Moving Least-squares Approximation Research Articles

The element-free Galerkin method for the dynamic Signorini contact problems with friction in elastic materials

The element-free Galerkin method is proposed for the dynamic Signorini 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 a complete Pascal's monomial basis in the moving least-squares approximation and the penalty factor. Numerical examples verify our theoretical results.

Axial vibration analysis of nanorods with variable density based on nonlocal elastic theory and high-order finite difference method

This paper proposes a novel high-order finite difference method (p-FDM) for predicting the axial vibration behaviour of elastic nanorods with variable density based on nonlocal elastic theory. For p-FDM, it first solves the nodal weighting coefficients for approximating the first-order derivatives of displacements at the discrete nodes by using moving least-squares approximation. Next, p-FDM recursively determines the nodal weighting coefficients for approximating the higher-order derivatives by using the obtained weighting coefficients for the first-order derivative. After verifying with exact solutions, results show that the proposed p-FDM yields accurate vibration solutions for a nanorod with either fixed–fixed ends or fixed–free ends. Only a small number of regularly or irregularly distributed grid nodes are required by a high order p-FDM to obtain a converged solution. p-FDM realises an exponential rate of convergence for frequency solution when setting p=n−2 where p is order of p-FDM and n is the number of grid nodes. New axial vibration solutions of nanorods with exponential density distribution are found by using p-FDM. It is shown that the axial frequencies are always larger for a nanorod with an exponential density distribution and no small-scale effect.

Numerical solution of the quasistatic contact problem with the Tresca friction in elastic-viscoplastic materials by the element-free Galerkin method

The element-free Galerkin method is presented for a quasistatic contact problem with the Tresca friction in elastic-viscoplastic materials. The elastic-viscoplastic constitutive model is considered in the problem and the clamped boundary conditions are imposed by the penalty method. The error estimates of the element-free Galerkin method are also obtained, which indicate that the convergence order depends on the nodal spacing, the time step, the largest degree of basis functions in the moving least-squares approximation and the penalty factor. Numerical examples demonstrate that the element-free Galerkin method provides an efficient numerical solution of the quasistatic contact problem with the Tresca friction in elastic-viscoplastic materials.

Bending and free vibration analyses of circular stiffened plates using the FSDT mesh-free method

Based on the first-order shear deformation theory, a meshless method for bending and free vibration analyses of circular stiffened plates is proposed. The circular stiffened plate is modelled as a composite structure that consists of a flat circular plate and stiffeners. A series of points are used to discretize the flat circular plate and stiffeners to obtain the mesh-free model. The moving least-squares approximation is used to construct the shape functions and to derive the displacement fields of the flat circular plate and stiffeners, respectively. The total potential energy and kinetic energy of the circular stiffened plate are obtained according to the compatibility condition between the flat circular plate and stiffeners. The equations governing the bending and free vibration behaviors of the circular stiffened plate are derived according to the principle of Minimum Potential Energy and Hamilton's Principle, respectively. The boundary conditions are enforced by the full transformation method. The convergence of the proposed method and the domain of influence (DOI) that affects the numerical results are studied. Several examples are calculated by the proposed method and compared with the results available in the literature or given by the finite element software ABAQUS. The results show that the proposed method frees the researchers from determining a nodal line on the plate along every stiffener, which is very beneficial for carrying out optimization studies on stiffener location.

Application of moving least squares algorithm for solving systems of Volterra integral equations

Abstract The numerical method developed in the current paper is based on the moving least squares (MLS) method. To this end, the MLS approximation method has been used, and a program has been made which can solve the system of Volterra integral equations (VIEs) with any number of equations and unknown functions. And then the proposed method is implemented on the system of linear VIEs with variable coefficients. The numerical examples are given that show the acceptable accuracy and efficiency of the proposed scheme.

Automatic mesh-free boundary analysis: Multi-objective optimization

The paper is concerned with the numerical solution of two-dimensional potential problems, through a mesh-free boundary model in a multi-objective optimization framework that automatically generates Pareto-optimal mesh-free discretization arrangements. This robust new strategy of analysis allows for simultaneously improving the solution accuracy, the conditioning of the numerical solver, the stability and efficiency of the mesh-free analysis.The boundary mesh-free model (BMFM) is built on the boundary integral equation of the Laplace potential, with a moving least squares (MLS) approximation of variables. The model considers independent MLS approximations in each boundary segment and performs integration with standard numerical quadrature.The main novelty of the paper is the automatic generation of Pareto-optimal nodal arrangements and corresponding compact supports of the mesh-free boundary model, by means of an evolutionary multi-objective optimization process, based on genetic algorithms, which uses reliable very efficient objective functions.A benchmark problem is presented to assess the accuracy and efficiency of the modeling strategy. The remarkably accurate results obtained, in perfect agreement with those of analytical solutions, make very reliable this robust new strategy of automatic mesh-free boundary analysis in a multi-objective optimization framework.

A dimensionless numerical mesh-free model for the compressible fluid flows

In this paper, we propose a dimensionless numerical mesh-free model for the simulation of the compressible isothermal viscous flows. The novelty of this work consists to formulate the Navier-Stokes equations under a dimensionless form and to solve them by a high order mesh-free algorithm to simulate the compressible fluid flows. This algorithm combines a classical implicit Euler scheme, a high order continuation with the Moving Least Squares (MLS) and a homotopy transformation. The MLS approximation and implicit Euler scheme are used respectively for the spatial and temporal discretizations of dimensionless Navier-Stokes equations. The homotopy transformation serves to introduce in dimensionless Navier Stokes equations an arbitrary operator and a parameter without physical dimension. The obtained equations are solved by a high order continuation. The performance of the presented model is tested on the standard benchmark lid-driven cavity problem. Then, the Mach and Reynolds numbers effect is discussed. The obtained results are compared with those of the Finite Difference Method (FDM) coupled with an explicit Runge-Kutta (R-K) scheme and those of literature. This comparison reveals that the results of the dimensionless model are obtained with a less expensive CPU time compared to that of the other algorithms.

A sharp interface direct-forcing immersed boundary method using the moving least square approximation

Based on the Moving Least Square (MLS) approximation, we propose a sharp interface direct-forcing immersed boundary method for incompressible fluid flows with fixed and moving boundaries. Since the domain of definition for the interpolation is highly flexible and the MLS approximation provides an accurate reconstructed approximation of the solution, the proposed method serves the precision and versatility required for a numerical framework to study the fluid-structure interaction problems. To alleviate the inherent spurious numerical oscillation that occurs in the calculated forces on moving boundary embedded objects, we use a two step predictor-corrector method in which the direct forcing terms are calculated after the predictor step and imposed on the whole solid domain as well as at the immediate vicinity of the solid boundary inside the fluid domain. To represent the arbitrary geometries, we adopt a signed distance function representation of the rigid body and an interpolation strategy to considerably reduce the computational cost of the re-initialization of the distance function at every time step. The potential capability of the method is demonstrated for both fixed and moving boundary problems. We also solve a sedimentation of a single cylinder to demonstrate the ability of the present method in solving fluid-structure interaction problems. These numerical experiments show that the proposed moving least square immersed boundary method can handle relatively complex moving problems while enjoying a versatile interpolation strategy and keeping the boundary conditions sharp with remarkable accuracy.

Two meshless methods for Dirichlet boundary optimal control problem governed by elliptic PDEs

In this paper, finite point method (FPM) and meshless weighted least squares (MWLS) method are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The FPM scheme uses shape function constructed by moving least square (MLS) approximation to discretize the equations, while the MWLS scheme employs both MLS approximation and penalty terms to solve the same problem. Error estimates for the FPM scheme are presented and numerical results are provided to examine the impact of parameters and validate the efficiency of the proposed schemes. The extended model (Navier–Stokes equations) shows the ability of our algorithm to handle complex problems. Our explorative work shows the flexibility and great potential of the meshless methods in optimal control problems.

Moving least squares approximation-based online control optimised by the team game algorithm for Duffing-Holmes chaotic problems

ABSTRACT In this paper, an online optimal adaptive robust fuzzy controller based on the Moving Least Squares (MLS) and Team Game Algorithm (TGA) is introduced to control uncertain chaotic nonlinear systems. At first, a robust supervisory stabiliser and a fuzzy adaptive PID controller are designed and combined to handle a Duffing-Holmes chaotic oscillator. Next, the TGA is utilised to optimise the parameters of the designed fuzzy adaptive robust controller for different values of system disturbances. Then, to adapt the optimum design parameters of the controller with the uncertain values of the external disturbances, the MLS approximation is implemented. In order to evaluate the performance of the proposed technique, it is applied on a Duffing-Holmes chaotic nonlinear oscillator with different initial conditions. The results and the analysis prove the efficiency of the proposed controller with regard to uncertain systems’ challenging external disturbances that impair the stability, accuracy and optimality of the system.

Codimensional surface tension flow using moving-least-squares particles

We propose a new Eulerian-Lagrangian approach to simulate the various surface tension phenomena characterized by volume, thin sheets, thin filaments, and points using Moving-Least-Squares (MLS) particles. At the center of our approach is a meshless Lagrangian description of the different types of codimensional geometries and their transitions using an MLS approximation. In particular, we differentiate the codimension-1 and codimension-2 geometries on Lagrangian MLS particles to precisely describe the evolution of thin sheets and filaments, and we discretize the codimension-0 operators on a background Cartesian grid for efficient volumetric processing. Physical forces including surface tension and pressure across different codimensions are coupled in a monolithic manner by solving one single linear system to evolve the surface-tension driven Navier-Stokes system in a complex non-manifold space. The codimensional transitions are handled explicitly by tracking a codimension number stored on each particle, which replaces the tedious meshing operators in a conventional mesh-based approach. Using the proposed framework, we simulate a broad array of visually appealing surface tension phenomena, including the fluid chain, bell, polygon, catenoid, and dripping, to demonstrate the efficacy of our approach in capturing the complex fluid characteristics with mixed codimensions, in a robust, versatile, and connectivity-free manner.

The interpolating element-free Galerkin method for elastic large deformation problems

This paper presents an interpolating element-free Galerkin (IEFG) method for solving the two-dimensional (2D) elastic large deformation problems. By using the improved interpolating moving least-squares method to form shape function, and using the Galerkin weak form of 2D elastic large deformation problems to obtain the discrete equations, we obtain the formulae of the IEFG method for 2D elastic large deformation problems. As the displacement boundary conditions can be applied directly, the IEFG method can acquire higher computational efficiency and accuracy than the traditional element-free Galerkin (EFG) method, which is based on the moving least-squares approximation and can not apply the displacement boundary conditions directly. To analyze the influences of node distribution, scale parameter of influence domain and the loading step on the numerical solutions of the IEFG method, three numerical examples are proposed. The IEFG method has almost the same high accuracy as the EFG method, and for some 2D elastic large deformation problems the IEFG method even has higher computational accuracy.

Error analysis of the meshless finite point method

Abstract The finite point method (FPM) is a notable truly meshless method based on the moving least squares (MLS) approximation and the point collocation technique. In this paper, the error of the FPM is analyzed theoretically. Theoretical results show that the present error bound is directly related to the nodal spacing and the order of basis functions used in the MLS approximation. The present error estimation is independent of the condition number of the coefficient matrix and improves the previously reported estimations. Numerical examples with more than 160000 nodes are given to confirm the theoretical result.

Convergence analysis and error estimates of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time

This paper is presented for the convergence analysis of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time, which arises from the theory of viscoelastic materials with edge friction. First, the existence and uniqueness of the solutions for the evolutionary variational inequality of the second-order in time are proved, which are mainly based on the fixed point theorem. Second, the convergence analysis of the interpolating element-free Galerkin method is presented for them. The error estimates show that the convergence order depends not only on the number of basis functions in the interpolating moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.

Error analysis of interpolating element free Galerkin method to solve non-linear extended Fisher–Kolmogorov equation

Nonlinear partial differential equations (PDEs) play an important role in the modeling of the natural phenomena as they have great significance in real-world applications. This investigation proposes a new algorithm to find the numerical solution of the non-linear extended Fisher–Kolmogorov equation. Firstly, the time variable is discretized by a second-order finite difference scheme. The rate of convergence and stability of the semi-discrete formulation are studied by the energy method. The existence and uniqueness of the solution of the weak form based on the proposed technique have been proved in detail. Furthermore, the interpolating element free Galerkin approach based on the interpolation moving least-squares approximation is employed to derive a fully discrete scheme. Finally, the error estimate of the full-discrete plan is proposed and its convergence order is O(τ2+δm+1) in which τ, δ and m denote the time step, the radius of the weight function and smoothness of the exact solution of the main problem, respectively.

Augmented moving least squares approximation using fundamental solutions

An augmented moving least squares (AMLS) approximation is presented in this paper for constructing approximate functions in meshless methods. The basis functions in the AMLS approximation, differ from monomial basis functions in the classical moving least squares (MLS) approximation, involve fundamental solutions of the underlying differential equations, and thus enhance the performance of the MLS approximation. As the MLS approximation, the AMLS approximation can also be used for curve and surface fitting. Numerical examples involving 2D and 3D Laplace, Helmholtz and modified Helmholtz equations in complicated domains reveal that the accuracy of the AMLS approximation is about 104 ~ 106 times higher than that of the MLS approximation.

Analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions

The localized method of fundamental solutions (LMFS) is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions (MFS). The resultant system of linear algebraic equations in the LMFS is sparse and banded and thus, drastically reduces the storage and computational burden of the MFS. In the LMFS, the moving least square (MLS) approximation, based on fundamental solutions, is used to construct approximate solution at each node. In this paper, this fundamental solutions-based MLS approximation, named as an augmented MLS (AMLS) approximation, is generalized to any point in the computational domain. Computational formulas, theoretical properties and error estimates of the AMLS approximation are derived. Then, taking Laplace equation as an example, this paper sets up a framework for the theoretical error analysis of the LMFS. Finally, numerical results are presented to verify the efficiency and theoretical results of the AMLS approximation and the LMFS. Convergence and comparison researches are conducted to validate the accuracy, convergence and efficiency.

A dual interpolation boundary face method for exterior acoustic problems based on the Burton–Miller formulation

A dual interpolation boundary face method (DiBFM) based on the Burton–Miller formulation for solving exterior acoustic problems is presented. The method is able to unify the conforming and nonconforming BFM. In DiBFM implementation, the nodes of a conforming element are divided into two categories: source nodes (inside the element) and virtual nodes (on the boundary of the element). Acoustic pressure and its derivative are approximated using conforming elements, in the same way as conforming BFM. However, the Burton–Miller integral equations are just collocated at source nodes, in the same way as nonconforming BFM. To arrive at a square linear system, additional constraint equations established by the moving least-squares approximation are provided to condense the degrees of freedom relating to virtual nodes. Several numerical examples demonstrate the accuracy and efficiency of the proposed method.

On the pricing of multi-asset options under jump-diffusion processes using meshfree moving least-squares approximation

The moving least-squares (MLS) approximation is a powerful numerical scheme widely used in the meshfree literature to construct local multivariate polynomial basis functions for expanding the solution of a given differential or integral equation. For partial integro-differential equations arising from the valuation of multi-asset options written on correlated Lévy-driven assets, we propose here an MLS-based collocation scheme in conjunction with implicit-explicit (IMEX) temporal discretization to numerically solve the problem. We apply the method to price both European and American options and compute the option hedge parameters. In the case of American options, we use an operator splitting approach to solve the linear complementarity formulation of the problem. Our numerical experiments show the efficiency of the proposed scheme in comparison with some competing approaches, specially finite difference methods.

Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.