Moving Least Squares Research Articles

An interface shell element for coupling non-matching quadrilateral shell meshes

In this study, a novel interface shell element (ISE) is developed based on a variable-node element formulation to couple non-matching quadrilateral shell meshes. Shape functions for ISEs are explicitly presented in a polynomial form with the use of appropriate supports of weight functions in moving least square (MLS) approximation. Assumed natural strains in the form of the mixed interpolation of tensorial components (MITC) approach are employed to avoid the transverse shear locking when the thickness of shell tends to zero. Moreover, an assumed membrane strain field defined over quadrilateral subdomains subdividing an ISE is used to alleviate the membrane locking in curved ISEs. Numerical experiments show the effectiveness and efficiency of ISE for connecting dissimilar quadrilateral shell meshes at a common interface.

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.

Convergence rate for the moving least-squares learning with dependent sampling

We consider the moving least-squares (MLS) method by the regression learning framework under the assumption that the sampling process satisfies the α-mixing condition. We conduct the rigorous error analysis by using the probability inequalities for the dependent samples in the error estimates. When the dependent samples satisfy an exponential α-mixing, we derive the satisfactory learning rate and error bound of the algorithm.

The MLPG for crack analyses in composites with flexoelectricity effects

The meshless Petrov-Galerkin (MLPG) method is developed to analyse 2-D crack problems where the electric field and displacement gradients exhibit a size effect. The size-effect phenomenon in micro/nano electronic structures is described by the strain- and electric field-gradients. Both the electric intensity vector and strain gradients are considered in the constitutive equations of the material and the governing equations are derived with the corresponding boundary conditions using the variational principle. The coupled governing partial differential equations (PDE) for stresses and electric displacement field are satisfied in a local weak-form on small fictitious subdomains. All field quantities are approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, we obtain a system of algebraic equations for the nodal unknowns.

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.

Tube hydroforming optimization using a surrogate modeling approach and Genetic Algorithm

ABSTRACTThe quality of a product obtained by hydroforming process is influenced by the geometrical, material and process parameters. In this paper, to predict an acceptable T-shaped tube with minimum wall thickness variations, and accomplishes the industrial requirements, a methodology based on the coupling of three-dimensional finite element incremental simulation based on Explicit Dynamic approach and an automatic surrogate model are proposed. The surrogate model is based on an adaptive moving target zone, and both Moving Least Square (MLS) and the Kriging technique. The optimization results are presented and compared in term of efficiency. Five quality criteria are used, an objective function defining the thickness variation with four nonlinear constraints functions, to reduce the risk of necking and to fulfil the industrial requirements. The proposed approach will provide a numerical estimation of the “best” tool dimensions and ideal punch stroke in order to obtain a final feasible workpiece.

Modified moving least squares method for two-dimensional linear and nonlinear systems of integral equations

We extended the moving least squares (MLS) and modified moving least squares (MMLS) methods for solving two-dimensional linear and nonlinear systems of integral equations. This modification is proposed on the quadratic base functions $$(m=2)$$ by imposing additional terms based on the coefficients of the polynomial base functions. This approach prevents the singular moment matrix in the context of MLS based on meshfree methods. Additionally, finding the optimum value for the radius of the domain influence is an open problem for MLS-based methods. So an efficient algorithm is introduced for computing a suitable value of dilatation parameter to determine the radius of the support domain. This algorithm able to prevent the singular matrix which is an outcome of adverse selection of the radius of influence domain. In numerical examples are provided to enable us to compare MMLS method and standard MLS method by the new proposed algorithm. Comparing the errors of MMLS and MLS method determines the capability and accuracy of applied techniques to solve systems of integral equation problems. This indicates the advantage of the proposed method respect to MLS method.

Block-based adaptive mesh refinement for fluid–structure interactions in incompressible flows

In this study, an immersed boundary (IB) approach on the basis of moving least squares (MLS) interpolation is proposed for analyzing the dynamic response of a rigid body immersed in incompressible flows. An improved mapping strategy is proposed for a quick update of the signed distance field. A CIP-CSL (constraint interpolation profile - semi-Lagrangian) scheme with a compact stencil is adopted for the convective term in momentum equation. Fluid–structure interaction problems can be solved by either the weak or the strong coupling schemes according to the density ratio of the solid and fluid. This research is based on our previous studies on block-structured adaptive mesh refinement (AMR) method for incompressible flows (Liu & Hu, 2018). Present AMR-FSI solver is proved to be accurate and robust in predicting dynamics of VIV (vortex induced vibration) problems. The efficiency of the adaptive method is demonstrated by the 2D simulation of a freely falling plate with the comparison to other numerical methods. Finally, the freely falling and rising 3D sphere are computed and compared with corresponding experimental measurement.

Static analysis of functionally graded nanocomposite sandwich plates reinforced by defected CNT

In this paper, stress distribution and deflection in sandwich plates with functionally graded nanocomposite face sheets have been carried out by a first order shear deformation theory (FSDT) based mesh-free method. The sandwich plates are assumed resting on Winkler-Pasternak elastic foundation and the nanocomposite is reinforced by three types of defected carbon nanotubes (CNTs) which are compared with pristine CNT-reinforced nanocomposite. A multiscale modeling with molecular dynamics (MD) simulations in the nanoscale and micromechanical approach of Halpin-Tsai are used to calculate the elastic constants of the nanocomposite. In the mesh-free analysis, moving least squares (MLSs) shape functions are used to approximate displacement field and the transformation method is used for imposition of essential boundary conditions. The effects of Stone-Wales (SW) and vacancy defect configurations of CNTs, number of defects, CNT distribution and volume fraction, boundary conditions and geometric dimensions are investigated on the static analysis of the sandwich plates. It is observed that when CNT volume fraction is 5%, six vacancy defects in CNT configuration can increase the deflection of the sandwich plate up to 41.67%.

Advanced Implicit Meshless Approaches for the Rayleigh–Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative

To solve the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in a bounded domain is important in the research for diffusion processes. In this paper, novel implicit meshless approaches based on the moving least squares (MLS) approximation for spatial discretization and two different time discrete schemes, which are the first-order semi-discrete scheme and the second-order semi-discrete scheme for time, are developed for the numerical simulation of the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in a bounded domain. Based on these two time discretization schemes, the newly developed meshless approaches will have the first-order and the second-order accuracy in time, respectively. The stability and convergence of the implicit MLS meshless approaches are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approaches. It has found that the newly developed meshless approaches are accurate and convergent for fractional partial differential equations (FPDEs). Most importantly, the meshless approaches are robust for arbitrarily distributed nodes and complex domains.

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

Enhanced particle method with stress point integration for simulation of incompressible fluid-nonlinear elastic structure interaction

A fully-Lagrangian particle-based computational method is developed for simulation of incompressible Fluid, non-linear Structure Interaction (FSI) with incorporation of stress point integration (Randles and Libersky, 2005) to resolve instabilities related to zero-energy modes. Structural dynamics is founded on discretization of the divergence of stress according to Moving Least Squares (MLS) method. The stress point integration is incorporated in calculation of structural dynamics, resulting in a Dual Particle Dynamics (DPD) structure model (Randles and Libersky, 2005). A structure model based on nodal integration is also considered for comparison and simply referred to as MLS. The DPD and MLS structure models are coupled with an enhanced projection-based Moving Particle Semi-implicit (MPS) method as the fluid model, resulting in DPD–MPS and MLS–MPS FSI solvers, respectively. The enhanced performance of DPD with respect to MLS is first shown through a set of tests for structure model. Then the superior performance of DPD–MPS FSI solver with respect to MLS–MPS one is demonstrated through a set of FSI benchmark tests. The present study also presents a new algorithm for fluid–structure coupling via components of stress tensors in surface boundary stress points.

High order approximation to non-smooth multivariate functions

Common approximation tools return low-order approximations in the vicinities of singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f=g+r+ where g,r∈CM+1(Rn) and the function r+ is defined byr+(y)={r(y),r(y)≥00,r(y)<0,∀y∈Rn. Given scattered (or uniform) data points X⊂Rn, we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.

A Local Galerkin Integral Equation Method for Solving Integro-differential Equations Arising in Oscillating Magnetic Fields

The current work investigates a computational method to solve a class of integro-differential equations which describes the charged particle motion for certain configurations of oscillating magnetic fields. To handle the method, we first convert these types of integro-differential equations to two-dimensional Volterra integral equations of the second kind. Afterward, the solution of the mentioned Volterra integral equations can be estimated using the Galerkin method based on the moving least squares (MLS) approach constructed on scattered points. The MLS methodology is an effective technique for approximating an unknown function that involves a locally weighted least squares polynomial fitting. Since the scheme does not need any background meshes, it can be identified as a meshless method. The scheme is simple and effective to solve integro-differential equations and its algorithm can be easily implemented. Finally, numerical examples are included to show the validity and efficiency of the new technique.

Bending of FGM plates under thermal load: Classical thermoelasticity analysis by a meshless method

Abstract Thermal stresses, especially at the interface between two different materials, often play an important role in the failure of laminated composite structures. Thus, it is a strong motivation to replace laminated plate structures by FGM ones if possible. The functional gradation of material coefficients, however, yields new coupling effects between the in-plane deformation and bending modes. Therefore the study of behaviour of FGM plates under thermal loadings has become important. In this paper, the bending of thin and/or thick FGM plates under thermal load is considered within the classical theory of thermoelasticity. The variable material properties of plate (such as the Young's modulus, thermal expansion coefficient, etc.) are allowed to be continuous functions of the position. The governing equations which are given by the 4th order partial differential equations (PDE) are decomposed into the 2nd order PDEs in order to overcome the inaccuracy of approximation of high order derivatives of field variables. The strong form meshless formulations for solution of thin plate bending problem is developed in combination with Moving Least Squares (MLS) approximation scheme. The attention is paid to the study of the influence of various parameters of gradations of material coefficients on bending of plates.

Multiview point clouds denoising based on interference elimination

Newly emerging low-cost depth sensors offer huge potentials for three-dimensional (3-D) modeling, but existing high noise restricts these sensors from obtaining accurate results. Thus, we proposed a method for denoising registered multiview point clouds with high noise to solve that problem. The proposed method is aimed at fully using redundant information to eliminate the interferences among point clouds of different views based on an iterative procedure. In each iteration, noisy points are either deleted or moved to their weighted average targets in accordance with two cases. Simulated data and practical data captured by a Kinect v2 sensor were tested in experiments qualitatively and quantitatively. Results showed that the proposed method can effectively reduce noise and recover local features from highly noisy multiview point clouds with good robustness, compared to truncated signed distance function and moving least squares (MLS). Moreover, the resulting low-noise point clouds can be further smoothed by the MLS to achieve improved results. This study provides the feasibility of obtaining fine 3-D models with high-noise devices, especially for depth sensors, such as Kinect.

An a posteriori, efficient, high-spectral resolution hybrid finite-difference method for compressible flows

A high-order hybrid method consisting of a high-accurate explicit finite-difference scheme and a Weighted Essentially Non-Oscillatory (WENO) scheme is proposed in this article. Following this premise, two variants are outlined: a hybrid made up of a Finite Difference scheme and a compact WENO scheme (CRWENO 5), and a hybrid made up of a Finite Difference scheme and a non-compact WENO scheme (WENO 5). The main difference with respect to similar schemes is its a posteriori nature, based on the Multidimensional Optimal Order Detection (MOOD) method. To deal with complex geometries, a multi-block approach using Moving Least Squares (MLS) procedure for communication between meshes is used. The hybrid schemes are validated with several 1D and 2D test cases to illustrate their accuracy and shock-capturing properties.

Numerical treatment for solving two-dimensional space-fractional advection–dispersion equation using meshless method

Space-fractional advection–dispersion equation (SFADE) can describe particle transport in a variety of fields more accurately than the classical models of integer-order derivative. Because of nonlocal property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model, and few have been reported in the literature. In this paper, a numerical analysis of the two-dimensional SFADE is carried out by the element-free Galerkin (EFG) method. The trial functions for the SFADE are constructed by the moving least-square (MLS) approximation. By the Galerkin weak form, the energy functional is formulated. Employing the energy functional minimization procedure, the final algebraic equations system is obtained. The Riemann–Liouville operator is discretized by the Grünwald formula. With center difference method, EFG method and Grünwald formula, the fully discrete approximation schemes for SFADE are established. Comparing with exact results and available results by other well-known methods, the computed approximate solutions are presented in the format of tables and graphs. The presented results demonstrate the validity, efficiency and accuracy of the proposed techniques. Furthermore, the error is computed and the proposed method has reasonable convergence rates in spatial and temporal discretizations.

A parametric study of the MLPG method for thermo-mechanical solidification analysis

Based on the meshless local Petrov–Galerkin (MLPG) method, a thermo-elasto-plastic analysis of solidification problem is presented. The effect of significant parameters of the MLPG method, including the size and shape of sub-domain and support domain, nodal arrangement, nodal density and Gaussian points on the solution accuracy of the problems is investigated to determine their optimal values. The local weak forms are derived by considering a Heaviside step function as the test function. To interpolate the solution variables, the moving least-squares (MLS) approximation is applied. Using the effective heat capacity method, thermal analysis of the solidification process is performed. The von-Mises yield criterion and isotropic hardening model are employed for the elasto-plastic behavior, and material parameters are assumed to be temperature-dependent. To demonstrate the capability of the present method in solving solidification problems, the obtained results have been compared with the analytical and accurate finite element method solutions.

Effect of thermal gradient load on thermo-elastic vibrational behavior of sandwich plates reinforced by carbon nanotube agglomerations

In this study, we investigated the effect of thermal gradient load on natural frequencies of sandwich plates with polymer-based nanocomposite face sheets reinforced by functionally graded (FG) single-walled carbon nanotubes (SWCNTs) agglomerations. Volume fractions and agglomerations of CNTs change across the thickness of each nanocomposite face sheet and temperature-dependent properties of polymer/CNT composite can be estimated with the Eshelby-Mori-Tanaka method. First-order shear deformation theory and a moving least square (MLS) shape function based mesh-free method have also been developed for free vibration and steady state thermal analysis on sandwich plates on two-parameter elastic foundations. We investigated the effects of orientation, aggregation, volume fraction and distribution of CNT. We also studied the effects of essential boundary conditions, sandwich dimensions, and elastic foundation coefficients on frequency behavior of sandwich plates.