Moving Least Squares Research Articles

The Improved Element-Free Galerkin Method for 3D Steady Convection-Diffusion-Reaction Problems with Variable Coefficients

In order to obtain the numerical results of 3D convection-diffusion-reaction problems with variable coefficients efficiently, we select the improved element-free Galerkin (IEFG) method instead of the traditional element-free Galerkin (EFG) method by using the improved moving least-squares (MLS) approximation to obtain the shape function. For the governing equation of 3D convection-diffusion-reaction problems, we can derive the corresponding equivalent functional; then, the essential boundary conditions are imposed by applying the penalty method; thus, the equivalent integral weak form is obtained. By introducing the IMLS approximation, we can derive the final solved linear equations of the convection-diffusion-reaction problem. In numerical examples, the scale parameter and the penalty factor of the IEFG method for such problems are discussed, the convergence is proved numerically, and the calculation efficiency of the IEFG method are verified by four numerical examples.

Numerical study of the variable-order time-fractional mobile/immobile advection-diffusion equation using direct meshless local Petrov-Galerkin methods

In this paper, we use direct meshless local Petrov-Galerkin (DMLPG) methods for solving the variable-order time-fractional mobile/immobile advection-diffusion equation in two dimensions. The basis of the DMLPG methods is on the generalized moving least-square (GMLS) approximation and local weak form of the equation. Since the GMLS approximation uses basic polynomials as shape functions, there is no need to construct complex shape functions in the moving least-square (MLS) approximation. In addition, the calculation of numerical integrals in the local weak form of the equation is simple. We also give proof of the convergence and stability of the time-discrete scheme. The analytical results show that the semi-discrete time scheme is unconditionally stable and the convergence order is O(Δt2). Moreover, to show the accuracy and efficiency of the methods, we use the regular and irregular domains with distributions of uniform and scattered nodes in numerical examples.

Investigations on a high order SPH scheme using WENO reconstruction

Theoretically, a 2nd order convergence can be reached with the Smoothed Particle Hydrodynamics (SPH) method. However, depending on the spatial disorder of particles, the order of convergence observed in practice can be lower than one. In this paper, a methodology is proposed for the reconstruction of high order numerical fluxes in Riemann-SPH formulations for weakly-compressible flows, so as to increase the global order of convergence of the scheme. This methodology is based on the use of one-dimensional Weighted Essentially Non-Oscillatory (WENO) reconstructions applied at each pair of interacting particles, and each 1D WENO stencil is completed using Moving-Least-Squares (MLS) reconstructions. It is shown that a 6th order convergence can be reached with the proposed SPH-WENO scheme. The gain in accuracy and convergence properties of this scheme is shown and discussed through its application to various one-dimensional and two-dimensional test cases. In particular, the influence of the number of neighbor particles and of the spatial particle disorder is studied. Finally, it is shown that the proposed high-order SPH scheme provides a better accuracy to CPU time ratio than usual Riemann-SPH schemes. The treatment of boundary conditions on rigid walls with the proposed scheme is also discussed.

The MLS-based numerical manifold method for Darcy flow in heterogeneous porous media

The moving least squares (MLS) interpolation gets rid of the shackles of meshes, while the numerical manifold method (NMM) can solve both continuity and discontinuity problems in a unified way. The MLS-based NMM (abbreviated as MLS-NMM) inherits the individual merits of MLS and NMM. In the present work, the MLS-NMM is extended to solve groundwater flow problems in heterogeneous porous media. To accurately simulate the flow velocity field at and in the vicinity of the material interfaces, the refraction law is introduced into the proposed model as a posteriori condition. Several examples are employed to verify the correctness and accuracy of the proposed model. The results indicate that the MLS-NMM can model the Darcy flow in heterogeneous porous media with high accuracy, as well as satisfy the refraction law rigorously. In addition, the smooth flow velocity field can be obtained directly without the need for extra post-processing.

Digital Image Correlation Measurement Based on Pointwise Moving Least-Square Fitting

Digital Image Correlation Measurement Based on Pointwise Moving Least-Square Fitting

Transition of thermally induced flow in a square cavity using an iterative predictor–corrector meshfree approach

AbstractThis paper aims to propose a new approach based on meshfree method to solve the dimensionless steady nonlinear natural convection equations. The solution methodology uses the moving least squares (MLS) shape functions to discretize the weak form of the governing equations and the Newton–Raphson algorithm to compute the nonlinear solution. Parametric studies of natural convection concerning closed two‐dimensional (2D) cavity with different operating conditions filled with air is used to assess the impact of the proposed approach on the performance of computed solutions. The cavity is formed by adiabatic and isothermal walls. For that, in all considered cases, the vertical sidewalls of the enclosure are heated differentially and maintained at hot and cold temperatures. The horizontal top and bottom walls are considered thermally insulated. The thermophysical properties of the working fluid are assumed to be constant except the density variations causing a body force term in the momentum equation. No‐slip boundary condition is applied at all enclosure boundaries and all internal body boundaries. The considered cases are described and analyzed in detail to check the validity of the proposed method. A comparison is performed between the results obtained from the present computer code and those available in the literature. The influence of various controlling parameters is analyzed and discussed graphically through the obtained streamlines, isotherms, and (maximum and average) Nusselt number. The numerical results computed by the proposed method are found to be in good agreement with the reference data.

Research on 3D curved weld seam trajectory position and orientation detection method

The "teaching-reproduction" mode and off-line programming mode are widely used in traditional welding robots. For curved welding workpieces, the operators need to visually inspect the position and orientation of the weld seam during processing to select the teaching points or the starting point for programming. The welding accuracy depends on the operators’ experience. Re-teaching or re-programming is required after adjusting the welding position or orientation, resulting in poor welding accuracy and efficiency. To enhance the adaptability of the robot to various welding conditions, we propose a 3D curved weld seam trajectory position and orientation detection method. A local welding workpiece point cloud is obtained by laser vision sensor scanning. The point cloud is smoothed by using the moving least squares (MLS) method. The surface variation of each point is calculated, and the surface variation threshold is set to extract the weld seam feature points. The weld seam orientation is calculated by solving the basic vectors of the spatial curve according to the parametric equation of the weld seam trajectory. The rotation angles of the robot are calculated by the transformation matrix between the weld seam local coordinate system and the robot base coordinate system. According to the experiments, the proposed method can adapt to various orientations of the curved welding workpiece and improve the robot's adaptability to orientations, therefore, the accuracy and efficiency of welding will be enhanced.

Numerical Solution of Shrödinger Equations Based on the Meshless Methods

In this work, two-dimensional time-dependent quantum equation problems are studied. We introduce a numerical algorithm for solving the two-dimensional nonlinear complex quantum system with MLS and FDM methods. An efficient and accurate computational algorithm based on both, the moving least squares (MLS) and the finite difference (FDM) methods is proposed for solving it. The results demonstrate that the proposed algorithm is a robust algorithm with good accuracy. This is developed on MLS and FDM methods using numerical simulation for solving these kind of problems.

An Improved Element-Free Galerkin Method Based on the Dimension Splitting Moving Least-Squares Method for 2D Potential Problems in Irregular Domains

By introducing the dimension splitting (DS) method into the moving least-squares (MLS) approximation, a dimension splitting moving least-squares (DS-MLS) method is proposed in this paper. In the DS-MLS method, the operator splitting and independent variable splitting of the DS method are used to reduce the dimension, thereby reducing the computational complexity of the matrix. The shape function of the DS-MLS method has the advantages of simple derivation and high computational efficiency. Then, by coupling DS-MLS method and Galerkin weak form, and performing the coordinate transformation, an improved element-free Galerkin method (IEFGM) based on the DS-MLS method is proposed for two-dimensional (2D) potential problems on irregular domains. The effectiveness of the method in this paper is verified by some numerical examples. The numerical results show that, compared with the element-free Galerkin (EFG) method, the IEFGM based on the DS-MLS method in this paper consumes less CPU time and has higher computational accuracy for some 2D potential problems on irregular domains.

Numerical Simulation of Rheological Models for Complex Fluids Using Hierarchical Grids.

In this work, we implement models that are able to describe complex rheological behaviour (such as shear-banding and elastoviscoplasticity) in the HiGTree/HiGFlow system, which is a recently developed Computational Fluid Dynamics (CFD) software that can simulate Newtonian, Generalised-Newtonian and viscoelastic flows using finite differences in hierarchical grids. The system uses a moving least squares (MLS) meshless interpolation technique, allowing for more complex mesh configurations while still keeping the overall order of accuracy. The selected models are the Vasquez-Cook-McKinley (VCM) model for shear-banding micellar solutions and the Saramito model for viscoelastic fluids with yield stress. Development of solvers and numerical simulations of inertial flows of these models in 2D channels and planar-contraction 4:1 are carried out in the HiGTree/HiGFlow system. Our results are compared with those predicted by two other methodologies: the OpenFOAM-based software RheoTool that uses the Finite-Volume-Method and an in-house code that uses the Vorticity-Velocity-Formulation (VVF). We found an excellent agreement between the numerical results obtained by these three different methods. A mesh convergence analysis using uniform and refined meshes is also carried out, where we show that great convergence results in tree-based grids are obtained thanks to the finite difference method and the meshless interpolation scheme used by the HiGFlow software. More importantly, we show that our methodology implemented in the HiGTreee/HiGFlow system can successfully reproduce rheological behaviour of high interest by the rheology community, such as non-monotonic flow curves of micellar solutions and plug-flow velocity profiles of yield-stress viscoelastic fluids.

Bi-iterative moving enhanced model for probability-based transient LCF life prediction of turbine blisk

The low cycle fatigue (LCF) life of aeroengine turbine blisk endures uncertain and transient loads from multi-physical fields, so that it is necessary to evaluate the LCF life for improving the safety of turbine blisk and aeroengine from a probabilistic perspective. In this paper, bi-iterative moving enhanced modeling (BIMEM) approach is developed to conduct the probability-based transient LCF prediction of turbine blisk with fluid-thermal-structural interaction, by integrating extremum thought, Kriging model, moving least squares (MLS) technique and modified particle swarm optimization (MPSO) algorithm. The extremum thought is used to simplify the processes of transient responses of turbine blisk into extremum values in time domain. Both MLS technique and Kriging model are employed to establish the relationship functions between output response and input parameters, for deriving transient reliability sensitivity analysis of turbine blisk LCF life. The MPSO algorithm is applied to find the optimal values of hyperparameters in Kriging model with the effective samples; from the probability-based transient LCF prediction of turbine blisk, it is illustrated that turbine blisk reliability degree is 0.9986, when the allowable value of LCF life is 2968 cycles; and the primary factor affecting turbine blisk LCF life is strength coefficient, followed by gas temperature, fatigue strength exponent, and so forth; the developed BIMEM approach is better in modeling precision and simulation efficiency in the probabilistic-based LCF life prediction, relative to other methods. The efforts of this paper offer a promising way for the health evaluation and prediction of complex structures besides turbine blisk.

Adaptive Numerical Method for Approximation of Traffic Flow Equations

For a long time now, traffic equations have been considered, and different modeling has been done for it. In this article, we work on the macroscopic model, especially the most famous light model. Because these models are among the stiff and shocking problems, theoretical methods do not give good answers to these problems. This paper describes a meshless method to solve the traffic flow equation as a stiff equation. In the proposed method, we also use the exponential time differencing (ETD) method and the exponential time differencing fourth-order Runge–Kutta (ETDRK4). The purpose of this new method is to use methods of the moving least squares (MLS) method and a modified exponential time differencing fourth-order Runge–Kutta scheme. To solve these equations, we use the meshless method MLS to approximate the spatial derivatives and then use method ETDRK4 to obtain approximate solutions. In order to improve the possible instabilities of method ETDRK4, approaches have been stated. The MLS method provided good results for these equations due to its high flexibility and high accuracy and has a moving window and obtains the solution at the shock point without any false oscillations. The technique is described in detail, and a number of computational examples are presented.

Analysis of the elasto-magnetic wave propagation in a smart magnetostrictive actuator using a mesh-free method

This work aims to propose an implicit algorithm for solving the magnetoelastic wave propagation in a magnetostrictive actuator. This algorithm is built by combining meshless methods with the Newmark implicit time scheme. The mathematical formulation of vibrations of the actuator is performed by modeling the experimental actuator device by a monodimensional magnetostrictive cylindrical rod attached to an elastic spring, subjected to an axial magnetic field. The formulation of the problem is carried out using the rod model. The study is carried out taking into account mechanical and magnetic effects described by a linear model of strain-stress magnetoelastic relations, whose coefficients depend on the square of the magnetic field, elastic, magnetostrictive, and magnetoelastic characteristics of the elastomagnetic material of which the rod is made. The dynamic equation which describes the magnetoelastic wave propagation in the actuator is obtained by applying the Lagrangian formalism based on the least action principle. The wave propagation is governed by a spatio-temporal partial differential equation whose coefficients are a function of on elastic, magnetostrictive, and magnetic characteristics and implicitly depends on time. The resolution of the dynamic equation is made by adopting two numerical approaches developed by combining the mesh-free collocation methods, the Moving Least Squares (MLS), and the Radial Point Interpolation Method (RPIM) with the Newmark implicit scheme temporal integration. By performing the spatial discretization of the displacement, we obtain a system of linear second-order ordinary differential equations (ODEs), then transform it into a system of first-order ordinary differential equations after integrating Newmark’s implicit time scheme. A comparison of results computed by the proposed numerical approaches RPIM and MLS with experimental data available in the literature is presented. The stability and the solutions qualities of two meshless methods, RPIM and MLS, are also studied. The influence of magnetical and mechanical parameters as well as the coupling coefficients on the performance of the magnetostrictive actuator is also made.

The MLS-based numerical manifold method for nonlinear transient heat conduction problems in functionally graded materials

In this study, we aim to numerically solve two-dimensional nonlinear transient heat conduction problems of functionally gradient materials (FGMs), where the governing equation is a quasilinear partial differential equation of second order. For this purpose, the weak form of the initial boundary value problem is first established. Then influence domains of nodes in the moving least squares (MLS) instead of finite elements are used as the mathematical patches to construct the mathematical cover of the numerical manifold method (NMM); while the shape functions of MLS-nodes as the weight functions subordinate to the mathematical cover, leading to the moving least squares based numerical manifold method (MLS-NMM). With the weak form, the spatial discretization is carried out using MLS-NMM, whereas temporal discretization follows the Euler backward difference. Finally, a series of numerical experiments are conducted concerning transient heat conduction problems of FGMs with heat source and heat convection, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving nonlinear transient heat conduction of FGMs.

Bending and free vibration analyses of ribbed plates with a hole based on the FSDT meshless method

A meshless method based on the first-order shear deformation (FSDT) theory and moving least-squares (MLS) approximation for bending and free vibration analyses of the ribbed plate with a hole is proposed. Firstly, a ribbed plate with a hole is regarded as a composite structure of a plate (with a hole) and ribs. Nodes are used to discretize the plate and ribs. Secondly, the displacement fields of the plate and ribs are established based on the FSDT and MLS, and their potential and kinetic energy functional are derived. The diffraction law is used to treat the discontinuity related to the hole on the plate. Thirdly, the total potential and kinetic energy of the ribbed plate is obtained by superposing the potential and kinetic energy of the plate and ribs and introducing the compatibility condition. Finally, the equations governing the elastic bending and free vibration of the ribbed plate with a hole are obtained according to the principle of Minimum Potential Energy and Hamilton’s Principle. The boundary conditions are enforced by the full transformation method. Thereafter, several ribbed plate with hole examples with different holes, rib arrangements, boundary conditions and load conditions are calculated. The results are compared with those given by ABAQUS and literature. The results show that the effectiveness and accuracy of the proposed method in analyzing the static and dynamic problems of the ribbed plate with a hole.

Valorization of Oil Palm Empty Fruit Bunch for Cellulose Fibers: A Reinforcement Material in Polyvinyl Alcohol Biocomposites for Its Application as Detergent Capsules

Cellulose fibers isolated from oil palm empty fruit bunches (OPEFB) have been studied as a potential reinforcement for polyvinyl alcohol (PVA) biocomposite. Analysis of variance (ANOVA) showed that all three parameters—hydrolysis temperature, time and acid concentration, as well as their interactions—significantly affected the yield of cellulose. Moving Least Squares (MLS) and Multivariable Power Least Squares (MPLS) models demonstrated good fitness. The model also proved that acid concentration was the dominant parameter, supported by the Fourier transform infrared spectroscopy (FTIR) analysis. Hydrolysis using 54% acid at 35 °C and 15 min achieved the highest cellulose yield of 80.72%. Cellulose-reinforced PVA biocomposite films demonstrated better mechanical strength, elongation at break, moisture barrier properties, thermal stability and poorer light transmission rate compared to neat PVA due to the high aspect ratio, crystallinity and good compatibility of cellulose fibers. These findings suggested the potential of cellulose fibers-reinforced PVA biocomposite film as water-soluble detergent capsules.

Buckling analysis of corrugated-core sandwich plates using a FSDT and a meshfree Galerkin method

A meshfree Galerkin method is proposed for the buckling analysis of corrugated-core sandwich plates (CSP). A CSP is a composite structure composed of three parts — one corrugated sheet in the middle of the structure which can be approximated by an orthotropic plate that has different elastic properties along the two perpendicular directions. The displacement fields of the component are given with the moving-least squares (MLS) approximation and the first-order shear deformation theory (FSDT). The equations governing the elastic buckling behaviors of the three components are derived and superposed to present the meshfree governing equation of the entire CSP. A technique that was originally used to enforce essential boundary conditions in meshfree methods is developed to treat the governing equations before the superposition according to the displacement compatibility. The program is compiled on the C++ platform, and several numerical examples are employed to demonstrate the convergence and accuracy of the proposed method. The results show that the proposed method has high accuracy for the stability analysis of CSP under certain conditions.

Evaluation of permeability of soil & rock aggregate using meshless numerical manifold method

Soil & Rock Aggregate (SRA) is ubiquitous in nature and an excellent fill material in engineering. The presence of SRA rock blocks in number and size-difference makes it impossible to simulate each rock block separately in engineering computations. A practical way is to regard SRA as an equivalent continuum and to evaluate properties of the equivalent continuum using numerical methods. In this way, it is hopeful to overcome big errors of laboratory tests due to removing big blocks. Randomness of SRA rock blocks in shape and orientation renders clumsy mesh dependent methods such as finite element methods. By using finite elements to form mathematical patches, the numerical manifold method (NMM), called FE-NMM, reduces to some degree mesh dependency, but still involves a huge number of operations in cover-cutting and information-rebuilding while generating physical covers. Specifying node influence domains in the moving least squares (MLS) as mathematical patches, and MLS shape functions as the weight functions of the physical patches, this study develops the procedure MLS-NMM to evaluate SRA permeability, completely avoiding extra labor in FE-NMM. Many complicated deployments of rock blocks in SRA are designed and analyzed, suggesting significant advantages of MLS-NMM over FE-NMM. Meanwhile, some interesting results are observed.

Terrain matching positioning method for underwater vehicles based on curvature discrimination

Terrain matching positioning technology is used to locate underwater vehicles by the terrain information. The traditional method uses all real-time bathymetric points to traverse the underwater prior map, which has a large calculation amount and high complexity. To improve computational efficiency while ensuring accuracy, we represent the terrain matching positioning method based on the local terrain features and the global search technology in this paper. Firstly, the Moving Least Square (MLS) method is used to construct the MLS surface of the multi-beam bathymetric terrain. Then, a shape operator related to the principal curvatures is employed to reflect the curvedness of the MLS surface at each bathymetric point. Next, a Local Curvature Binary Discrimination (LCBD) method is proposed to describe the shape of local MLS surface with a LCBD descriptor, and divide the points satisfying the LCBD situations into the sampling point set. Finally, the Genetic Algorithm (GA) is introduced to replace the traditional traversal method (TM) to search the optimal estimated position of vehicle from the underwater prior map. The experimental results show that the proposed method can quickly locate the vehicle without affecting the accuracy, and the efficiency is improved by more than 80% compared with the traditional method.

A continuum-discrete multiscale coupling method for pristine and defected single-walled carbon nanotubes

A continuum-discrete multiscale coupling (CDMC) method is developed for predicting the large deformation and buckling behaviors of pristine and defected single-walled carbon nanotubes (SWCNTs). With the use of the moving least squares (MLS) approximation as the linkage for the deformations of the discrete atomic structure and the corresponding continuum model, the delicate microstructures of pristine or defected SWCNTs which possess abnormal properties can be involved accurately in the mechanic model built by this CDMC method. Based on the proposed CDMC method, a variationally consistent meshless computational scheme is developed. As the degree of freedom of SWCNTs can be chosen freely in the context of CDMC method, the numerical computational efficiency is higher than that of the fully atomistic simulations. Various numerical tests are carried out for simulating the nonlinear large and buckling deformations of SWCNTs. The results compared with those of the fully atomistic simulations show that CDMC method is accurate and efficient for predicting the nonlinear mechanical properties of pristine and defective SWCNTs and has great potential for dealing with large scale boundary problems.