Interpolating Moving Least-squares Research Articles

Moving least‐squares aided finite element method: A powerful means to predict flow fields in the presence of a solid part

AbstractWith the assistance of the moving least‐squares (MLS) interpolation functions, a two‐dimensional finite element code is developed to consider the effects of a stationary or moving solid body in a flow domain. At the same time, the mesh or grid is independent of the shape of the solid body. We achieve this goal in two steps. In the first step, we use MLS interpolants to enhance the pressure (P) and velocity (V) shape functions. By this means, we capture different discontinuities in a flow domain. In our previous publications, we have named this technique the PVMLS method (pressure and velocity shape functions enhanced by the MLS interpolants) and described it thoroughly. In the second step, we modify the PVMLS method (the M‐PVMLS method) to consider the effect of a solid part(s) in a flow domain. To evaluate the new method's performance, we compare the results of the M‐PVMLS method with a finite element code that uses boundary‐fitted meshes.

Mass lumping schemes fitted to MLS-based numerical manifold method in vibration of plates with cutouts using CPT and FSDT

It is critical to accurately extract the natural frequencies in the transverse vibration of plates using the classical plate theory (CPT) and the first-order shear deformation plate theory (FSDT). The moving least squares (MLS) based numerical manifold method (MLS-based NMM) in conjunction with a suitable diagonally lumped mass matrix is a reasonable choice because it can naturally treat plates with cutouts and easily formulate an H2 regular approximation based on MLS interpolation, as well as mitigate shear locking issues for vibration analysis of thin plates based on FSDT. Moreover, the mass lumping techniques involving the row-sum method, the diagonal scaling method, and the mathematically rigorous manifold-based method are extended and derived in the unified framework of MLS-based NMM for transverse vibration analysis of plates. Furthermore, the positive-definiteness of the lumped mass matrix (LMM) generated by the manifold-based mass method is explained in MLS settings, and the row-sum method is proven to be a special case of the manifold-based method. A comprehensive comparative study of different LMMs is performed in accordance with the consistent mass matrix (CMM) on extensive numerical benchmarks. Numerical results demonstrate the convergence properties of LMMs compared with CMM in MLS-based NMM for plate vibration based on CPT and FSDT. It can be observed that LMMs yield comparable performance compared with CMM. The row-sum method obtains accurate results in most cases but can not guarantee the positivity. In contrast, the manifold method can guarantee the positive-definiteness of LMM and is more accurate than the diagonal scaling method in most cases.

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 hybrid smoothed moving least-squares interpolation method for acoustic scattering problems

A hybrid smoothed moving least-squares interpolation method for acoustic scattering problems

Theoretical study of the CO2-O2 van der Waals complex: potential energy surface and applications.

A four-dimensional-potential energy surface (4D-PES) of the atmospherically relevant carbon dioxide-oxygen molecule (CO2-O2) van der Waals complex is mapped using the ab initio explicitly correlated coupled cluster method with single, double, and perturbative triple excitations (UCCSD(T)-F12b), and extrapolation to the complete basis set (CBS) limit using the cc-pVTZ-F12/cc-pVQZ-F12 bases and the l-3 formula. An analytic representation of the 4D-PES was fitted using the method of interpolating moving least squares (IMLS). These calculations predict that the most stable configuration of CO2-O2 complex corresponds to a planar slipped-parallel structure with a binding energy of V ∼ -243 cm-1. Another isomer is found on the PES, corresponding to a non-planar cross-shaped structure, with V ∼ -218 cm-1. The transition structure connecting the two minima is found at V ∼ -211 cm-1. We also performed comparisons with some CO2-X van der Waals complexes. Moreover, we provide a SAPT analysis of this molecular system. Then, we discuss the complexation induced shifts of CO2 and O2. Afterwards, this new 4D-PES is employed to compute the second virial coefficient including temperature dependence. A comparison between quantities obtained in our calculations and those from experiments found close agreement attesting to the high quality of the PES and to the importance of considering a full description of the anisotropic potential for the derivation of thermophysical properties of CO2-O2 mixtures.

Moving least-squares aided finite element method (MLS-FEM): A powerful means to consider simultaneously velocity and pressure discontinuities of multi-phase flow fields

We suggest a strategy to consider discontinuities in the two-phase flow calculations. Our approach comprises enhancing the shape functions of the finite element method (FEM) with the moving least-squares (MLS) interpolation functions. The new shape functions correlate a parameter at any arbitrary point to its surrounding nodes instead of an element's nodes. We name this strategy the "moving least-squares" aided "finite element method" (MLS-FEM).In a previous paper Mostafaiyan et al., we have employed the concept of the MLS-FEM to develop the PMLS (pressure shape function enhanced by the MLS interpolation functions) method, which predicts pressure discontinuities due to the surface tension forces. We take another additional step to consider velocity and pressure discontinuities in a domain with two different fluids. Therefore, we use the MLS technique to enhance the pressure (P) and velocity (V) shape functions. We name this scheme PVMLS (pressure and velocity shape functions enhanced by the MLS technique).We validate the PVMLS method with a stationary drop problem (as a benchmark). We show that the previous PMLS method fails to predict an accurate pressure or smooth streamlines by increasing the Capillary number and deviating from the unit viscosity ratio. However, the PVMLS method provides more accurate results since it considers velocity as a discontinuous parameter. Also, we compare the results of the PVMLS method at very high viscosity ratios (undeformable fluid) with a boundary-fitted single-phase problem. The reported similarities between the predicted pressure values and streamlines in both strategies demonstrate the reliability of the PVMLS method. Finally, we discuss that by employing the PVMLS method to calculate the velocity field, the interface advection will be more mass-conservative.

Error compensation based on surface reconstruction for industrial robot on two-dimensional manifold

PurposeThis paper aims to present error compensation based on surface reconstruction to improve the positioning accuracy of industrial robots.Design/methodology/approachIn previous research, it has been proved that the positioning error of industrial robots is continuous on the two-dimensional manifold of six-joint space. The point cloud generated by positioning error data can be used to fit the continuous surfaces, which makes it possible to apply surface reconstruction on error compensation. The moving least-squares interpolation and the B-spline method are used for the error surface reconstruction.FindingsThe results of experiments and simulations validate the effectiveness of error compensation by the moving least-squares interpolation and the B-spline method.Practical implicationsThe proposed methods can control the average of compensated positioning error within 0.2 mm, which meets the requirement of a tolerance (±0.5 mm) for fastener hole drilling in aircraft assembly.Originality/valueThe error surface reconstruction based on the B-spline method has great superiority because fewer sample points are needed to use this method than others while keeping the compensation accuracy at the same level. The control points of the B-spline error surface can be adjusted with measured data, which can be applied for the error prediction in any temperature field.

The numerical approximation for the solution of linear and nonlinear integral equations of the second kind by interpolating moving least squares

In this paper, the interpolating moving least-squares (IMLS) method is discussed. The interpolating moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. Then we apply the IMLS method for the numerical solution of Volterra–Fredholm integral equations, and finally some examples are given to show the accuracy and applicability of the method.

A reduced-order variational multiscale interpolating element free Galerkin technique based on proper orthogonal decomposition for solving Navier–Stokes equations coupled with a heat transfer equation: Nonstationary incompressible Boussinesq equations

In the recent decade, meshless methods have been handled for solving some PDEs due to their easiness. One of the most efficient meshless methods is the element free Galerkin (EFG) method. The test and trial functions of the EFG are based upon the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG (VMEFG) procedure. In the current article, the shape functions of interpolating moving least squares (IMLS) approximation are applied to the variational multiscale EFG technique to numerical study the Navier–Stokes equations coupled with a heat transfer equation such that this model is well-known as two-dimensional nonstationary Boussinesq equations. In order to reduce the computational time of simulation, we employ a reduced order model (ROM) based on the proper orthogonal decomposition (POD) technique. In the current paper, we developed a new reduced order model based on the meshless numerical procedure for solving an important model in fluid mechanics. To illustrate the reduction in CPU time as well as the efficiency of the proposed method, we investigate two-dimensional cases.

Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

A Study on Stable Regularized Moving Least-Squares Interpolation and Coupled with SPH Method

The smoothed particle hydrodynamics (SPH) method has been popularly applied in various fields, including astrodynamics, thermodynamics, aerodynamics, and hydrodynamics. Generally, a high-precision interpolation is required to calculate the particle physical attributes and their derivatives for the boundary treatment and postproceeding in the SPH simulation. However, as a result of the truncation of kernel function support domain and irregular particle distribution, the interpolation using conventional SPH interpolation experiences low accuracy for the particles near the boundary and free surface. To overcome this drawback, stable regularized moving least-squares (SRMLS) method was introduced for interpolation in SPH. The surface fitting studies were performed with a variety of polyline bases, spatial resolutions, particle distributions, kernel functions, and support domain sizes. Numerical solutions were compared with the results using moving least-squares (MLS) and three SPH methods, including CSPH, K2SPH, and KGFSPH, and it was found that SRMLS not only has nonsingular moment matrix, but also obtains high-accuracy result. Finally, the capability of the algorithm coupled with SRMLS and SPH was illustrated and assessed through several numerical tests.

Analysis and application of the interpolating element free Galerkin (IEFG) method to simulate the prevention of groundwater contamination with application in fluid flow

We develop a meshless numerical procedure to simulate the groundwater equation (GWE). The used technique is based on the interpolating element free Galerkin (IEFG) method. The interpolating moving least squares (IMLS) approximation produces a set of functions such that they are well-known as “shape functions”. The IEFG technique employs the shape functions of IMLS approximation. The shape functions of IMLS approximation vanish on the boundary and also they satisfy the property of the Kronecker Delta function. Thus, Dirichlet boundary conditions can be exactly imposed. In this paper, we check the unconditional stability and convergence of the proposed numerical scheme based on the energy method. The numerical results confirm the theoretical analysis.

An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

Purpose This paper aims to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability and convergence of the numerical solutions. Design/methodology/approach In the time-fractional diffusion equation, the time fractional derivatives are approximated by L1 method, and the shape functions are constructed by the interpolating moving least-squares (IMLS) method. The final system equations are obtained by using the Galerkin weak form. Because the shape functions have the interpolating property, the unknowns can be solved by the iterative method after imposing the essential boundary condition directly. Findings Both theoretical and numerical results show that the IEFG method for the time-fractional diffusion equation has high accuracy. The stability of the fully discrete scheme of the method on the time step is stable unconditionally with a high convergence rate. Originality/value This work will provide an interpolating meshless method to study the numerical solutions of the time-fractional diffusion equation using the IEFG method.

Isomerism Effects in the Collisional Excitation of Cyanoacetylene by Molecular Hydrogen

Rotational excitation of the interstellar HC2NC and HNC3 molecules, two isomers of HC3N, induced by collisions with H2 is investigated at low collision energy using a quantum time-independent approach. The scattering calculations are based on new high-level ab initio four-dimensional (4D) potential energy surfaces (PESs) computed at the explicitly correlated coupled cluster with single, double, and perturbative triple excitations [CCSD(T)-F12b] level of theory. The method of interpolating moving least squares (IMLS) was used to construct 4D analytical PESs. Rotationally inelastic cross sections among the low-lying rotational levels of HC2NC and HNC3 were obtained using a pure quantum close-coupling approach for total energies up to ∼100 cm–1. The corresponding thermal rate coefficients were computed for temperatures ranging from 1 to 20 K. Propensity rules in favor of even Δj1 transitions were found for both HC2NC and HNC3 in collisions with para-H2(j2 = 0), with j1 being the rotational level of HC2NC and HNC3 molecules. The new rate coefficients were compared to previously published HC3N–para-H2(j2 = 0) rate coefficients. As expected, differences were found, especially for the rate coefficients corresponding to Δj1 = 1 transitions. Such a comparison confirms the importance of having specific collisional data for the different isomers of a molecule. The new rate coefficients will be crucial to improve the estimation of the HC3N/HC2NC/HNC3 abundance ratio in the interstellar medium.

An improved mixed Lagrangian–Eulerian (IMLE) method for modelling incompressible Navier–Stokes flows with CUDA programming on multi-GPUs

In this study, a GPU-accelerated improved mixed Lagrangian–Eulerian (IMLE) method is proposed to solve the three-dimensional incompressible Navier–Stokes equations. To improve the prediction accuracy, the proposed IMLE method approximates the total derivative term in Lagragian sense, and the spatial derivative terms are approximated on Eulerian coordinates. Transfer of data from Lagrangian particles to data on Eulerian grids is accurately carried out by adopting moving least squares (MLS) interpolation method. The velocity-pressure decoupling issue is overcome by adopting pressure-free projection method in which the pressure field is calculated by solving a pressure Poisson equation (PPE). It is noted that the MLS interpolation is time consuming since this procedure belongs to a pointwise scheme in which a local matrix equation shall be solved on each grid point. In addition, the discretized PPE forms a large sparse matrix and it is computationally intensive to solve by using the conjugate gradient (CG) method. Therefore, we are aimed to resort to CUDA- and OpenMP-programming means to accelerate the computation. In this study, the performance of the multiple GPUs code can reach up to 27 times faster with respect to multi-threads CPU performance.

The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems

In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is applied to obtain the approximation function. The penalty method is applied to impose the displacement boundary condition, and Galerkin weak form of elastic large deformation problems based on total Lagrange formulation is used to form the final equations which is solved with the Newton–Raphson iteration method, then the improved element-free Galerkin (IEFG) method based on a nonsingular weight function for elastic large deformation problems is presented. The IMLS method can overcome the disadvantage of singular weight functions in the traditional MLS method, then the IEFG method in this paper has high computational accuracy and efficiency, which are shown by numerical examples of elastic large deformation problems. And the influences of the weight functions, scale parameter of influence domain, step number and penalty factor on the numerical results are discussed.

Time‐varying magnetic field analysis using an improved meshless method based on interpolating moving least squares

In this study, analysis of 2D time-varying magnetic field using an improved meshless method based on interpolating moving least squares (IMLS) is proposed. In this study, solving the time-domain magnetic field equation in a meshless method carried out by selection of magnetic field intensity as a variable instead of magnetic vector potential which is widely used as a variable. Using this approach, the post-processing process will be fast and the boundary condition is implemented simply. For performance evaluation of the proposed method in eddy current analysis, two other meshless techniques, i.e. moving least squares (MLS) and radial point interpolation method (RPIM) have been considered. The results of improved IMLS are compared with MLS, RPIM and finite element method (FEM) results. Verification of improved meshless results is performed by FEM.

Accelerating multi‐dimensional interpolation using moving least‐squares on the GPU

SummaryThis paper focuses on designing and implementing parallel Moving Least Squares (MLS) interpolation algorithms by exploiting the Graphics Processing Unit (GPU) for the usage in Meshfree methods. The MLS method is an approach for scattered points' approximation / interpolation, which is commonly employed as the shape functions in various Meshfree methods. To improve the computational efficiency in building stiffness matrices in Meshfree methods, we are specifically interested in parallelizing the MLS interpolation on the GPU. The accelerated Meshfree methods can be employed to numerically analyze large deformations of soil and rock masses such as the landslides. In this paper, we first introduce our previously proposed method for finding the k nearest neighboring points located within the local region of the interest point in MLS. We then develop one sequential and three parallel implementations for each of three variations of the MLS interpolation, including the serial implementation, the parallel implementation on the multi‐core CPU, the parallel implementation on a single GPU, and the parallel implementation on multi‐GPUs. To evaluate the computational performance of our GPU implementations, five groups of benchmark tests are conducted in both two‐dimensions and three‐dimensions. We observe that our GPU implementations can achieve satisfied speedups over the sequential CPU implementation for varied sizes of testing data. To benefit the community, all source code and testing data related to the presented parallel MLS interpolation are publicly available.

Upon Element-Free Galerkin Method and its Using in Static and Dynamic Structure Analysis

Abstract This paper presents, in a synthetically way, the fundamentals of the element-free Galerkin (EFG) method - a meshfree method - under development but with many capabilities for solving complex problems in mechanical engineering, like impact problems etc. For interpolation, the EFG method uses moving least-squares (MLS) interpolants in curve and surface fitting. Unlike other interpolants, the MLS interpolants do not pass through the data because the Dirac function properties are not available. This aspect could be a disadvantage of the EFG method but next to it, there are many advantages. Upon these issues a discussion exists in this paper. Finally, some applications of the EFG method are presented referring to static and dynamic analysis of structures. The examples and conclusions can be useful for knowing and using of the EFG method

The Improved Element-Free Galerkin Method Based on the Nonsingular Weight Functions for Inhomogeneous Swelling of Polymer Gels

In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is used to construct the approximation function, the weak form of the problem of inhomogeneous swelling of polymer gels is used to obtain the final discretized equations, and penalty method is applied to impose the displacement boundary condition, then an improved element-free Galerkin (IEFG) method for the problem of the inhomogeneous swelling of polymer gels is presented. Three selected examples of inhomogeneous swelling of polymer gels solved with the IEFG method are given in this paper. The accuracy of the numerical solutions of the IEFG method are discussed by using different weight functions, penalty factor, scale parameter of influence domain, node distribution and step number. Numerical results of the IEFG method for inhomogeneous swelling of polymer gels show that this method has great precision, and it can solve large deformation problems of polymer gels effectively.