Improved Moving Least-squares Research Articles

Solving heat conduction problems in the start-up stage of direct chill casting processes via a temperature-enthalpy mixed formulation based on the improved element-free Galerkin method

A novel approach to solve the transient heat conduction problem with moving boundary and solid↔liquid phase change involved in the start-up stage of direct chill casting (DCC) processes is introduced in this work, which is based on the improved element-free Galerkin (IEFG) method. The procedure is developed under a temperature-enthalpy mixed formulation of the internal energy balance, but the construction of improved moving least squares (IMLS) is required only to approximate temperature. The reliability and suitability of this approach have been first proven in a 1-D alloy solidification benchmark problem with exact solution, and subsequently used to solve a more complex three-dimensional applied thermal problem concerning the start-up stage of the DCC process of an aluminium alloy slab. The domain growth conditioned by the bottom block (moving boundary) is modelled moving down the nodes used to represent the problem domain at each time step, and adding a new nodes plane at the domain upper boundary. Outcomes have revealed the usefulness of this technique to overcome the difficulties concerning solid↔liquid phase change marked non-linearities and moving boundaries involved in the transient analysis of DCC heat transfer problems, allowing the achievement of accurate and stable results in a remarkably simple manner.

The Improved Element-Free Galerkin Method for Anisotropic Steady-State Heat Conduction Problems

In this paper, we considered the improved element-free Galerkin (IEFG) method for solving 2D anisotropic steady-state heat conduction problems. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty method is applied to enforce the boundary conditions, thus the final discretized equations of the IEFG method for anisotropic steady-state heat conduction problems can be obtained by combining with the corresponding Galerkin weak form. The influences of node distribution, weight functions, scale parameters and penalty factors on the computational accuracy of the IEFG method are analyzed respectively, and these numerical solutions show that less computational resources are spent when using the IEFG method.

The Improved Element-Free Galerkin Method for 3D Helmholtz Equations

The improved element-free Galerkin (IEFG) method is proposed in this paper for solving 3D Helmholtz equations. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty technique is used to enforce the essential boundary conditions. Thus, the final discretized equations of the IEFG method for 3D Helmholtz equations can be derived by using the corresponding Galerkin weak form. The influences of the node distribution, the weight functions, the scale parameters of the influence domain, and the penalty factors on the computational accuracy of the solutions are analyzed, and the numerical results of three examples show that the proposed method in this paper can not only enhance the computational speed of the element-free Galerkin (EFG) method but also eliminate the phenomenon of the singular matrix.

The Improved Element-Free Galerkin Method for Diffusional Drug Release Problems

By using the improved moving least-square (IMLS) approximation to present the shape function, the improved element-free Galerkin (IEFG) method is investigated to solve diffusional drug release problems in this paper. In order to get the discretized equation system, Galerkin weak form of a diffusional drug release problem is used with applying essential boundary conditions using the penalty method. The difference method is applied for discretization of time domain. Then the formulae of IEFG method for solving diffusional drug release problems are presented. Three numerical example problems are given to study the convergence of solutions of IEFG method in this paper. The influences of scale parameters of influence domain, penalty factor and node distribution on the accuracy of the solutions of IEFG method are discussed. Compared with finite element method, the correctness of IEFG method in this paper is shown.

A plausible extension of standard penalty, streamline upwind and immersed boundary techniques to the improved element-free Galerkin-based solution of incompressible Navier–Stokes equations

The present work has been conducted in order to propose the extension of standard penalty and stabilization techniques to the improved element-free Galerkin (IEFG) method, for the numerical solution of incompressible fluid flow problems. In principle, the numerical procedures to be implemented in this communication have been conceived for finite element method (FEM)-based solutions, and these include the reduced integration penalty method (RIPM), the streamline upwind Petrov–Galerkin (SUPG) scheme, and a penalty-based immersed boundary method (PBIBM) for the imposition of essential boundary conditions along internal fluid–solid interfaces. The linear momentum balance and mass conservation equations have been coupled via the RIPM, in order to obtain a global weak formulation where the IEFG model is entirely developed in terms of improved moving least squares (IMLS) approximations of the velocity field. A detailed explanation concerning the appropriate extension of both the RIPM and SUPG procedures to the context of IEFG formulations, has also been provided. The resulting formulation has been applied to the solution of two well-known benchmark problems: i) Lid-driven square cavity flow, and ii) Flow past a fixed cylinder. Regarding the flow past a fixed cylinder benchmark problem, the fluid–solid interaction has been imposed as an internal immersed boundary condition via the PBIBM. The feasibility and reliability of implementing the RIPM, SUPG and PBIBM procedures in the IEFG formulation, have been proven by comparison with experimental and mesh-based numerical results reported in the literature. The results obtained in this study have revealed that a proper extension of the aforementioned penalty and stabilization techniques to the IEFG formulation, allows the achievement of accurate and stable numerical results during the solution of incompressible fluid-dynamics problems.

Analyzing 3D Advection-Diffusion Problems by Using the Improved Element-Free Galerkin Method

In this paper, the improved element-free Galerkin (IEFG) method is used for solving 3D advection-diffusion problems. The improved moving least-squares (IMLS) approximation is used to form the trial function, the penalty method is applied to introduce the essential boundary conditions, the Galerkin weak form and the difference method are used to obtain the final discretized equations, and then the formulae of the IEFG method for 3D advection-diffusion problems are presented. The error and the convergence are analyzed by numerical examples, and the numerical results show that the IEFG method not only has a higher computational speed but also can avoid singular matrix of the element-free Galerkin (EFG) method.

Multiscale modeling of crystal plastic deformation of polycrystalline titanium at high temperatures

This paper presents a multiscale modeling framework based on microscale crystal plasticity theory and macroscale element-free methodology for computational simulation of polycrystalline metals. The element-free method employs an improved moving least squares (IMLS) shape function in the Ritz procedure for computational simulation. In the multiscale modeling scheme, the macroscale strain increments of polycrystalline metals are solved by the element-free method and transferred into the mesoscale grains through a scale transition law, and further transferred into the microscale slip systems through the Schmid’s law. The shear strain increments of all grains are then summed up and averaged to calculate the macroscale plastic strain increment. Functions of the temperature-related hardening parameters of the slip systems are determined by calibrating with experimental data. Then, tensile tests are performed on polycrystalline sheets to study their deformation behavior at elevated temperatures. The plasticity of polycrystalline sheets is found to be significantly affected by the grain size. The concentration of shear stress and additional hardening are occurred at the grain boundaries of soft and hard grains. Moreover, it is revealed that the yield strength of the polycrystalline sheets is greatly reduced as the hardening parameters and thermal softening factor decrease at elevated temperatures.

An IB-LBM implementation for fluid-solid interactions with an MLS approximation for implicit coupling

• Extend the implicit velocity correction method to simulate complex fluid solid interaction. • Implicit immersed boundary method to combined IMLS is first presented. • IMLS approximation with the weight orthogonal basis functions for stable solution. This paper presents a combination of the implicit immersed boundary (IB) method and the lattice Boltzmann method (LBM) for a fluid-solid interaction involving an immersed body with moving boundaries and complex geometries. In the implicit IB-LBM method, the flow is computed using LBM. The body force caused by the immersed body is not pre-calculated, but implicitly determined using a corrected velocity field that accurately satisfies the no-slip and no-penetration conditions on the interface between the fluid and solid. The information transfer is not using the traditional Dirac delta function and moving-least-square (MLS). But an improved MLS (IMLS) approximation has been developed to combined the implicit IB-LBM method, where the orthogonal function system with a weight function is used as the basis function and overcome these disadvantages of traditional MLS, in which it may cause numerical instabilities because the matrix inversion must be solved and the final equations system is easily ill-conditioned or singular. Several different flow problems (a two-dimensional flow past a static, rotating and oscillating cylinder, and the vortex-induced vibration of a cylinder as well as the free movement of flapping foil) are simulated to validate the accuracy and efficiency of the present implicit IB-LBM method. The simulation results show good agreement with previous numerical and experimental results. It is found that the present IB-LBM is efficient and reliable in dealing with fluid–solid interaction problems.

The dimension split element-free Galerkin method for three-dimensional potential problems

This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.

The improved element-free Galerkin method for elastoplasticity large deformation problems

Based on the improved moving least-squares (IMLS) approximation, the improved element-free Galerkin (IEFG) method for elastoplasticity large deformation problems is presented. Compared with the moving least-squares (MLS) approximation, by orthogonalizing the basis function, the IMLS approximation can overcome the disadvantage of ill-conditional or singular equations in the MLS approximation. Compared with the meshless methods based on the MLS approximation, under the similar computational accuracy, the ones using the IMLS approximation have higher computational efficiency. The IMLS approximation is used to form the approximation function, the Galerkin weak form of elastoplasticity large deformation problem is used to form the discretized equation system, and the penalty method is used to apply the displacement boundary conditions, then the formulae of the improved element-free Galerkin (IEFG) method for elastoplasticity large deformation problems are obtained, and Newton-Raphson method is used to obtain the solution of the final equation system. Some numerical examples are given to discuss the influences of the weight function, the scale parameter, the penalty factor, the node distribution and the load step number on the computational precision of the numerical solutions. Considering the relative errors when the cubic, quartic and quintic spline functions are used as weight functions, it is shown that the numerical solution using the cubic spline function has higher computational precision. About the scale parameter of the influence domain and the penalty factor, the numerical results show that d max=3.6 and α =1010× E will obtain the solutions with higher precision. The convergence of the method in this paper is analyzed by considering the influence of the node distribution on the computational precision of the solutions, and it is shown that the method in this paper is convergent. And the influence of the total load step number on the computational precision of the solutions is also discussed. Numerical examples show that the method in this paper is effective, and it has the advantage of improving the computational efficiency of the EFG method.

Meshless analysis of two-dimensional two-sided space-fractional wave equation based on improved moving least-squares approximation

ABSTRACTThe Space-fractional wave equations (SFWE) have been found to be very adequate in describing anomalous transport and dispersion phenomena. Due to the non-local property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model. In this paper, a meshless analysis of two-dimensional two-sided SFWE is proposed based on the improved moving least-squares (IMLS) approximation. The trial function for the SFWE is constructed by the IMLS approximation, where the resulting algebraic equation system to obtain the shape functions is no more ill conditioned and has high computational efficiency. The Riemann–Liouville operator is discretized by the Grünwald formula. The centre difference method and the strong-forms of the SFWE are used to obtain the final fully discrete algebraic equation. And the essential boundary conditions can be directly and easily imposed on as a finite element method. Due to the adoption of IMLS approximation and strong-forms, this method will be highly accurate and efficient. Numerical results demonstrate that this method is highly accurate and computationally efficient for SFWE. Moreover, the convergence and error estimate have been analysed in our study.

Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems**Project supported by the National Natural Science Foundation of China (Grant No. 11471063), the Chongqing Research Program of Basic Research and Frontier Technology, China (Grant No. cstc2015jcyjBX0083), and the Educational Commission Foundation of Chongqing City, China (Grant No. KJ1600330).

We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

Geometrically nonlinear large deformation of CNT-reinforced composite plates with internal column supports

Abstract The geometrical nonlinear analysis of internally supported nanocomposite plates subjected to a uniformly distributed load is carried out. This study investigates the effects of internal point/column supports on the large deformation bending of nanocomposite plates reinforced by carbon nanotubes (CNTs) with different types of distributions, namely, uniform and two kinds of functionally graded distributions through the thickness of the plates. Two-dimensional displacement field of the plate is approximated by a set of Improved Moving Least Squares (IMLS) functions. The arc-length iterative algorithm with the modified Newton method is employed to obtain the nonlinear response of nanocomposite plates. Convergence studies indicate the validity and effectiveness of the element-free IMLS-Ritz method. The effects of plate thickness-to-width ratio, volume fraction ratio, and plate aspect ratio on the large deformation behavior of nanocomposite plates under various boundary conditions are examined. To the best of the authors’ knowledge, the problem has not been attempted in the open literature.

Analysis of fracture problems of airport pavement by improved element-free Galerkin method

Using the improved element-free Galerkin (IEFG) method, in this paper we introduce the characteristic parameter r which can reflect the singular stress near the crack tip into the basic function of the improved moving least-squares (IMLS) approximation. Combining fracture theory with the IEFG method, we present an IEFG method of treating the elastic fracture problems, and analyze a numerical example of two-dimensional layered system of airport composite pavement with reflective crack. In the IEFG method, the IMLS approximation is used to form the shape function. The IMLS approximation is presented from the moving least-squares (MLS) approximation, which is the basis of the element-free Galerkin (EFG) method. Compared with the MLS approximation, the IMLS approximation uses the orthonormal basis functions to obtain the shape function, which leads to the fact that the matrices for obtaining the undetermined coefficients are diagonal. Then the IMLS approximation can obtain the solutions of the undetermined coefficients directly without the inverse matrices. The IMLS approximation can overcome the disadvantages of the MLS approximation, in which the ill-conditional or singular matrices are formed sometimes. And it can also improve the computational efficiency of the MLS approximation. Because of the advantages of the IMLS approximation, the IEFG method has greater computational efficiency than the EFG method which is based on the MLS approximation, and can obtain the solution for arbitrary node distribution, even though the EFG method cannot obtain the solution due to the ill-conditional or singular matrices in the MLS approximation. Paving the asphalt concrete layer on the cement concrete pavement is an effective approach to improving the structure and service performance of an airport pavement, which is called airport composite pavement. The airport composite pavement has the advantages of rigid pavement and flexible pavement, but there are various forms of joints or cracks of cement concrete slab, which makes the crack reflect into the asphalt overlay easily under the plane load and environmental factors. Reflective crack is one of the main failure forms of the airport composite pavement. Therefore, it is of great theoretical significance and engineering application to study the generation and development mechanism of reflective crack of the airport composite pavement. For the numerical methods of solving the fracture problems, introducing the characteristic parameter r which can reflect the singular stress near the crack tip into the basic function is a general approach. In this paper, we use this approach to obtain the IEFG method for fracture problems, and the layered system of airport composite pavement with reflective crack is considered. The numerical results of the displacements and stresses in the airport composite pavement are given. And at the tip of the crack, the stress is singular, which makes the crack of the airport composite pavement grow. This paper provides a new method for solving the reflective crack problem of airport composite pavement.

Solving unsteady Schrödinger equation using the improved element-free Galerkin method**Project supported by the National Natural Science Foundation of China (Grant No. 11171208), the Natural Science Foundation of Zhejiang Province, China (Grant No. LY15A020007), the Natural Science Foundation of Ningbo City (Grant No. 2014A610028), and the K. C. Wong Magna Fund in Ningbo University, China.

By employing the improved moving least-square (IMLS) approximation, the improved element-free Galerkin (IEFG) method is presented for the unsteady Schrödinger equation. In the IEFG method, the two-dimensional (2D) trial function is approximated by the IMLS approximation, the variation method is used to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. Because the number of coefficients in the IMLS approximation is less than in the moving least-square (MLS) approximation, fewer nodes are needed in the entire domain when the IMLS approximation is used than when the MLS approximation is adopted. Then the IEFG method has high computational efficiency and accuracy. Several numerical examples are given to verify the accuracy and efficiency of the IEFG method in this paper.

Postbuckling behavior of bi-axially compressed arbitrarily straight-sided quadrilateral functionally graded material plates

In this paper, the problem of the postbuckling behavior of biaxial compressed straight-sided, functionally graded material (FGM) plates of quadrilateral shape is studied. The plate considered is subjected to in-plane loads on all four edges. A computational framework based on an improved moving least-squares (IMLS) approximation for the field variables is developed for the analysis. The solution procedures involved a transformation from the physical domain to a computational domain and then discrete the nonlinear governing equation using the IMLS-Ritz method. The first-order shear deformation theory (FSDT) with the von Kármán nonlinearity is employed. A nonlinear solution to the postbuckling of quadrilateral FGM plates is computed through the modified Newton–Raphson method combined with the arc-length iterative algorithm. A stabilized conforming nodal integration scheme is employed to improve computational efficiency and eliminate shear and membrane locking. The validity and accuracy of the numerical results are established through convergence studies. To the best of the authors’ knowledge, the problem has not been attempted in the open literature.

Postbuckling analysis of axially compressed CNT reinforced functionally graded composite plates resting on Pasternak foundations using an element-free approach

A postbuckling analysis of functionally graded carbon nanotube (FG-CNT) reinforced composite plates resting on Pasternak foundations is presented. The plate under consideration is subjected to axial compression. The improved moving least-squares (IMLS) approximations are employed for the field variables in the Ritz method to derive the discretized governing equations. The plates are reinforced by single-walled carbon nanotubes (SWCNTs) which are assumed to be graded through the thickness direction with different types of distributions. The effective material properties of FG-CNT plates are estimated through a micromechanical model based on the extended rule of mixture. The bending stiffness is evaluated using the stabilized conforming nodal integration scheme. The membrane and shear terms in the governing equation are determined using the direct nodal integration technique. The postbuckling path is traced using the arc-length method combined with the modified Newton–Raphson technique. Parametric studies are conducted to examine the effects of CNT volume fraction, plate thickness-to-width ratio, plate aspect ratio and elastic foundation modulus on the postbuckling behaviors of FG-CNT plates under various boundary conditions.

Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving least-square approximation

This paper combines variational formulations of boundary integral equations (BIEs) and the improved moving least-square (IMLS) approximation to develop a symmetric meshless method, the improved Galerkin boundary node method (IGBNM) for boundary-only analysis of boundary value problems in potential theory and viscous fluid flow. The IMLS approximation is used in the IGBNM to construct meshless shape functions. In the IMLS approximation, the system of algebra equations can be solved without the inverse matrix. Compared with other MLS-based meshless methods, the IGBNM is a direct numerical method in which the basic unknown quantities are the actual nodal values. Besides, boundary conditions in the IGBNM are implemented directly and easily, and the resulting ‘stiffness’ matrices conserve the symmetry and positive definiteness of the variational formulations. Thus, it gives a higher computational efficiency. Total details of numerical implementation and error analysis of the IGBNM are first given for general BIEs. Then, taking potential problems and viscous fluid flow problems as examples, we set up a framework for numerical implementation and asymptotic error estimates of the IGBNM. Finally, some numerical tests are presented to demonstrate the efficiency of the method.

An element-free IMLS-Ritz method for numerical solution of three-dimensional wave equations

Abstract This paper presents an element-free based numerical approach for solving three-dimensional wave equations based on the Ritz minimization procedure. The method involves the use of a set of orthogonal shape functions to approximate its field variables. In this study, an improved moving least-squares (IMLS) is used to generate sets of orthogonal shape functions that reduced the number of unknown coefficients in the trial functions. The entire approximation procedure can be easily implemented numerically. The accuracy of the approximation can be enhanced by increasing the number of nodes used in the computation. As the result of the above procedures, a final algebraic equation system is derived through discretizing the constructed functional. The functional is established by enforcing the Dirichlet boundary conditions via the penalty approach. The simplicity and applicability of the element-free method are demonstrated by solving several selected linear and nonlinear three-dimensional wave equations. Besides the convergence study, the present results are compared with the available published solutions from the literature, where possible, verifying the accuracy, efficiency and reliability of the IMLS-Ritz method.

Geometrically nonlinear large deformation analysis of functionally graded carbon nanotube reinforced composite straight-sided quadrilateral plates

Abstract An improved moving least-squares (IMLS) approximation for the field variables is proposed for geometrically nonlinear large deformation analysis of functionally graded carbon nanotube (FG-CNT) reinforced composite quadrilateral plates. The plate considered is of moderate thickness and, hence, the first-order shear deformation theory (FSDT) and Von Karman assumption are adopted to incorporate the transverse shear strains, rotary inertia and moderate rotations. The CNTs are assumed to be uniaxially aligned in the axial direction and functionally graded in the plate thickness direction. The discrete nonlinear governing equation is derived based on the IMLS-Ritz method. The modified Newton–Raphson method combined with the arc-length iterative algorithm is employed to solve the nonlinear deformation of the FG-CNT reinforced composite quadrilateral plates. Improvements in computational efficiency and elimination of shear and membrane locking are achieved using a stabilized conforming nodal integration scheme to evaluate the system’s bending stiffness. Through detailed parametric studies, CNT distribution, CNTs volume fraction, aspect ratio and thickness-to-width ratio and different boundary conditions are demonstrated to effect significantly on the large deflection behaviors of the quadrilateral FG-CNT reinforced composite plates.