Moving Least-squares Approximation Research Articles

The complex variable meshless local Petrov–Galerkin method for elastodynamic analysis of functionally graded materials

As an improvement of the meshless local Petrov–Galerkin (MLPG), the complex variable meshless local Petrov–Galerkin (CVMLPG) method is extended here to dynamic analysis of functionally graded materials (FGMs). In this method, the complex variable moving least-squares (CVMLS) approximation is used instead of the traditional moving least-squares (MLS) to construct the shape functions. The main advantage of the CVMLS approximation over MLS approximation is that the number of the unknown coefficients in the trial function of the CVMLS approximation is less than that of the MLS approximation, thus higher efficiency and accuracy can be achieved under the same node distributions. In implementation of the present method, the variations of the FGMs properties are computed by using material parameters at Gauss points, so it totally avoids the issue of the assumption of homogeneous in each element in the finite element method (FEM) for the FGMs. Several numerical examinations for dynamic analysis of FGMs are carried out to demonstrate the accuracy and efficiency of the CVMLPG.

Moving Least Squares (MLS) Method for the Nonlinear Hyperbolic Telegraph Equation with Variable Coefficients

Telegraph equation which widely used for modeling many engineering and physical phenomena has considered by some researchers in recent years. In this paper, a numerical scheme based on the moving least squares (MLS) approximation and finite difference method (FDM) is proposed for solving a class of the nonlinear hyperbolic telegraph equation with variable coefficients. In the new developed scheme, we use collocation points and approximate solution of the problem under study by using MLS approximation. The MLS method is a meshless approach and does not need any background mesh structure. A time stepping approach is employed for the first- and second-order time derivatives. The proposed method provides a semi-discrete solutions for the problems under study. In space domain, the MLS approximation and in time domain, the finite difference technique are employed. This method after discretization leads to a linear system of algebraic equations. Some numerical results are given and compared with analytical solutions to demonstrate the validity and efficiency of the proposed technique.

Dynamic Analysis of Functionally Graded Fiber Reinforced Cylinders under an Impact Load by a Three Dimensional Mesh Free Approach

In this paper three dimensional (3D) moving least squares (MLS) shape functions in cylindrical coordinates are developed and used for the dynamic analysis of functionally graded fiber reinforced cylinders via a mesh free method. The solution is based on the weak form of the equations of motion and the MLS approximation is used to construct the 3D shape functions in cylindrical coordinates. Essential boundary conditions are imposed using transformation method. Two kinds of variations of the mechanical properties in the thickness direction are considered. In the first model the fibers are assumed to be oriented in the axial direction and the volume fractions of the fibers and matrix are changed continuously according to the power-law distribution. In the second model the volume fractions of constituents are constant and the fibers orientation is changed continuously in the thickness direction according to the power-law distribution. The resulted set of differential equations is solved by the Wilson method. Effects of fiber volume fraction, geometry of cylinder, fiber orientation and boundary conditions on dynamic response of these cylinders are investigated by the proposed model.

A meshless method for solving two-dimensional variable-order time fractional advection–diffusion equation

Several physical phenomena such as transformation of pollutants, energy, particles and many others can be described by the well-known convection–diffusion equation which is a combination of the diffusion and advection equations. In this paper, this equation is generalized with the concept of variable-order fractional derivatives. The generalized equation is called variable-order time fractional advection–diffusion equation (V-OTFA–DE). An accurate and robust meshless method based on the moving least squares (MLS) approximation and the finite difference scheme is proposed for its numerical solution on two-dimensional (2-D) arbitrary domains. In the time domain, the finite difference technique with a θ-weighted scheme and in the space domain, the MLS approximation are employed to obtain appropriate semi-discrete solutions. Since the newly developed method is a meshless approach, it does not require any background mesh structure to obtain semi-discrete solutions of the problem under consideration, and the numerical solutions are constructed entirely based on a set of scattered nodes. The proposed method is validated in solving three different examples including two benchmark problems and an applied problem of pollutant distribution in the atmosphere. In all such cases, the obtained results show that the proposed method is very accurate and robust. Moreover, a remarkable property so-called positive scheme for the proposed method is observed in solving concentration transport phenomena.

Multiscale simulation of mechanical properties and microstructure of CNT-reinforced cement-based composites

In the hydration process of a cementitious matrix, considerable interactive physical and chemical changes take place inside the material that affect both the composition and morphology of cement paste during its early stages. These changes are properly considered within a multiscale model that comprises length-scale integration and gives access to the effective properties via upscaling. In this paper, the kinetics laws of the induction period, nucleation, and growth-controlled and diffusion-controlled hydration are considered, and the evolutions of volume fractions of clinker phases and various hydration products with respect to the hydration degrees are simulated. Based on the microstructural evolution of cement hydration, the properties of cement paste are estimated with a combination of self-consistent and Mori–Tanaka schemes, and the mechanical properties of carbon nanotube-reinforced cement-based composites on the macroscale are then predicted with a meshless method on the basis of the moving least-squares approximation. Finally, the accuracy and efficiency of the proposed model are verified by comparisons with experiments and finite element model simulations.

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.

Moving least squares collocation method for Volterra integral equations with proportional delay

ABSTRACTIn this work, we apply the moving least squares (MLS) method for numerical solution of Volterra integral equations with proportional delay. The scheme utilizes the shape functions of the MLS approximation constructed on scattered points as a basis in the discrete collocation method. The proposed method is meshless, since it does not require any background mesh or domain elements. An error bound is obtained to ensure the convergence and reliability of the method. Numerical results approve the efficiency and applicability of the proposed method.

A meshless method and stability analysis for the nonlinear Schrödinger equation

In this paper, the non-linear Schrödinger (NLS) equation was solved by a meshless method based on the moving least-squares approximation. Single-soliton solution and interaction of two colliding solitons were studied to validate the accuracy of the proposed method. In calculations, different weight functions in which Gaussian and Wendland’s compactly supported functions were used. For the single-soliton solution of NLS equation whose analytical solution was known , error norms and invariants were calculated for two test problems. The obtained results were compared with both each other and some earlier results. Stability analysis of difference equation was done by applying the proposed method for NLS equation.

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.

An interpolating element-free Galerkin scaled boundary method for the elasticity problem

As a newly-developed semi-analytical method, the scaled boundary finite element method is very powerful to deal with singular and unbounded problems. By combining the element-free Galerkin (EFG) method with the scaled boundary method in the frame of improved interpolating moving least-squares (IIMLS) method, an interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is firstly proposed to solve elasticity problems in this paper. In the IEFG-SBM, the solution in the radial direction is obtained analytically and only nodes are required to discretize the boundaries of the computational domain. In addition, higher accuracy and faster convergence are obtained due to the higher continuity of the shape functions in the circumferential direction. The IEFG-SBM does not need the fundamental solution and thus no singular integrations are involved. The shape function of the IIMLS method satisfies the Kronecker delta function property and thus the essential boundary conditions can be imposed directly as in the traditional finite element method. In comparison with the interpolating moving least-squares (IMLS) method proposed by Lancaster and Salkauskas, the key advantage of the IIMLS method is that it does not require singular weight function and thus any weight function used in the MLS approximation can also be applied in the IIMLS method. In addition, there are less unknown coefficients in the IIMLS method than in the conventional moving least-squares (MLS) approximation. Thus fewer nodes are required in the local influence domain and a higher computational accuracy can be reached in the IIMLS-based meshless method. At last, several numerical examples are presented to verify the effectiveness and accuracy of the developed method for the elasticity problem.

Steady heat conduction analyses using an interpolating element-free Galerkin scaled boundary method

Through use of the improved interpolating moving least-squares (IIMLS) shape functions in the circumferential direction of the scaled boundary method based on the Galerkin approach, an interpolating element-free Galerkin scaled boundary method (IEFG–SBM) is developed in this paper for analyzing steady heat conduction problems, which weakens the governing differential equations in the circumferential direction and seeks analytical solutions in the radial direction. The IIMLS method exhibits some advantages over the moving least-squares approximation and the interpolating moving least-squares method because its shape functions possess the delta function property and the involved weight function is nonsingular. In the IEFG–SBM, only a nodal data structure on the boundary is required and the primary unknown quantities are real solutions of nodal variables. Higher accuracy and faster convergence are obtained due to the increased smoothness and continuity of shape functions. Based on the IEFG–SBM, the steady heat conduction problems with thermal singularities and unbounded domains can be ideally modeled. Some numerical examples are presented to validate the availability and accuracy of the present method for steady heat conduction analysis.

Analysis of the Time Fractional 2-D Diffusion-Wave Equation via Moving Least Square (MLS) Approximation

In this paper, the time fractional two-dimensional diffusion-wave equation defined by Caputo sense for ( $$1<\alpha <2$$ ) is analyzed by an efficient and accurate computational method namely meshless local Petrov–Galerkin (MLPG) method which is based on the Galerkin weak form and moving least squares (MLS) approximation. We consider a general domain with Dirichlet boundary conditions further to given initial values as continuous functions. Meshless Galerkin weak form is adopted to the interior nodes while the meshless collocation technique is applied for the nodes on the boundaries of the domain. Since Dirichlet boundary condition is imposed directly therefore the general domains are also applicable easily. In MLPG method, the MLS approximation is usually used to construct shape functions which plays important rule in the convergence and stability of the method. It is proved the method is unconditionally stable in some sense. Two numerical examples are presented, one of them with the regular domain and the other one with non-regular domain, and satisfactory agreements are achieved.

On the stability of the moving least squares approximation and the element-free Galerkin method

In this paper, the stability of the moving least squares (MLS) approximation and a stabilized MLS approximation is analyzed theoretically and verified numerically. It is shown that the stability of the MLS approximation deteriorates severely as the nodal spacing decreases, while the stability of the stabilized MLS approximation is not affected by the nodal spacing. The stabilized MLS approximation is introduced into the element-free Galerkin (EFG) method to produce a stabilized EFG method. Theoretical error analysis of the stabilized EFG method is provided for boundary value problems with mixed boundary conditions of Dirichlet and Robin type. Numerical examples confirm the theoretical results, and show that the stabilized EFG method has higher computational precision and better stability than the original EFG method.

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

In this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge–Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on three-dimensional PDEs, we select some equations and system of PDEs such as Allen–Cahn, Gray–Scott, Ginzburg–Landau, Brusselator models, predator–prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme.

Optimal design and system characterization of graphene sheets in a micro/nano actuator

In this article, an element free Galerkin method (EFGM) is used for dynamic characterization of a capacitive nanoactuator as subjected to a DC voltage by using a proposed nonlocal plate theory. The system governing equations of an electrostatic nanoactuator are first derived based upon the theory of nonlocal Kirchhoff plate. The intermolecular forces, such as Casimir and van der Waals forces, are included in the proposed model. The weak form representation of equilibrium equations is presented based on Hamilton principle and a discrete moving least squares (MLS) approximation for the shape function. Since the MLS approximation does not satisfy the principle of Kronecker delta, a penalty method is then imposed upon to equip as auxiliary boundary conditions. The discrete weak form is adopted to solve for eigenvalue solutions and natural frequencies of the plate. Since numerical experimental results indicate that the number of nodes that scattered in the working domain can affect final solutions dramatically, a calibration scheme is introduced into the proposed modeling system beforehand by using some referred known functions. Eventually, the characterization dynamic behaviors for a nanoactuator with designated DC voltage is made possible. The results indicate that the proposed modeling approach and computational program are accurate and feasible for design and system characterization of single-layered graphene sheets (SLGS) by the adjustment of scale effect to allow the associated pull-in voltage of the device to be optimally designated.

A meshless method based on moving least squares for the simulation of free surface flows

In this paper, a meshless method based on moving least squares (MLS) is presented to simulate free surface flows. It is a Lagrangian particle scheme wherein the fluid domain is discretized by a finite number of particles or pointset; therefore, this meshless technique is also called the finite pointset method (FPM). FPM is a numerical approach to solving the incompressible Navier-Stokes equations by applying the projection method. The spatial derivatives appearing in the governing equations of fluid flow are obtained using MLS approximants. The pressure Poisson equation with Neumann boundary condition is handled by an iterative scheme known as the stabilized bi-conjugate gradient method. Three types of benchmark numerical tests, namely, dam-breaking flows, solitary wave propagation, and liquid sloshing of tanks, are adopted to test the accuracy and performance of the proposed meshless approach. The results show that the FPM based on MLS is able to simulate complex free surface flows more efficiently and accurately.

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.

The Improved Direct Collocation Meshless Approach for Radiative Heat Transfer in Participating Media

The moving least-squares (MLS) direct collocation meshless method (DCM) is an effective numerical scheme for solving the radiative heat transfer in participating media. In this method the trial function is constructed by a MLS approximation and the radiative transfer equation (RTE) is discretized directly at nodes by collocation. The main drawback of this method is that, like most of the other numerical methods, the solution to the RTE by the DCM also suffers much from nonphysical oscillations in some cases caused by the convection-dominated property of the RTE. To overcome the numerical oscillations, special stabilization techniques are usually adopted, which increases the complexity and computation time of problem. In the present work a new scheme based on the outflow-boundary intensity interpolation correction is proposed that can easily ensure a large reduction in numerical oscillations of results without any complex stabilization technique. Adaptive support domain technique is also adopted, and the size of the support domain of each evaluated point changes with the density of nodes with irregular distribution. Five cases are studied to illustrate the numerical performance of these improvements. The numerical results compare well with the benchmark approximate solutions, and it is shown that the improved moving least-square direct collocation meshless method (iDCM) is easily implemented, efficient, of high accuracy, and excellent stability, to solve radiative heat transfer in homogeneous participating media.

Error estimates for the moving least-square approximation and the element-free Galerkin method in n-dimensional spaces

The moving least square (MLS) approximation is one of the most important methods to construct approximation functions in meshless methods. For the error analysis of the MLS-based meshless methods it is fundamental to have error estimates of the MLS approximation in the generic n-dimensional Sobolev spaces. In this paper, error estimates for the MLS approximation are obtained in the Wk,p norm in arbitrary n dimensions when weight functions satisfy certain conditions. The element-free Galerkin (EFG) method is a typical Galerkin method combined with the use of the MLS approximation. The error results of the MLS approximation are then used to yield error estimates of the EFG method for solving both Neumann and Dirichlet boundary value problems. Finally, some numerical examples are given to confirm the theoretical analysis.

Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method**Project supported by the National Natural Science Foundation of China (Grant Nos. 11171208 and U1433104).

In this paper, based on the conjugate of the complex basis function, a new complex variable moving least-squares approximation is discussed. Then using the new approximation to obtain the shape function, an improved complex variable element-free Galerkin (ICVEFG) method is presented for two-dimensional (2D) elastoplasticity problems. Compared with the previous complex variable moving least-squares approximation, the new approximation has greater computational precision and efficiency. Using the penalty method to apply the essential boundary conditions, and using the constrained Galerkin weak form of 2D elastoplasticity to obtain the system equations, we obtain the corresponding formulae of the ICVEFG method for 2D elastoplasticity. Three selected numerical examples are presented using the ICVEFG method to show that the ICVEFG method has the advantages such as greater precision and computational efficiency over the conventional meshless methods.