells is calculated using the modified solid isotropic material with penalisation (SIMP) method. A parametric study is also conducted on the degree of proportionality and history dependence of both PTOc and PTOs algorithms. A variety of numerical examples with simple and complex geometries, and structured and unstructured discretisations are presented to show the accuracy, efficiency, and robustness of the developed computational algorithms. Both PTOc and PTOs algorithms can handle large topological changes, and provide excellent optimisation convergence characteristics.

A space-time generalized finite difference method for solving unsteady double-diffusive natural convection in fluid-saturated porous media

In this paper, the space-time generalized finite difference scheme is proposed to effectively solve the unsteady double-diffusive natural convection problem in the fluid-saturated porous media. In such a case, it is mathematically described by nonlinear time-dependent partial differential equations based on Darcy's law. In this work, the space-time approach is applied using a combination of the generalized finite difference, Newton-Raphson, and time-marching methods. In the space-time generalized finite difference scheme, the spatial and temporal derivatives can be performed using the technique for spatial discretization. Thus, the stability of the proposed numerical scheme is determined by the generalized finite difference method. Due to the property of this numerical method, which is based on the Taylor series expansion and the moving-least square method, the resultant matrix system is a sparse matrix. Then, the Newton-Raphson method is used to solve the nonlinear system efficiently. Furthermore, the time-marching method is utilized to proceed along the time axis after a numerical process in one space-time domain. By using this method, the proposed numerical scheme can efficiently simulate the problems which have an unpredictable end time. In this study, three benchmark examples are tested to verify the capability of the proposed meshless scheme.

Analysis of a superconvergent recursive moving least squares approximation

The recursive moving least squares (MLS) approximation is a superconvergent technique for constructing shape functions in meshless methods. Computational formulas, properties and theoretical error of the recursive MLS approximation are analyzed in this paper. Theoretical results reveal that high order derivatives of the approximation have the same convergence order as the first order derivative. Numerical results confirm the superconvergence of the recursive MLS approximation.

A discrete-continuum mosaic model for the buckling of inner tubes of double-walled carbon nanotubes under compression

In this paper, the discrete-continuum mosaic (DCM) method is applied for predicting buckling of inner tubes of double-walled carbon nanotubes (DWNTs). The moving least squares (MLS) approximation is used for linking discrete atomic model and the corresponding continuum counterpart. Actual atomic positions are used as interaction points for the interlayer van der Waals (vdW) force of DWNTs, which can accurately capture the discreteness of non-bonded interactions, and a DCM contact model is constructed. A variationally consistent meshless computational scheme based on the DCM contact model is developed for simulating the buckling of inner tubes of DWNTs upon compression. Results show that both the lengths and radii of outer tubes can influence the buckling patterns of the corresponding inner tubes. The inner tubes of DWNTs with the interlayer spacing of 0.2034 nm have strongest buckling resistance for the case that the outer tubes are not extremely short. The local critical axial compression ratio of the inner tube increases with the lengths of the inner and outer tube increasing given that the interlayer spacing of a DWNT is less than 0.34 nm.

High-order finite volume method for linear elasticity on unstructured meshes

This paper presents a high-order finite volume method for solving linear elasticity problems on two-dimensional unstructured meshes. The method is designed to increase the effectiveness of finite volume methods in solving structural problems affected by shear locking.The particular feature of the proposed method is the use of Moving Least Squares (MLS) and Local Regression Estimators (LRE). Unlike other approaches proposed before, these interpolation schemes lead to a natural and simple extension of the classical finite volume method to arbitrary order. The unknowns of the problem are still the nodal values of the displacement which are obtained implicitly in a direct solution strategy.Some canonical tests are performed to demonstrate the accuracy of the method. An analytical example is considered to evaluate the sensitivity of the solution concerning the parameters of the algorithm. A thin curved beam and a crack problem are considered to show that the method can deal with the shear locking effect, stress concentrations, and geometries where unstructured meshes are required. An overall better behavior of the LRE is observed. A comparison between low and high-order schemes is presented, and a set of parameters for the interpolation method is found, delivering good results for the proposed cases.

An Improved Supervoxel Clustering Algorithm of 3D Point Clouds for the Localization of Industrial Robots

Supervoxels have a widespread application of instance segmentation on account of the merit of providing a highly approximate representation with fewer data. However, low accuracy, mainly caused by point cloud adhesion in the localization of industrial robots, is a crucial issue. An improved bottom-up clustering method based on supervoxels was proposed for better accuracy. Firstly, point cloud data were preprocessed to eliminate the noise points and background. Then, improved supervoxel over-segmentation with moving least squares (MLS) surface fitting was employed to segment the point clouds of workpieces into supervoxel clusters. Every supervoxel cluster can be refined by MLS surface fitting, which reduces the occurrence that over-segmentation divides the point clouds of two objects into a patch. Additionally, an adaptive merging algorithm based on fusion features and convexity judgment was proposed to accomplish the clustering of the individual workpiece. An experimental platform was set up to verify the proposed method. The experimental results showed that the recognition accuracy and the recognition rate in three different kinds of workpieces were all over 0.980 and 0.935, respectively. Combined with the sample consensus initial alignment (SAC-IA) coarse registration and iterative closest point (ICP) fine registration, the coarse-to-fine strategy was adopted to obtain the location of the segmented workpieces in the experiments. The experimental results demonstrate that the proposed clustering algorithm can accomplish the localization of industrial robots with higher accuracy and lower registration time.

Mesh‐free model for Hopf's bifurcation points in incompressible fluid flows problems

AbstractIn this article, a new numerical model is proposed to detect numerically the Hopf bifurcation point of incompressible fluid flows problems. The concept of this model consists to propose a Hopf bifurcation indicator which is solved by a high order mesh free algorithm (HO‐MFA) using the moving least squares (MLS) approximation. This numerical model is based on a strong formulation of Navier–Stokes equations. This procedure allowed us to recover the results of the literature in using only a modelization for low Reynolds number. This new model is tested on the classical flow around a cylindrical obstacle and the backward‐facing step flow to show the advantage and ability of the proposed model to detect the bifurcation points and to determinate the flow periodicity.

The space–time generalized finite difference scheme for solving the nonlinear equal-width equation in the long-time simulation

In this research, the space–time (ST) coupled generalized finite difference method (GFDM) was extended to cooperate with the Newton–Raphson and time-marching methods for stably solving the nonlinear equal-width equation in long-time simulation. The equal-width equation is a time-dependent nonlinear partial differential equation and has a third-order derivation that is mixed space and time. In such a case, the action of fluid flow is over a long period, and the numerical scheme needs to rely on the state-steady condition or set up a specific time to stop the simulation. In the ST-GFDM, the temporal and spatial discretizations can be performed using the GFDM in an ST-domain. That makes the proposed scheme can easily deal with the mixed derivative. By applying the GFDM, which is advanced from the Taylor series expansion and the moving-least square method, the process of the numerical discretizations is only related to nearby nodes on the central node. Thus, the Jacobian matrix is sparse and can be solved by the Newton–Raphson method efficiently. Furthermore, the time-marching method is used to proceed with an ST-domain along the time axis. In this paper, two numerical examples are solved to verify the capability of the proposed ST-GFDM.

Intelligent moving extremum weighted surrogate modeling framework for dynamic reliability estimation of complex structures

To improve the dynamic reliability analyses of complex structures, intelligent weighted Kriging-based moving extremum framework (IWKMEF) is developed by absorbing moving least squares (MLS) thought, Gaussian weight, particle swarm optimization (PSO) method and Kriging model into extremum response surface method (ERSM). ERSM method is employed to convert the dynamic output response into extremum values. MLS thought is used to find effective samples. Gaussian weight is to improve modeling precision. PSO method is applied to optimize the local compact support region radius of MLS. The radial deformation of turbine blisk is conducted to verify the effectiveness of IWKMEF method. The results show that the reliability degree of turbine blisk is 0.9984 when the allowable value is 1.9217 × 10−3 m; The developed IWKMEF holds high performance by comparing direct simulation, ERSM and traditional Kriging model. The efforts of this study provide a useful insight for the dynamic reliability analysis of complex structure.

A local gradient smoothing method for solving the free vibration model of functionally graded coupled structures

In this paper, a meshless method for analyzing the free vibration of four-parameter functionally graded composite shells is presented, which is based on moving least square (MLS) approximation and local gradient smoothing method (LGSM). Firstly, the displacement field model of the substructure is established by using the first order shear deformation theory (FSDT), and the governing equation of the substructure is obtained by using Hamilton principle. Starting from the MLS approximation the displacement field of substructure, the derivative of the shape function is approximated by the LGSM, and the vibration analysis models of each substructure are established respectively. Then, the vibration model of the whole coupling structure is established by coupling each substructure with the displacement coordination relation of the substructure, and the vibration model of the whole coupling structure is established. For the vibration problems considering different boundary conditions, the results obtained from the meshless vibration model are compared with the FEM results to verify the accuracy and applicability of the established analysis model. At the same time, in order to improve the accuracy of the solution, the convergence analysis is also carried out. Finally, in order to enrich the research content, the influence of geometric parameters and material parameters of the structure on the vibration characteristics of the whole structure is studied. The results show that this method can effectively analyze the free vibration of the coupled structure.

An efficient meshfree computational approach to the analyze of thermoelastic waves of functionally graded materials in a two-dimensional space

In this paper, a meshless local radial point interpolation method (RPIM) is proposed for the thermoelastic analysis of functionally graded materials (FGMs) under thermal shocks. The properties of such materials are temperature-dependent varying and change gradually through thickness based on the Mori–Tanaka scheme and the volume fraction power law distribution (Appendix A). The characterization and visualization of the FGMs under a prescribed temperature gradient or heat flux are then detailed. The proposed method does not need any background cell-based integration and is thus very suitable to deal with complex geometries and scattered data. Moreover, the shape functions of this method possess the properties of the Kronecker delta, wherein the essential boundary conditions can be analytically satisfied. The numerical results are fully discussed and validated by the corresponding outcomes of the moving least squares (MLS) method, demonstrating the high capability and efficiency of RPIM proposed here in simulating the dynamic thermoelastic analysis of FGMs.

An arbitrary Lagrangian-Eulerian SPH-MLS method for the computation of compressible viscous flows

In this work we present a high-accurate discretization to solve the compressible Navier-Stokes equations using an Arbitrary Lagrangian-Eulerian meshless method (SPH-MLS), which can be seen as a general formulation that includes some well-known meshfree methods as a particular case. The formulation is based on the use of Moving Least Squares (MLS) approximants as weight functions on a Galerkin formulation and to accurate discretize the convective and viscous fluxes. This formulation also verifies the discrete partition of unity and reproduces the zero-gradient condition for constant functions. Convective fluxes are discretized using Riemann solvers. In order to obtain high accuracy MLS is also used for the high-order reconstruction of the Riemann states.The accuracy and performance of the proposed method is demonstrated by solving different steady and unsteady benchmark problems.

Impact and contact problems of explosives by ‘Mixed’ meshless local Petrov-Galerkin finite volume method

The ‘Mixed’ Meshless Local Petrov-Galerkin Finite Volume method (MLPG5) is developed for solving low-velocity impact problems of plastic bonded explosives (PBX). Due to the ductility and plasticity of PBX, there is large deformation instead of pulverization or fragmentation. In this paper, the impact dynamics equations are solved by the meshless approach (MLPG5), both stress as well as displacements are interpolated, through the moving least squares (MLS) interpolation schemes, which eliminates the expensive process of domain integrals and differentiating the shape function. The collocation method is applied to enforce strain-displacement relationships only at each nodal point, instead of the subdomain. In the present mixed method, the independent interpolation of stress improves accuracies of both the plastic stress and contact stress. The contact force is obtained by penalty function method via iteration scheme. Due to the non-linear constitutive relation, the incremental stress is obtained via radial return schemes through Newton Raphson iteration. Finally, several numerical examples are given to demonstrate the feasibility and the accuracy of the present numerical approach compared with the finite element method. Numerical examples also demonstrate the advantages of the present methods: (i) smaller support sizes can be used; (ii) higher accuracies of stress are obtained.

Generalized finite difference method for solving the bending problem of variable thickness thin plate

In this work, a meshless generalized finite difference method (GFDM), based on the Taylor series expansion and weighted moving-least square approximation, is proposed to solve the bending problem of variable thickness thin plate under various combinations of boundary conditions. The proposed method can treat the plates of complex shapes with an arbitrary continuous varying thickness in a simple and modular way. Besides, the method proposed a new strategy to create a well-determined linear system by introducing supplementary nodes on the boundary. Numerical examples are provided to demonstrate the accuracy and stability of the proposed method.

Surface reconstruction method for measurement data with outlier detection by using improved RANSAC and correction parameter

The moving least squares (MLS) and moving total least squares (MTLS) are two of the most popular methods used for reconstructing measurement data, on account of their good local approximation accuracy. However, their reconstruction accuracy and robustness will be greatly reduced when there are outliers in measurement data. This article proposes an improved MTLS method (IMTLS), which introduces an improved random sample consensus (RANSAC) algorithm and a correction parameter in the support domain, to deal with the outliers and random errors. Based on the nodes within the support domain, firstly the improved RANSAC is used to generate a model for establishing the group of pre-interpolation and calculating the residual of each node. Subsequently, the abnormal degree of the node with the largest residual is evaluated by the correction parameter associated with the node residual and random errors. The node with certain abnormal degree will be eliminated and the remaining nodes are used to obtain the approximation coefficients. The correction parameter can be used for data reconstruction without insufficient or excessive elimination. The results of numerical simulation and measurement experiment show that the reconstruction accuracy and robustness of the IMTLS method is superior to the MLS and MTLS method.

Numerical solution of an inverse source problem for a time-fractional PDE via direct meshless local Petrov–Galerkin method

In this paper, we employ an improved Meshless Local Petrov–Galerkin (MLPG), namely Direct Meshless Local Petrov–Galerkin (DMLPG) method, for solving an inverse time-dependent source problem for two-dimensional fractal mobile/immobile solute transport equation on regular and irregular domains. In the weak form DMLPG method, numerical integrations apply over low-degree polynomial basis functions instead of complicated moving least squares (MLS) shape functions of MLPG method. Therefore, the computational costs often reduce in the DMLPG method. In space domain, we employ the DMLPG method and the time derivatives of problem are discretized based on finite difference schemes. Numerical results demonstrate efficiency and accuracy of the proposed algorithm for solving the inverse source problem.

Comparison of meshfree displacement and stress error recovery of finite element solutions using moving least squares interpolation

The moving least square (MLS) interpolation based the recovery procedures have been successfully applied to recover the finite element solution errors in the analysis of elastic plates and pipes problems and can be advantageously applied for large deformation and fracture problems. The study presents the displacement and stress error recovery characteristics in the error estimation analysis employing Moving Least Squares interpolation approach. The study considers quartic spline, cubic spline, and exponential weight function with three different order of basis function in Moving Least Squares interpolation based error recovery analysis. The displacement/stress errors in finite element solution are quantified in energy norm. The cylinder and plate benchmark examples using triangular and quadrilateral elements are analyzed to compare the convergence, effectivity and adaptively improved meshes obtained using the various displacement/stress recovery procedures. The study shows that cubic spline weight function and quadratic basis function found to perform better in MLS based meshfree recovery technique for stress as well as displacement errors recovery of finite element solution. It is observed from the study that increasing the order of basis function will enhance the error estimation quality that is, rate of convergence become faster with improved effectivity of the results. The increase in convergence rate with the increase of the order of basis function is more in displacement recovery technique as compared to stress recovery technique. It is observed in the analysis of benchmark example with linear triangular meshing that the error reduction using meshfree MLS interpolation based displacement and stress recovery is about 10% and 150% respectively for the displacement and stress recovery over the mesh dependent least square based displacement and stress recovery. The study concludes that the effectiveness and efficiency of meshfree displacement/stress error recovery technique strongly depends on the weight and basis functions of MLS method to recover the errors.

SARS-CoV-2 rate of spread in and across tissue, groundwater and soil: A meshless algorithm for the fractional diffusion equation

The epidemiological aspects of the viral dynamic of the SARS-CoV-2 have become increasingly crucial due to major questions and uncertainties around the unaddressed issues of how corpse burial or the disposal of contaminated waste impacts nearby soil and groundwater. Here, a theoretical framework base on a meshless algorithm using the moving least squares (MLS) shape functions is adopted for solving the time-fractional model of the viral diffusion in and across three different environments including water, tissue, and soil. Our computations predict that by considering the α (order of fractional derivative) best fit to experimental data, the virus has a traveling distance of 1mm in water after 22, regardless of the source of contamination (e.g., from tissue or soil). The outcomes and extrapolations of our study are fundamental for providing valuable benchmarks for future experimentation on this topic and ultimately for the accurate description of viral spread across different environments. In addition to COVID-19 relief efforts, our methodology can be adapted for a wide range of applications such as studying virus ecology and genomic reservoirs in freshwater and marine environments.

Enhanced moving least square method for the solution of volterra integro-differential equation: an interpolating polynomial

This paper presents an enhanced moving least square method for the solution of volterra integro-differential equation: an interpolating polynomial. It is a numerical scheme that utilizes a modified shape function of the conventional Moving Least Square (MLS) method to solve fourth order Integro-differential equations. Smooth orthogonal polynomials have been constructed and used as the basis functions. A robust and unrestricted trigonometric weight function, along with the basis function, drives the shape function and facilitates the convergence of the scheme. The choice of the support size and some controlling parameters ensures the existence of the moment matrix inverse and the MLS solution. Valid explanation and illustration were made for the existence of the inverse linear operator. To overcome problems of near-singularity, the singular value decomposition rule is used to compute the inverse of the moment matrix. Gauss quadrature rule is used to compute the integral at the initial test points when the exact solution is unknown. Some tested problems were solved to show the applicability of the method. The results obtained compare favourable with the exact solutions. Finally, a highly significant interpolating polynomial is obtained and used to reproduce the solutions over the entire problem domain. The negligible magnitude of the error at each evaluation knot demonstrates the reliability and effectiveness of this scheme.

Generalized finite difference method for three-dimensional eigenproblems of Helmholtz equation

In this paper, a meshless numerical procedure, based on the generalized finite difference method (GFDM) is proposed to efficiently and accurately solve the three-dimensional eigenproblems of the Helmholtz equation. The eigenvalues and eigenvectors are very important to various engineering applications in three-dimensional acoustics, optics and electromagnetics, so it is essential to develop an efficient numerical model to analyze the three-dimensional eigenproblems in irregular domains. In the GFDM, the Taylor series and the moving-least squares method are used to derive the expressions at every node. By enforcing the satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes, the resultant system of linear algebraic equations can be expressed as the eigenproblems of matrix and then the eigenvalues and eigenvectors can be efficiently acquired. In this paper, four numerical examples are provided to validate the accuracy and simplicity of the proposed numerical scheme. Furthermore, the numerical results are compared with analytical solutions and other numerical results to verify the merits of the proposed method.