Moving Least-squares Approximation 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.

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.

Modeling nonlinear deformation of slender auxetic structures under follower loads with complex variable meshfree methods

The auxetic structure with negative Poisson’s ratio has broad application prospects in many engineering fields. Although it is often simplified as a linear elastic problem in theoretical models, the generally used slender structure, such as the reentrant honeycombs, will inevitably undergo large deformation resulting in significant non-conservative load effects in the service conditions. Therefore, a numerical framework for modeling nonlinear deformation of the auxetic structures under follower loads with the complex variable element-free Galerkin method is developed in this paper. The application of the complex variable meshfree method is to deal with the numerical difficulties caused by mesh distortion that may occur in large deformation problems, and at the same time, it can improve the construction efficiency of meshfree shape functions through the complex variable moving least-squares approximation. The Galerkin weak form of the incremental total Lagrangian formula for large deformation problems with the enforced essential boundary conditions using the penalty method is derived and then discretized in the complex variable meshless implementation. Five numerical examples are presented with detailed convergence study to demonstrate the accuracy of the proposed approach in dealing with non-conservative large deformation of the slender auxetic structures.

Numerical solution of Allen–Cahn model on surfaces via an effective method based on generalized moving least squares (GMLS) approximation and the closest point approach

A numerical method has been designed for solving the surface Allen–Cahn model. The proposed numerical procedure is based on the generalized moving least-squares approximation and the closest point method. This approach is free from the structure of the underlying surface. It only needs a set of arbitrarily distributed mesh-free points on the surface.

Development of a Moving-Least-Square-based Immersed Boundary Method for the simulation of viscous Compressible Flows

In this paper, a Moving-Least-Squares (MLS)-based immersed boundary method (IBM) for simulating viscous compressible flows is proposed. For this method, the compressible lattice Boltzmann flux solver (LBFS) is used to solve the flow field on the Eulerian mesh and the MLS-based IBM introduced in present paper is used to implement boundary condition. The discrete Lagrangian points are used to represent the Solid boundaries. A second-order MLS interpolation is used for numerical exchange between two grids of IBM. The forced implicit boundary condition is adopted to ensure that the physical boundary condition can be accurately satisfied. The present method is able to achieve a higher numerical accuracy than traditional IBM, which use the Delta function for interpolation. Some numerical validation tests are used for the validation of proposed method. The order and accuracy of the MLS approximation are tested by using a classical example of flow past a cylinder. Test of flow past the NACA0012 airfoil at different Mach numbers, angles of attack and Reynolds numbers is applied to verified the accuracy.

A coupling approach of the isogeometric–meshfree method and peridynamics for static and dynamic crack propagation

A coupling approach of the isogeometric–meshfree method and the peridynamic method is developed for static and dynamic crack propagation. The coupling approach exhibits advantages in the flexibility of modeling cracks and the exactness of geometry representation. The isogeometric–meshfree method, which adopts the moving least-squares approximations to establish the equivalence between meshfree shape functions and isogeometric basis functions, is capable of obtaining the exact geometry. The isogeometric–meshfree nodes located on the geometry can be directly coupled with peridynamic points by the balanced force principle, which is straightforward and effective. With this approach, the boundary conditions can be enforced directly using the isogeometric–meshfree method, and the surface effects of peridynamics are effectively eliminated. Moreover, the present approach is flexible in switching the isogeometric–meshfree nodes to peridynamic points and achieving adaptive coupling with the same convergence rate as the one for the isogeometric–meshfree method, which is higher than the one for the original peridynamic method. The dual-horizon peridynamic method is adopted for non-uniform discretization. Based on the exact geometry representation, the coupling approach is further extended to crack problems with contact loading. Several representative examples are presented to validate the effectiveness of the present approach in solving static/dynamic crack problems and studying fractures caused by contact.

An improved meshless method based on the dimension splitting moving least-squares method for elasticity problems

An improved element-free Galerkin method (IEFGM) is proposed in this paper to solve the two-dimensional elasticity problems. In the IEFGM, the dimension-splitting moving least squares (DS-MLS) method is used to construct the trial functions, and the Galerkin variational weak form coupled with the integral coordinate transformation is applied to derive the final discrete equations of the elastic problems. The DS-MLS method is developed from the dimensional splitting method and the moving least squares (MLS) approximation. Since the shape function of the DS-MLS method is derived independently from the direction of dimensional splitting and the splitting subdivision surface, the dimension and complexity of matrix operations are greatly reduced in solving the shape function of the MLS approximation, thereby improving the calculation efficiency. Some typical examples are discussed to show the effectiveness of the improved meshless method in this paper. From the numerical results, because the DS-MLS method reduces the dimensionality when solving the shape functions, the improved meshless method in this paper can consume less CPU time and acquire a higher accuracy solution than the EFG method.

American options pricing under regime-switching jump-diffusion models with meshfree finite point method

In an incomplete market construction and by no-arbitrage assumption, the American options pricing problem under the jump-diffusion regime-switching process is formulated by a variational form of coupled partial integro-differential equations. In this paper, a valuation algorithm is developed for American options when the dynamics of underlying assets follow the regime-switching jump-diffusion processes. Using the fact that the price of an American option under jump-diffusion regime-switching processes is formulated by a collection of coupled variational partial integro-differential equations with the free boundary characteristic, we combine the moving least-squares approximation with an operator splitting method to treat American constraints. Numerical experiments with American options under three, five, and seven regimes demonstrate the efficiency and effectiveness of our computational scheme for pricing American options under the regime-switching models.

A complex variable EFG method for hyperelastic large deformation analysis under non-conservative loads

The finite element method is extremely popular in almost all fields of engineering and scientific research due to its powerful and accessible industrial software for computational modelling. However, alternatives including the meshless methods, have received increasing attention on the large deformation problems of hyperelastic materials that need to avoid mesh distortion. In this paper, based on the complex variable moving least-squares approximation, the complex variable element-free Galerkin (CVEFG) method for the large deformation problem of hyperelastic materials under non-conservative loads is established. In the present method, the Newton-Raphson method is used for solving the nonlinear equations. The algorithm for hyperelastic large deformation problems under non-conservative loads are established. Three numerical examples are used to verify the effectiveness of the CVEFG method in this paper. Numerical results show that the solution obtained using the CVEFG method is closer to that obtained by ABAQUS and is more efficient than the element-free Galerkin method.

Interpolating Modified Moving Least Squares based element free Galerkin method for fracture mechanics problems

Interpolating Modified Moving Least Squares (IMMLS) based element-free Galerkin method (EFGM) is a simple, robust technique which addresses the limitations of classical Moving Least Squares Approximation (MLSA) based EFGM and is well-demonstrated for tissue deformation. However, its application to fracture mechanics featuring strong discontinuities has not been explored. To address this gap, the present work explores isotropic and orthotropic thermoelastic Linear Elastic Fracture Mechanics (LEFM) cases — with initial isotropic and orthotropic LEFM formulations extended to respective thermoelastic cases — and an isotropic Elastoplastic Fracture Mechanics (EPFM) case employing isotropic bilinear stress–strain curve. The results including field-variables and stress intensity factors (SIFs) are corroborated with converged finite element method (FEM) results; in addition, the results also include classical-MLSA results for a comparative study. L2 norms with respect to FEM indicate an overall close correspondence with a better match for primary variables than secondary variables — indicating the efficacy of IMMLS technique for fracture mechanics. In comparison with classical MLSA-based EFGM results, IMMLS-based EFGM (IN-EFGM) results are nearly similar. However, the advantages demonstrated by IMMLS technique over the classical MLSA-based EFGM in enforcing essential boundary conditions, a larger set of admissible nodal configuration and lesser computational time render IN-EFGM as a potential tool for fracture mechanics.

A meshless method for geometric nonlinear analysis of arbitrary polygonal and circular stiffened plates

Based on the first-order shear deformation theory, a meshless Galerkin method for geometric nonlinear analysis of arbitrary polygonal and circular stiffened plates is proposed. The arbitrary polygonal and circular stiffened plates are modeled as composite structures that consist of flat plates and stiffeners. A series of points are used to discretize the flat plate and stiffeners to obtain the meshless model of the stiffened plates. The moving least-squares approximation is used to construct the shape functions and the displacement fields. The large deflection theory of von Karman is adopted for the geometric nonlinear analysis of the stiffened plates, and flexure and extensional properties of the stiffened plates are derived. The stiffeners are fitted to the flat plate by implementing displacement compatibility conditions between the flat plates and the stiffeners. The boundary conditions are enforced by the full transformation method. To demonstrate the convergence and accuracy of the proposed method, several numerical examples are employed. The solutions that are computed by the proposed method are compared with the finite element solutions that are given by ABAQUS and with the results of other research work. The results show that the agreement is good. Because no mesh is required in the model, the stiffeners can be placed anywhere on the flat plates, and changes to the positions of the stiffeners do not entail the remeshing of the plates.

The element-free Galerkin method for the variational–hemivariational inequality of the dynamic Signorini–Tresca contact problems with friction in elastic materials

The element-free Galerkin method is presented for the variational–hemivariational inequality of the dynamic Signorini–Tresca contact problems with friction in elastic materials. The Dirichlet boundary conditions and the constrained conditions are imposed by the penalty method. The error estimates of the element-free Galerkin method indicate that the convergence order depends on the spatial step, the time step, the largest degree of basis functions in the moving least-squares approximation and the penalty factor. Numerical examples verify our theoretical results.

On tracking arbitrary crack path with complex variable meshless methods

This study presents a numerical modelling framework based on complex variable meshless methods, which can accurately and efficiently track arbitrary crack paths in two-dimensional linear elastic solids. The key novelty of this work is that the proposed meshless modelling scheme enables a direct element-free approximation for the solutions of linear elastic fracture mechanics problems. The complex variable moving least-squares approximation with a group of simple complex polynomial basis is applied to implement this meshless model, with which the fracture problems with both stationary or progressive cracks are considered and studied. The effects of choosing different definitions of weighted complex variable error norm and different forms of complex polynomial basis on the computational accuracy of crack tip fields and crack paths prediction are analysed and discussed. Five benchmark numerical examples were studied to demonstrate the superiority of the present complex variable meshless framework over a standard element-free Galerkin method in tracking arbitrary crack paths in two-dimensional elastic solids.

Fully implicit, stabilised, three-field material point method for dynamic coupled problems

This study presents the formulation and implementation of a fully implicit stabilised Material Point Method (MPM) for dynamic problems in two-phase porous media. In particular, the proposed method is built on a three-field formulation of the governing conservation laws, which uses solid displacement, pore pressure and fluid displacement as primary variables (u–p–U formulation). Stress oscillations associated with grid-crossing and pore pressure instabilities near the undrained/incompressible limit are mitigated by implementing enhanced shape functions according to the Generalised Interpolation Material Point (GIMP) method, as well as a patch recovery of pore pressures – from background nodes to material points – based on the same Moving Least Square Approximation (MLSA) approach investigated by Zheng et al. [1]. The accuracy and computational convenience of the proposed method are discussed with reference to several poroelastic verification examples, spanning different regimes of material deformation (small versus large) and dynamic motion (slow versus fast). The computational performance of the proposed method in combination with the PARDISO solver for the discrete linear system is also compared to explicit MPM modelling [1] in terms of accuracy, convergence rate, and computation time.

A new approach based on moving least square method for calculating the Weibull coefficients

AbstractIn this study, the graphical method, which is widely used to find the coefficients of the Weibull distribution function (WDF), has been improved by using the moving least squares approximation (MLSA) replace least squares method (LSM). The accuracy of the results obtained by the moving least squares method is shown by two different error analysis tests. In order to implement this method, Osmaniye region, which is rich in wind potential maintenance, has been selected. The root mean square error (RMSE) and mean percentage error (MPE) are used to compare the efficiency of proposed method. In this study, the results of the graphical method developed by the MLSA were compared monthly and yearly with the results obtained according to the least squares method. It was observed that the results of the proposed method in the two error analysis tests were much better. Also, the mean wind speed and average wind power values of the selected region were calculated with the coefficients found by MLSA. These calculated values were compared with actual values.

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.

Difference between smoothed particle hydrodynamics and moving particle semi-implicit operators

Smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods are representative meshfree particle methods used to compute Lagrangian mechanics. The approximations of differential operators in the SPH and MPS methods have several similarities, but the theoretical discussion of the difference between them is limited. This study mathematically describes the difference via a comprehensive derivation of the first- and second-order derivative operators for each method. The comprehensive derivation indicates that the SPH and MPS operators are consistent with the pressure Poisson equation and moving least-squares approximation, respectively. The variation in consistency can explain the difference in the schemes of the incompressible flow problem. Additionally, the comprehensive derivation of the MPS operators can result in novel second-order and anisotropic operators. This study strengthens the theoretical understanding of the SPH and MPS methods and facilitates the selection of the appropriate method by users. Furthermore, the proposed MPS operators contribute to developing methods with adaptive or multiscale particle distributions.

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.

Analysis of the MLS variants in the meshless local Petrov-Galerkin method for a solution to the 2D Laplace equation

We investigated the performance of various moving least squares (MLS) approximation schemes in surface fitting and in a partial-differential-equations solver based on the meshless local Petrov-Galerkin (MLPG) method. A modified weight functions in the MLS approximation and the orthogonal basis functions-based MLS ensures the Kronecker delta property. Variants of the MLS were first compared for the surface-fitting test cases. Then, the MLPG method, based on improved versions of the MLS approximations, was developed. The convergence rate, accuracy and computational time of the MLPG method were compared for the 2D Laplace equation. The MLPG method with the Kronecker delta property makes it easy to impose the essential boundary conditions, as with the finite element method (FEM), while on an orthogonal basis the function based MLS approximation increases the computational efficiency of the MLPG method, which both narrow the gap between the FEM and the MLPG.

On the static analysis of inhomogeneous magneto-electro-elastic plates in thermal environment via element-free Galerkin method

The inhomogeneous magneto-electro-elastic (MEE) coupling element-free Galerkin method (IMC-EFGM) is proposed for solving static behaviors of MEE structures. Plates made of homogeneous or inhomogeneous MEE materials are analyzed by IMC-EFGM, and the mechanical behaviors in different temperature fields are simulated. Displacement, magnetic potential and electric potential are studied using moving least squares (MLS) approximation. For inhomogeneous MEE materials, the constitutive equation in the MEE-thermal environment is first obtained by using Gibbs energy, and then used together with the Hamiltonian principle to determine the system governing equation. Since the MLS approximation does not satisfy the principle of Kronecker delta, the penalty function method is used to impose the natural boundary. The value of material properties at the point of integration is taken into account when calculating inhomogeneous materials. Compared with the analytical solution and finite element method (FEM), IMC-EFGM is straightforward and has greater precision than FEM. Some numerical examples are shown for the static FG-MEE plate with different hole shapes. Homogeneous MEE materials and inhomogeneous MEE materials are mainly used in intelligent structures and have good application perspectives. This research will promote the future use of inhomogeneous MEE materials.