Moving Least-squares Method Research Articles

Parabolic-cylindrical moving least squares surfaces

Moving least squares (MLS) surface approximation is a popular tool for the processing and reconstruction of non-structured and noisy point clouds. This paper introduces a new variant improving the approximation quality when the underlying surface is assumed to be locally developable, which is often the case in point clouds coming from the acquisition of manufactured objects. Our approach follows Levin׳s classical MLS procedure: the point cloud is locally approximated by a bivariate quadratic polynomial height-field defined in a local tangent frame. The a priori developability knowledge is introduced by constraining the fitted polynomials to have a zero-Gaussian curvature leading to the actual fit of the so-called parabolic cylinders. When the local developability assumption cannot be made unambiguously, our fitted parabolic cylinders seamlessly degenerate to linear approximations. We show that our novel MLS kernel reconstructs more locally developable surfaces than previous MLS methods while being faithful to the data.

On the study of elastic properties of CNT-reinforced composites based on element-free MLS method with nanoscale cylindrical representative volume element

This paper presents a micromechanical element-free method to study elastic properties of carbon nanotubes (CNTs) reinforced composites. A three-dimensional nanoscale cylindrical representative volume element (RVE) has been considered in the analysis, in which CNT and matrix are both modeled as elastic materials based on continuum mechanics. The proposed model is used to investigate the behaviors of CNT-reinforced composites with respect to their effective mechanical properties. The relevant formulas to extract the effective material constants for three loading cases are derived in detail, based on the elasticity theory. Besides, the meshless method based on the moving least-squares (MLS) approximation and element-free Galerkin (EFG) method is implemented to evaluate the established model. The computed results are compared with the available solutions obtained from other analytical and numerical methods. Moreover, these results show that the presence of CNTs can significantly improve the longitudinal and transverse characteristics of the composite materials.

Force identification of dynamic systems using virtual work principle

One of the key inverse problems for estimating dynamic forces acting on a structure is to determine the force expansion and the corresponding solving method. This paper presents a moving least square (MLS) method for fitting dynamic forces, which improves the existing traditional methods. The simulation results show that the force expansion order has a tiny effect on the types of forces, which indicates the MLS method׳s excellent ability for local approximation and noise immunity as well as good fitting function. Then, the differential equation of motion for the system is transformed into an integral equation by using the virtual work principle, which can eliminate the structural acceleration response without introducing the calculation error. Besides, the transformation derives an expression of velocity by integrating by parts, which diminishes the error propagation of the velocity. Hence, the integral equation of motion for the system has a strong constraint to noise with zero mean value. Finally, this paper puts forward an optimization method to solve the equation. The numerical stability can be enhanced as the matrix inversion calculation is avoided. Illustrative examples involving different types of forces demonstrate that the transformation of the differential equation proposed through virtual work principle can eliminate interference efficiently and is robust for dynamic calculation.

Adaptive response surface based efficient Finite Element Model Updating

The requirement of repeated evaluation of structural responses in typical sensitivity based Finite Element Model Updating (FEMU) procedure limits its popular applications for large structures. The least-squares method (LSM) based response surface method (RSM) is applied as a potential alternative for responses approximations in iterative model updating procedure. However, the LSM is a major source of error in response prediction and the moving least-squares method (MLSM) is found to be more efficient in this regard. An attempt has been made in the present study to explore the effectiveness of MLSM based RSM in FEMU. A comparative assessment is performed between the MLSM based and the conventional LSM based RSM for model parameter updating. The comparative study is being illustrated with the help of two example problems using artificially generated input responses. It is generally observed that the MLSM based RSM identifies better than the LSM based approach.

An adaptive moving total least squares method for curve fitting

The moving least squares (MLS) method and the moving total least squares (MTLS) method have been developed to deal with the measured data contaminated with random error. The local approximants of MLS method only take into account the error of dependent variable, whereas MTLS method considers the errors of all the variables, which determines the local approximants in the sense of the total least squares. MTLS method is more reasonable than MLS method for dealing with errors-in-variables (EIV) model. But because of the weight function with compact support, it is complicated to choose fitting method for the best performance. This paper presents an Adaptive Moving Total Least Squares (AMTLS) method for EIV model. In AMTLS method, a parameter λ associated with the direction of local approximants is introduced. MLS method and MTLS method can be considered as special cases of AMTLS method. Curve fitting examples are given to prove the better performance of AMTLS method than MLS method and MTLS method.

Moving Least-Squares Method for Interlaced to Progressive Scanning Format Conversion

In this paper, we introduce an efficient intra-field deinterlacing algorithm based on moving least squares (MLS). The MLS algorithm has proven successful for approximating scattered data by minimizing a weighted mean-square error norm. In order to estimate the value of the missing point using the given data, we utilize MLS to generate a generic local approximation function about this point. In the MLS method, we adopt trigonometric functions to approximate the local function. This method is compared to other benchmark algorithms in terms of peak signal-to-noise ratio and structural similarity objective quality measures and deinterlacing speed. It was found to provide excellent performance and the best quality-speed tradeoff among the methods studied.

Approximate multi-objective optimization using conservative and feasible moving least squares method: application to automotive knuckle design

The original version of the moving least squares method (MLSM) does not always ensure solution feasibility for nonlinear and/or non-convex functions in the context of meta-model-based approximate optimization. The paper explores a new implementation of MLSM that ensures the conservative feasibility of Pareto optimal solutions in non-dominated sorting genetic algorithm (NSGA-II)-based approximate multi-objective optimization. We devised a `conservative and feasible MLSM' (CF-MLSM) to realize the conservativeness and feasibility of multi-objective Pareto optimal solutions for both unconstrained and constrained problems. We verified the usefulness of our proposed approach by exploring strength-based sizing optimization of an automotive knuckle component under bump and brake loading constraints.

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.

An Improved Moving Least Squares Method for Curve and Surface Fitting

The moving least squares (MLS) method has been developed for the fitting of measured data contaminated with random error. The local approximants of MLS method only take the error of dependent variable into account, whereas the independent variable of measured data always contains random error. Considering the errors of all variables, this paper presents an improved moving least squares (IMLS) method to generate curve and surface for the measured data. In IMLS method, total least squares (TLS) with a parameterλbased on singular value decomposition is introduced to the local approximants. A procedure is developed to determine the parameterλ. Numerical examples for curve and surface fitting are given to prove the performance of IMLS method.

보수적 근사모델을 적용한 신뢰성 기반 강건 최적설계 방법

The methods of robust design optimization (RDO) and reliability-based robust design optimization (RBRDO) were implemented in the present study. RBRDO is an integrated method that accounts for the design robustness of an objective function and for the reliability of constraints. The objective function in RBRDO is expressed in terms of the mean and standard deviation of an original objective function. Thus, a multi-objective formulation is employed. The regressive approximate models are generated via the moving least squares method (MLSM) and constraint-feasible moving least squares method (CF-MLSM), which make it possible to realize the feasibility regardless of the multimodality/nonlinearity of the constraint function during the approximate optimization processes. The regression model based RBRDO is newly devised and its numerical characteristics are explored using the design of an actively controlled ten bar truss structure.

The regular hybrid boundary node method in the bending analysis of thin plate structures subjected to a concentrated load

This paper proposes a computational procedure based on the regular hybrid boundary node method (RHBNM) for solving the thin plate subjected to a concentrated load. The solution is decomposed into the particular solution arising from the concentrated load and the complementary solution for the homogeneous equation . In the solution procedure, the particular solution is first obtained by the fundamental solution. For the latter, we employ the RHBNM to solve. The RHBNM is a promising boundary type meshless method based on a modified variational principle and the moving least squares (MLS) approximation, and exploits the meshless attributes of the MLS and the reduced dimensionality advantages of the boundary element method (BEM). In this paper, a modified variational functional of the thin plate is developed, in which the independent variables are the generalized displacements and generalized tractions on the boundary and the lateral deflection in the domain. The MLS method is employed to approximate the boundary variables whereas the domain variables are interpolated by a linear combination of fundamental solutions of both the biharmonic equation and Laplace's equation . Some numerical tests illustrate the validity and efficiency of the present method. ► The meshless RHBNM is extended to thin plate subjected to a concentrated load. ► Particular solution arising from concentrated load is obtained by fundamental solution. ► The complementary solution for the homogeneous equation is obtained by the RHBNM. ► A modified variational is developed to derive the system of equations. ► The numerical tests show that the accuracy and convergence rate are very high.

A Study on the Novel Coefficient Modeling for a Skewed Permanent Magnet and Overhang Structure for Optimal Design of Brushless DC Motor

A novel coefficient modeling technique, using a coefficient estimation method, is introduced for a brushless DC (BLDC) motor with a skewed permanent magnet and overhang structure. To construct the novel coefficient, the 3-D FEM and a design of experiment (DOE) are utilized along with the moving least square method (MLSM). Also, to verify the accuracy of the novel coefficient modeling, the conventional coefficient modeling using LSM is constructed and compared. Then the novel coefficient is combined with an approximate model constructed by 2-D FEM for use with an optimal design algorithm. For the searching process to obtain an optimal point, the genetic algorithm (GA) is used. Three-dimensional FEM simulation is used to verify the accuracy of the novel coefficient for the optimal design.

MLS 차분법을 이용한 고체역학 문제의 동적해석

The MLS(Moving Least Squares) Difference Method is a numerical scheme that combines the MLS method of Meshfree method and Taylor expansion involving not numerical quadrature or mesh structure but only nodes. This paper presents an dynamic algorithm of MLS difference method for solving transient solid mechanics problems. The developed algorithm performs time integration by using Newmark method and directly discretizes strong forms. It is very convenient to increase the order of Taylor polynomial because derivative approximations are obtained by the Taylor series expanded by MLS method without real differentiation. The accuracy and efficiency of the dynamic algorithm are verified through numerical experiments. Numerical results converge very well to the closed-form solutions and show less oscillation and periodic error than FEM(Finite Element Method).

Role of Conservative Moving Least Squares Methods in Reliability Based Design Optimization: A Mathematical Foundation

The response surface method (RSM) and the moving least squares method (MLSM) are extensively used due to their computational efficiency in solving reliability based design optimization (RBDO) problems. Since traditional RSM and MLSM are described by second-order polynomials, approximated constraints are sometimes unable to ensure feasibility when highly nonlinear and/or nonconvex constraint functions are approximated in RBDO. We explore the development of a new MLSM based meta-model that ensures the constraint feasibility of an optimal solution in RBDO. A constraint-feasible MLSM (CF-MLSM) is devised to realize feasibility regardless of the multimodality/nonlinearity of the constraint function in all approximation processes as well as the variation of random characteristics in RBDO. The usefulness of the proposed approach is verified by examining a nonlinear function problem and an actively controlled structure problem.

A Meshless Method Based on the Improved Interpolating Moving Least-Squares Method for the Regularized Long Wave Equation

This paper presents an improved interpolating moving least-squares (IIMLS) method, in which orthogonal functions system is used as the basis functions. In the IIMLS method, the final algebra equation system is not ill-conditioned, and can be solved without obtaining the inverse matrix. Hence, the computing speed and efficiency are improved. Then based on the IIMLS method, a meshless method is presented for the numerical solution of the regularized long wave (RLW) equation, which can be used to describe phenomena with weak nonlinearity and dispersion waves. And a numerical example is given to confirm the IMLS method.

A Simplified Interpolating Moving Least-Squares Method and its Error Estimates

A disadvantage of the MLS approximation is that the shape function of this method does not satisfy the property of Kronecker Delta function. Thus developing an interpolating MLS approximation is very important. In this paper, the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas is discussed in detail and a simplified expression of the approximation function of the IMLS method is given. The simpler expression makes it more convenient to use this method. The error estimate of the approximation function also is discussed. And a numerical example is given to confirm the results.

구조 최적설계를 위한 다양한 근사 최적화기법의 적용 및 비교에 관한 연구

본 논문에서는 범프 및 브레이크 하중조건 하에서 자동차 넉클의 강도설계에 관한 다양한 회귀모델 기반근사최적화 기법 및 그 성능을 비교하고자 한다. 최적설계문제는 응력, 변형 및 진동주파수의 제한조건 하에서 중량을 최소화하여 설계변수인 단면치수가 결정되도록 정식화 된다. 비교 연구를 위해 사용된 근사화 기법은 순차적 근사최적화(SAO), 순차적 이점대각이차 근사최적화(STDQAO), 그리고 개선된 이동최소 자승법(MLSM) 기반 근사최적화 기법인 CF-MLSM 와 Post-MLSM 이다. SAO 와 STDQAO 적용을 위하여 상용프로세스통합 설계최적화(PIDO) 코드를 사용하였다. 본 연구에 적용한 MLSM 기반 근사최적화 기법들은 제한조건의 가용성을 보장할 수 있도록 새롭게 개발되었다. 다양한 근사최적화 기법에 의한 설계결과는 설계 해의 개선 및 수렴속도등 수치적 성능을 기준으로 실제 비근사최적화 결과와 비교검토 되었다.

Reliability-based design optimization of knuckle component using conservative method of moving least squares meta-models

This paper discusses reliability-based design optimization (RBDO) of an automotive knuckle component under bump and brake loading conditions. The probabilistic design problem is to minimize the weight of a knuckle component subject to stresses, deformations, and frequency constraints in order to meet the given target reliability. The initial design is generated based on an actual vehicle specification. The finite element analysis is conducted using ABAQUS, and the probabilistic optimal solutions are obtained via the moving least squares method (MLSM) in the context of approximate optimization. For the meta-modeling of inequality constraint functions, a constraint-feasible moving least squares method (CF-MLSM) is used in the present study. The method of CF-MLSM based RBDO has been shown to not only ensure constraint feasibility in a case where the meta-model-based RBDO process is employed, but also to require low expense, as compared with both conventional MLSM and non-approximate RBDO methods.

Comparison of improved meshless interpolation schemes for SPH method and accuracy analysis

In the smoothed particle hydrodynamics (SPH) method, a meshless interpolation scheme is needed for the unknown function in order to discretize the governing equation. A particle approximation method has so far been used for this purpose. Traditional particle interpolation (TPI) is simple and easy to do, but its low accuracy has become an obstacle to its wider application. This can be seen in the cases of particle disorder arrangements and derivative calculations. There are many different methods to improve accuracy, with the moving least square (MLS) method one of the most important meshless interpolation methods. Unfortunately, it requires complex matrix computing and so is quite time-consuming. The authors developed a simpler scheme, called higher-order particle interpolation (HPI). This scheme can get more accurate derivatives than the MLS method, and its function value and derivatives can be obtained simultaneously. Although this scheme was developed for the SPH method, it has been found useful for other meshless methods.

Moving least-square method in learning theory

Moving least-square (MLS) is an approximation method for data interpolation, numerical analysis and statistics. In this paper we consider the MLS method in learning theory for the regression problem. Essential differences between MLS and other common learning algorithms are pointed out: lack of a natural uniform bound for estimators and the pointwise definition. The sample error is estimated in terms of the weight function and the finite dimensional hypothesis space. The approximation error is dealt with for two special cases for which convergence rates for the total L 2 error measuring the global approximation on the whole domain are provided.