Moving Least-squares Approximation Research Articles

Local Kronecker delta property of the MLS approximation and feasibility of directly imposing the essential boundary conditions for the EFG method

The element-free Galerkin (EFG) method is a promising method for solving many engineering problems. Because the shape functions of the EFG method obtained by the moving least-squares (MLS) approximation, generally, do not satisfy the Kronecker delta property, special techniques are required to impose the essential boundary conditions. In this paper, it is proved that the MLS shape functions satisfy the Kronecker delta property when the number of nodes in the support domain is equal to the number of the basis functions. According to this, a local Kronecker delta property, which is satisfying the Kronecker delta property only at boundary nodes, can be obtained in one- and two-dimension. This local Kronecker delta property is an inherent property of the one-dimensional MLS shape functions and can be obtained for the two-dimensional MLS shape functions by reducing the influence domain of each boundary node to weaken the influence between them. The local Kronecker delta property provides the feasibility of directly imposing the essential boundary conditions for the EFG method. Four numerical examples are computed to verify this feasibility. The coincidence of the numerical results obtained by the direct method and Lagrange multiplier method shows the feasibility of the direct method.

Identification of elastic properties from full-field measurements: a numerical study of the effect of filtering on the identification results

The use of full-field displacement measurements in mechanical testing has increased dramatically over the last two decades. This is a result of the very rich information they provide, which is enabling new possibilities for the characterization of material constitutive parameters for inhomogeneous tests often based upon inverse approaches. Nonetheless, the measurement errors limit the accuracy of the identification of the constitutive parameters and their possible spatial resolution. The question addressed by this work is the following: can a filtering of the displacement measurement improve the results of the identification of elastic properties? The discussion is based on the study of a numerical example where the elastic parameters of an elastic structure with inhomogeneous properties are sought from synthetic data representative of in-plane full-field data. The displacement data are first filtered through a diffuse approximation algorithm, based on a moving least-squares approximation. Then, the identification of the elastic parameters is performed by an inverse approach based on the minimization of a cost function, defined as the least-squares gap between the experimental data and their numerical counterpart (finite element model updating). Within this framework, a first-order analysis is proposed in order to characterize the errors in the identified parameters, the measurement error characteristics being known. Results from raw and filtered displacement data are compared and discussed, filtering improving the identification for lower spatial resolution. The choice of the norm to define the gap between the experiment and the calculation is also discussed. For practical use and to take advantage of the proposed first-order methodology, two different ways can be considered: applying the methodology to a numerical example, representative of the experimental setup, to determine whether or not a filtering is valuable, and estimating the uncertainties of the identified parameters at the end of the identification process of an experimental characterization.

A Kriging interpolation-based boundary face method for 3D potential problems

Abstract In this paper, a new implementation of the boundary face method (BFM) is presented and developed for solving 3D potential problems. The BFM is implemented directly based on the boundary representation data structure for geometry modeling to eliminate geometry errors. This study combines the BFM with Kriging interpolation method and the corresponding formulae are derived. The Kriging interpolation is applied instead of the traditional moving least squares (MLS) approximation to overcome the lack of Kronecker delta function property, then essential boundary conditions can be imposed directly and easily. Several numerical examples with different geometry and boundary conditions are presented to illustrate the performance of the present method. The comparisons of accuracy between MLS approximation and Kriging interpolation are studied.

The MLPG with improved weight function for two-dimensional heat equation with non-local boundary condition

In this paper, a meshless local Petrov–Galerkin (MLPG) method is presented to treat the heat equation with the Dirichlet, Neumann, and non-local boundary conditions on a square domain. The Moving Least Square (MLS) approximation is a classical MLS method, in which the Gaussian weight function is the most common shape function. However, shape functions for the classical MLS approximation lack the Kronecker delta function property. Thus in this method, the boundary conditions cannot have a penalty parameter imposed easily and directly. In the method we choose a weight function that leads to the MLS approximation shape functions approximating the Kronecker delta function property, and nodes on the Dirichlet boundary conditions, which enables a direct application of essential boundary conditions without the additional numerical method. The improved weight function in MLS approximation has been successfully implemented in solving the diffusion equation problem. Two test problems are presented to verify the efficiency, easiness and accuracy of the method. Also Ne and root mean square errors are obtained to show the convergence of the method.

Virtual boundary meshless least square integral method with moving least squares approximation for 2D elastic problem

Abstract Moving least squares approximation (MLSA) has been widely used in the meshless method. The singularity should appear in some special arrangements of nodes, such as the data nodes lie along straight lines and the distances between several nodes and calculation point are almost equal. The local weighted orthogonal basis functions (LWOBF) obtained by the orthogonalization of Gramm–Schmidt are employed to take the place of the general polynomial basis functions in MLSA. In this paper, MLSA with LWOBF is introduced into the virtual boundary meshless least square integral method to construct the shape function of the virtual source functions. The calculation format of virtual boundary meshless least square integral method with MLSA is deduced. The Gauss integration is adopted both on the virtual and real boundary elements. Some numerical examples are calculated by the proposed method. The non-singularity of MLSA with LWOBF is verified. The number of nodes constructing the shape function can be less than the number of LWOBF and the accuracy of numerical result varies little.

Stress intensity factor analyses of three-dimensional interface cracks using tetrahedral finite elements

A numerical method is proposed for evaluating the stress intensity factors of a three-dimensional bimaterial interfacial crack using tetrahedral finite elements. This technique is based on the M1-integral method and employs the moving least-squares approximation. Stress or strain in the M1-integral equation is automatically approximated from the nodal displacements obtained by the finite element analysis using the moving least-squares method. Therefore, the presented method needs no elemental information from the finite element analysis. In this study, stress intensity factor analyses of some typical three-dimensional interface crack problems using the tetrahedral finite elements are demonstrated.

LEAST-SQUARES MESHFREE COLLOCATION METHOD

A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.

Analyzing interaction between coplanar square cracks using an efficient boundary element-free method

This paper examines the interaction between coplanar square cracks by combining the moving least-squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element-free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.

The improved element-free Galerkin method for three-dimensional wave equation

The paper presents the improved element-free Galerkin (IEFG) method for three-dimensional wave propagation. The improved moving least-squares (IMLS) approximation is employed to construct the shape function, which uses an orthogonal function system with a weight function as the basis function. Compared with the conventional moving least-squares (MLS) approximation, the algebraic equation system in the IMLS approximation is not ill-conditioned, and can be solved directly without deriving the inverse matrix. Because there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the IEFG method than in the element-free Galerkin method. Thus, the IEFG method has a higher computing speed. In the IEFG method, the Galerkin weak form is employed to obtain a discretized system equation, and the penalty method is applied to impose the essential boundary condition. The traditional difference method for two-point boundary value problems is selected for the time discretization. As the wave equations and the boundary-initial conditions depend on time, the scaling parameter, number of nodes and the time step length are considered for the convergence study.

Slosh Dynamics of Liquid-Filled Rigid Containers: Two-Dimensional Meshless Local Petrov-Galerkin Approach

This paper presents some of the interesting effects arising from the nonlinear motion of the liquid-free surface, due to sloshing, in a partially filled rigid container subjected to forced excitation. A two-dimensional meshless local Petrov-Galerkin method is used for the numerical simulation of the problem. A local symmetric weak form (LSWF) for nonlinear sloshing of liquid is developed, and a truly meshless method, based on LSWF and moving least squares (MLS) approximation, is presented for the solution of the Laplace equation with the requisite time-dependent boundary conditions. The MLS approximation with linear basis as well as Gaussian type weight function is employed in the computation. At every instant of time, velocity potential is computed at each node and the nodal positions are updated. The choice of a suitable scaling parameter value in the MLS approximation is discussed in this study. The effectiveness of the developed algorithm is demonstrated through a few numerical examples. The accuracy and stability of the numerical method introduced are verified from the comparison with the existing reference solutions.

Analyzing modified equal width (MEW) wave equation using the improved element-free Galerkin method

Abstract The element-free Galerkin (EFG) method is a promising method for solving partial differential equations in which trial and test functions employed in the discretization process result from moving least-squares (MLS) approximation. In this paper, by employing the improved moving least-squares (IMLS) approximation, we derive formulae for an improved element-free Galerkin (IEFG) method for the modified equal width (MEW) wave equation. A variation of the method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method. Therefore, the IEFG method may result a better computing speed. In this paper, the effectiveness of the IEFG method for modified equal width (MEW) wave equation is investigated by numerical examples.

A contribution on the optimization strategies based on moving least squares approximation for sheet metal forming design

Computer-aided procedures to design and optimize forming processes are, nowadays, crucial research topics since industrial interest in costs and times reduction is always increasing. Many researchers have faced this research challenge with various approaches. Response surface methods (RSM) are probably the most known approaches since they proved their effectiveness in the recent years. With a peculiar attention to sheet metal forming process design, RSM should offer the possibility to reduce the number of numerical simulations which in many cases means to reduce design times and complexity. Actually, the number of direct problems (FEM simulations) to be solved in order to reach good function approximations by RSM is a key aspect of their application in sheet metal forming operations design. In this way, the possibility to build response surfaces basing on moving least squares approximations (MLS) by utilizing a moving and zooming region of interest can be considered a very attractive methodology. In this paper, MLS is utilized to solve two optimization problems for sheet metal forming processes. The influence on the optimization results was analyzed basing on MLS peculiarities. The idea is to utilize these peculiarities and make the MLS approximation as flexible as possible in order to reduce the computational effort of an optimization strategy. An innovative optimization method is proposed and the results show it is possible to strongly reduce the computational effort of sheet metal forming processes optimization. In particular, the advantages, in terms of computational effort reduction, with respect to classical RSM approaches have been demonstrated and quantified.

Extraction of ridge and valley lines from unorganized points

Given an unstructured point set, we use an MLS (moving least-squares) approximation to estimate the local curvatures and their derivatives at a point by means of an approximating surface. Then, we compute neighbor information using a Delaunay tessellation. Ridge and valley points can then be detected as zero-crossings, and connected using curvature directions. We demonstrate our method on several large point-sampled models, rendered by point-splatting, on which the ridge and valley lines are rendered with line width determined from curvatures.

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L2-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.

Smoothed Galerkin methods using cell-wise strain smoothing technique

A smoothed Galerkin method (SGM) using cell-wise strain smoothing operation is formulated in this paper. In present method, the field nodes can be divided into two types: boundary field nodes and interior field nodes. The background cells are divided into SC smoothing cells, and the strains in each smoothing cell are obtained using a gradient smoothing technique which can avoid evaluating derivatives of shape functions at integration point. The field variables of boundary points are approximated using linear interpolation of neighbour boundary field nodes, and the shape functions possess the Kronecker Delta property and facilitate the impositions of essential boundary conditions. The field variables of interior points are approximated using moving least-squares approximation using the support field nodes around them. A number of numerical examples are studied and confirm the significant features of the present methods: (1) can pass the standard patch test; (2) can easily impose essential boundary conditions as those in finite element method; (3) can avoid evaluating derivatives of shape functions; (4) no numerical parameter is required.

Elastoplastic Large Deformation Using Meshless Integral Method

In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integral equation is obtained from the weak form of elastoplasticity based on Green-Naghdi’s theory over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation. Green-Naghdi’s theory starts with the additive decomposition of the Green-Lagrange strain into elastic and plastic parts and considers aJ2elastoplastic constitutive law that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. A simple, generalized collocation method is proposed to enforce essential boundary conditions straightforwardly and accurately, while natural boundary conditions are incorporated in the system governing equations and require no special handling. The solution algorithm for large deformation analysis is discussed in detail. Numerical examples show that meshless integral method with large deformation is accurate and robust.

The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems

In this paper, the moving least-squares (MLS) approximation and the interpolating moving least-squares (IMLS) method proposed by Lancaster are discussed first. A new method for deriving the MLS approximation is presented, and the IMLS method is improved. Compared with the IMLS method proposed by Lancaster, the shape function of the improved IMLS method in this paper is simpler so that the new method has higher computing efficiency. Then combining the shape function of the improved IMLS method with Galerkin weak form of the potential problem, the interpolating element-free Galerkin (IEFG) method for the two- dimensional potential problem is presented, and the corresponding formulae are obtained. Compared with the conventional element-free Galerkin (EFG) method, the boundary conditions can be applied directly in the IEFG method, which makes the computing efficiency higher. For the purposes of demonstration, some selected numerical examples are solved using the IEFG method.

Meshless Galerkin algorithms for boundary integral equations with moving least square approximations

In this paper, we first give error estimates for the moving least square (MLS) approximation in the H k norm in two dimensions when nodes and weight functions satisfy certain conditions. This two-dimensional error results can be applied to the surface of a three-dimensional domain. Then combining boundary integral equations (BIEs) and the MLS approximation, a meshless Galerkin algorithm, the Galerkin boundary node method (GBNM), is presented. The optimal asymptotic error estimates of the GBNM for three-dimensional BIEs are derived. Finally, taking the Dirichlet problem of Laplace equation as an example, we set up a framework for error estimates of the GBNM for boundary value problems in three dimensions.

Stochastic subset optimization incorporating moving least squares response surface methodologies for stochastic sampling

The design of engineering systems under probabilistic model uncertainty using stochastic simulation techniques is discussed in this paper. A recently developed algorithm, called Stochastic Subset Optimization (SSO), is adopted for solution of the associated optimization problem. SSO is based on formulation of an augmented stochastic design problem and relies on simulation of samples from an auxiliary probability density function to establish a sensitivity analysis for the design variables. This stochastic sampling requires repeated evaluation of system model response and corresponds to the computationally most intensive part of the algorithm. A Moving Least Squares (MLS) response surface approximation is introduced here for reduction of this computational burden. The paper focuses on the efficient integration of the MLS approximation within SSO. Specifically, adaptive selection of the MLS weights is proposed exploiting information directly available from SSO to quantify the importance of the uncertain model parameters and the design variables in affecting the system response. A measure based on relative information entropy concepts is introduced for this purpose and its efficient evaluation is examined in detail. The selection of support points for generating the response surface is also addressed, and a novel measure is finally introduced for evaluating the efficiency of the response surface approximation in terms of the stochastic sampling accuracy.

A meshless local moving Kriging method for two-dimensional solids

Abstract An improved meshless local Petrov–Galerkin method (MLPG) for stress analysis of two-dimensional solids is presented in this paper. The MLPG method based on the moving least-squares approximation is one of the recent meshless approaches. However, accurate imposition of essential boundary conditions in the MLPG method often presents difficulties because the MLPG shape functions does not possess the Kronecker delta property. In order to eliminate this shortcoming, this approach uses the moving Kriging interpolation instead of the traditional moving least-square approximation to construct the MLPG shape functions, and then, the Heaviside step function is used as the test function over a local sub-domain. In this method, the essential boundary conditions can be enforced as the FEM, no domain integration is needed and only regular boundary integration is involved. In addition, the sensitivity of several important parameters of the present method is mainly studied and discussed. Comparing with the original meshless local Petrov–Galerkin method, the present method has simpler numerical procedures and lower computation cost. The effectiveness of the present method for two-dimensional solids problem is investigated by numerical examples in this paper.