Moving Least-squares Method Research Articles

Determination of Accuracy and Usability of a SLAM Scanner GeoSLAM Zeb Horizon: A Bridge Structure Case Study

The presented paper focuses on testing the performance of a SLAM scanner Zeb Horizon by GeoSLAM for the creation of a digital model of a bridge construction. A cloud acquired using a static scanner Leica ScanStation P40 served as a reference. Clouds from both scanners were registered into the same coordinate system using a Trimble S9 HP total station. SLAM scanner acquisition was performed independently in two passes. The data acquired using the SLAM scanner suffered from relatively high noise. Denoising using the MLS (Moving Least Squares) method was performed to reduce noise. An overall comparison of the point clouds was performed on both the original and MLS-smoothed data. In addition, the ICP (Iterative Closest Point) algorithm was also used to evaluate local accuracy. The RMSDs of MLS-denoised data were approximately 0.02 m for both GeoSLAM passes. Subsequently, a more detailed analysis was performed, calculating RMSDs for several profiles of the construction. This analysis revealed that the deviations of SLAM data from the reference data did not exceed 0.03 m in any direction (longitudinal, transverse, elevation) which is, considering the length of the bridge of 133 m, a very good result. These results demonstrate a high applicability of the tested scanner for many applications, such as the creation of digital twins.

Radial basis point interpolation for strain field calculation in digital image correlation

In order to extract smooth and accurate strain fields from the noisy displacement fields obtained by digital image correlation (DIC), a point interpolation meshless (PIM) method with a radial basis function (RBF) is introduced for full-field strain calculation, which overcomes the problems of slow calculation speed and unstable matrix inverse calculation of the element-free Galerkin method (EFG). The radial basis point interpolation method (RPIM) with three different radial basis functions and the moving least squares (MLS) and pointwise least squares (PLS) methods are compared by analyzing and validating the strain fields with high-strain gradients in simulation experiments. The results indicate that the RPIM is nearly 80% more computationally efficient than the MLS method when a larger support domain is used, and the efficiency of the RPIM is nearly 26% higher than that of the MLS method when a smaller support domain is used; the strain calculation accuracy is slightly lower than that of the MLS method by 0.3–0.5%, but the stability of the calculation is significantly improved. In contrast with the PLS method, which is easily affected by the noise and the size of the strain calculation window, the RPIM is insensitive to the displacement noise and the size of the support domain and can obtain a similar calculation accuracy. The RPIM with multiquadric (MQ) radial basis functions performs well in balancing the computational accuracy and efficiency and is insensitive to shape parameters. The application cases show that the method can effectively compute the strain field at the crack tip, validating its applicability to the study of the plastic region at the crack tip. In conclusion, the proposed RPIM-based method provides an accurate, practical, and robust approach for full-field strain measurements.

Crashworthiness analysis and structural optimization of shrink tube under interference condition

This paper aims to study the crashworthiness of shrink tubes for rail vehicles under axial impact loading. A theoretical model of shrink tube for platform force prediction under interference fit conditions was proposed. The trolley crash test was used to experimentally study the structural energy absorption and characteristics during collision, and a corresponding finite element model was constructed and validated. The finite element model validated the accuracy of the theoretical model. The influence of structural parameters on the crashworthiness of shrink tube under interference conditions was then studied. Structural design variables, including the magnitude of interference (int_pre), thickness (T), long right-angle edge (tri_x) and short right-angle edge (tri_z) were sampled using a full factorial and a Latin hypercube design method. Based on the above samples, the moving least squares method (MLSM) was used to construct an approximate model of peak force (Fmax), specific energy absorption (SEA), and platform force (Fplt). The main effects analysis illustrated that the tri_z had the most significant effects on the peak force, specific energy absorption and platform force. To minimize Fmax and maximize SEA, the global response surface method (GRSM) was employed for parameter optimization. Finally, the optimized configurations were obtained with int_pre=0.067mm, T=10.003mm, tri_x=21.485mm, and tri_z=6.101mm. Compared to the initial design, the SEA value was increased by 16.41 %, while the Fmax value was decreased by 2.21%. The error of platform force derived from the optimal solution and theoretical prediction was 3.62%, which verified that the theoretical prediction model was credible within an acceptable range.

On the numerical solution of a population growth model of a species living in a closed system based on the moving least squares scheme

In this research paper, we introduce a numerical approach to solve a particular type of nonlinear integro-differential equations derived from Volterra's population model. This model characterizes the growth of a biological species in a closed system and includes an integral term to consider the influence of toxin accumulation on the species, along with the conventional terms found in the logistic equation. The proposed technique estimates the solution of integro-differential equations utilizing the discrete Galerkin scheme using the moving least squares (MLS) algorithm. The locally weighted least squares polynomial fitting, known as the MLS method, is a valuable approach for approximating unknown functions. Since the offered scheme does not require any cell structures, it can be known as a meshless local discrete Galerkin method. Moreover, we obtain the error estimate of the proposed approach. The validity and efficiency of the newly developed technique are assessed over several nonlinear integro-differential equations.

Analysis and Optimization of an Expansion Energy-Absorbing Anti-Crawler for Rail Vehicles

This paper describes the crashworthiness optimization of an intumescent energy-absorbing anti-crawler, which was applied to anti-crawling devices for rail vehicles. The energy absorption characteristics of the expansion-type energy-absorbing anti-crawler were studied experimentally, a finite element model (FEM) was established, and the finite element simulation results were verified with the experimental results. In this paper, the response of the expansion structure was predicted using a validated finite element model. Then, the effects of the variables (expansion tube thickness (T), friction coefficient (μ), and slope angle of conical mandrel (α)) on the response were sampled using the design-of-experiments (DOE) method, including a full factorial design and a central composite. Based on these samples, an alternative model was developed using the moving least-squares method (MLSM). Using the results from the full factorial design for main effects analysis, T was found to have the most significant effect on the average force (Favg), while α had the greatest effect on the specific energy absorption (SEA). The Favg, fracture strain, thickness, taper, and friction coefficient of the structure were used as constraints, and the multiobjective genetic algorithm (MOGA) method was used for parameter optimization to obtain a higher SEA. Finally, the best parameters (T = 5.76 mm, μ = 0.178, α = 25°) with an SEA value of 36.52 kJ/kg were obtained. The SEA value was increased by 31.70% compared to the initial results.

Numerical Solution of Shrödinger Equations Based on the Meshless Methods

In this work, two-dimensional time-dependent quantum equation problems are studied. We introduce a numerical algorithm for solving the two-dimensional nonlinear complex quantum system with MLS and FDM methods. An efficient and accurate computational algorithm based on both, the moving least squares (MLS) and the finite difference (FDM) methods is proposed for solving it. The results demonstrate that the proposed algorithm is a robust algorithm with good accuracy. This is developed on MLS and FDM methods using numerical simulation for solving these kind of problems.

An Improved Element-Free Galerkin Method Based on the Dimension Splitting Moving Least-Squares Method for 2D Potential Problems in Irregular Domains

By introducing the dimension splitting (DS) method into the moving least-squares (MLS) approximation, a dimension splitting moving least-squares (DS-MLS) method is proposed in this paper. In the DS-MLS method, the operator splitting and independent variable splitting of the DS method are used to reduce the dimension, thereby reducing the computational complexity of the matrix. The shape function of the DS-MLS method has the advantages of simple derivation and high computational efficiency. Then, by coupling DS-MLS method and Galerkin weak form, and performing the coordinate transformation, an improved element-free Galerkin method (IEFGM) based on the DS-MLS method is proposed for two-dimensional (2D) potential problems on irregular domains. The effectiveness of the method in this paper is verified by some numerical examples. The numerical results show that, compared with the element-free Galerkin (EFG) method, the IEFGM based on the DS-MLS method in this paper consumes less CPU time and has higher computational accuracy for some 2D potential problems on irregular domains.

The effect of water and sodium hydrogen carbonate content on flexible polyurethane's microstructure, water, and mechanical properties

AbstractIt is well known that blowing agents (BAs) and polyol are essential components in polyurethane (PU) composition. Utilizing renewable sources in the material's formulation might reduce its environmental hazards while extending its possible engineering applications. In this study, the samples have been synthesized by using palm kernel oil‐based polyol (PKOP). Water and sodium hydrogen carbonate (SHB) have been used as BAs. Scanning electron microscopy (SEM) has shown that adding the mix of BAs is causing the cells' average size to increase up to 227% and have reduced the lamellae width by up to 2% in comparison with the reference sample. The water tests have illustrated that combining two parts per hundred polyol by weight (php) water and 25 php SHB to the sample has increased its water capacity up to 617%. However, the samples are only able to retain 6% of the absorbed water at the 7th day. It has been also found that porosity has affected the water uptake and all the samples are following Fick's diffusion law, and diffusion is correlated to the square root of time. Multivariable power least squares method (MPLSM) and moving least squares method (MLSM) have been applied to find the relation between tear resistance value and BAs ratio. It is found that both methods have a dominant variable compared to the other variables, but MLSM provided optimizable equation with better R2.

Adaptive Numerical Method for Approximation of Traffic Flow Equations

For a long time now, traffic equations have been considered, and different modeling has been done for it. In this article, we work on the macroscopic model, especially the most famous light model. Because these models are among the stiff and shocking problems, theoretical methods do not give good answers to these problems. This paper describes a meshless method to solve the traffic flow equation as a stiff equation. In the proposed method, we also use the exponential time differencing (ETD) method and the exponential time differencing fourth-order Runge–Kutta (ETDRK4). The purpose of this new method is to use methods of the moving least squares (MLS) method and a modified exponential time differencing fourth-order Runge–Kutta scheme. To solve these equations, we use the meshless method MLS to approximate the spatial derivatives and then use method ETDRK4 to obtain approximate solutions. In order to improve the possible instabilities of method ETDRK4, approaches have been stated. The MLS method provided good results for these equations due to its high flexibility and high accuracy and has a moving window and obtains the solution at the shock point without any false oscillations. The technique is described in detail, and a number of computational examples are presented.

A Novel Progressive Response Surface Method for High-order polynomial Metamodel

The response surface method (RSM) model as a general metamodel has already widely been applied to many practical problems. However, the majority of studies only focus on the 2nd-order terms RSM model instead of how to determine the high-order terms. This paper introduced a novel progressive response surface method (NPRSM) combining optimal Latin hypercube design (OLHD), moving least square method (MLSM) and statistical test. This NPRSM model can decide the high-order terms by progressive procedure. Then three numerical function can be proposed to validate the accuracy and the approximated performance of the NPRSM model.

Surface reconstruction method for measurement data with outlier detection by using improved RANSAC and correction parameter

The moving least squares (MLS) and moving total least squares (MTLS) are two of the most popular methods used for reconstructing measurement data, on account of their good local approximation accuracy. However, their reconstruction accuracy and robustness will be greatly reduced when there are outliers in measurement data. This article proposes an improved MTLS method (IMTLS), which introduces an improved random sample consensus (RANSAC) algorithm and a correction parameter in the support domain, to deal with the outliers and random errors. Based on the nodes within the support domain, firstly the improved RANSAC is used to generate a model for establishing the group of pre-interpolation and calculating the residual of each node. Subsequently, the abnormal degree of the node with the largest residual is evaluated by the correction parameter associated with the node residual and random errors. The node with certain abnormal degree will be eliminated and the remaining nodes are used to obtain the approximation coefficients. The correction parameter can be used for data reconstruction without insufficient or excessive elimination. The results of numerical simulation and measurement experiment show that the reconstruction accuracy and robustness of the IMTLS method is superior to the MLS and MTLS method.

Comparison of meshfree displacement and stress error recovery of finite element solutions using moving least squares interpolation

The moving least square (MLS) interpolation based the recovery procedures have been successfully applied to recover the finite element solution errors in the analysis of elastic plates and pipes problems and can be advantageously applied for large deformation and fracture problems. The study presents the displacement and stress error recovery characteristics in the error estimation analysis employing Moving Least Squares interpolation approach. The study considers quartic spline, cubic spline, and exponential weight function with three different order of basis function in Moving Least Squares interpolation based error recovery analysis. The displacement/stress errors in finite element solution are quantified in energy norm. The cylinder and plate benchmark examples using triangular and quadrilateral elements are analyzed to compare the convergence, effectivity and adaptively improved meshes obtained using the various displacement/stress recovery procedures. The study shows that cubic spline weight function and quadratic basis function found to perform better in MLS based meshfree recovery technique for stress as well as displacement errors recovery of finite element solution. It is observed from the study that increasing the order of basis function will enhance the error estimation quality that is, rate of convergence become faster with improved effectivity of the results. The increase in convergence rate with the increase of the order of basis function is more in displacement recovery technique as compared to stress recovery technique. It is observed in the analysis of benchmark example with linear triangular meshing that the error reduction using meshfree MLS interpolation based displacement and stress recovery is about 10% and 150% respectively for the displacement and stress recovery over the mesh dependent least square based displacement and stress recovery. The study concludes that the effectiveness and efficiency of meshfree displacement/stress error recovery technique strongly depends on the weight and basis functions of MLS method to recover the errors.

Optimal Design of an Inverter-Fed PMSM for a Brake System Considering Variation in Motor Control Parameters and Input Voltage

This study proposes an optimal design approach for an inverter-fed permanent magnet synchronous motor (PMSM) considering the variation in motor control parameters and input voltage (inverter output voltage), which vary with respect to the temperature and loading conditions, in an integrated brake system. In an integrated brake system, a quick response to load changes is crucial. Therefore, in this study, to consider the fluctuation in control variables and input voltage, the motor control parameters and input voltage were first calculated according to the operating temperature and loading condition. Subsequently, based on the calculated conditions and the approximated motor input voltage and control parameters, the objective functions corresponding to motor characteristics in transient and steady states were formulated using design-of-experiment (DOE), the moving least square method (MLSM), and the finite element method (FEM). Finally, the optimal design was performed using the genetic algorithm (GA). The validity of the proposed optimal design approach was verified by comparing its optimization results with the FEM analysis results. Thus, it was confirmed that a motor can be designed for an integrated brake system by considering the motor control parameters and motor input voltage, which vary with the driving conditions.

不連続面の発生と進展に対するメッシュ依存性を軽減するための数値解析

This study is present a numerical method to suppress the mesh dependency in strong discontinuity analysis by the finite cover method (FCM) that is known as a generalized finite element method. To realize fracture process from nucleation to propagation of discontinuity, a seamless crack propagation analysis is combined with the moving least squares method (MLSM), by which standard shape functions are enriched for nodes outside the associated physical cover. Thus, the displacement field is determined by not only the nodes associated but also the non-associated ones. When the integral domain of a finite cover is separated by the discontinuity surface, it sometimes becomes too small to calculate a proper solution. Thus, the displacement field is determined by not the nodes associated with the finite cover but ones around it. In other words, the nodal degrees of freedoms corresponding to the small integral domain of the cover is cut off from the global stiffness matrix. The capabilities of the developed numerical method are demonstrated throughout simple numerical examples.

Optimal Design of Thinned Array Using a Hybrid Genetic Algorithm

In this paper, a hybrid genetic algorithm (GA) is proposed for thinning a two-dimensional planar array by combining the conventional GA with moving least squares (MLS). This enhances the convergence rate and the global search performance. The MLS method is used to estimate local interpolation functions from non-uniform sample data (the population and the value of the objective function in the GA), and to find new and better populations from the interpolated functions. By incorporating these improved populations into the next generation, the MLS-GA achieves improved search performance of the global optimum and a faster convergence rate compared to conventional GA alone. Moreover, a nonlinear chirp function is used for an efficient thinning design. To verify the proposed MLS-GA, it is applied to a test function and the results are compared to that of the GA. The algorithm is then applied to thin an array with a rectangular grid and circular boundary. The design objectives are to minimize the peak side-lobe level and gain loss while satisfying a given thinning coefficient and to compare the results with the GA.

Gaussian Radial Basis Function interpolation in vertical deformation analysis

In many deformation analyses, the partial derivatives at the interpolated scattered data points are required. In this paper, the Gaussian Radial Basis Functions (GRBF) is proposed for the interpolation and differentiation of the scattered data in the vertical deformation analysis. For the optimal selection of the shape parameter, which is crucial in the GRBF interpolation, two methods are used: the Power Gaussian Radial Basis Functions (PGRBF) and Leave One Out Cross Validation (LOOCV) (LGRBF). We compared the PGRBF and LGRBF to the traditional interpolation methods such as the Finite Element Method (FEM), polynomials, Moving Least Squares (MLS), and the usual GRBF in both the simulated and actual Interferometric Synthetic Aperture Radar (InSAR) data. The estimated results showed that the surface interpolation accuracy was greatly improved by LGRBF and PGRBF methods in comparison withFEM, polynomial, and MLS methods. Finally, LGRBF and PGRBF interpolation methods are used to compute invariant vertical deformation parameters, i.e., changes in Gaussian and mean Curvatures in the Groningen area in the North of Netherlands.

Moving Least Squares Method and its Improvement: A Concise Review

The concise review systematically summarises the state-of-the-art variants of Moving Least Squares (MLS) method. MLS method is a mathematical tool which could render cogent support in data interpolation, shape construction and formulation of meshfree schemes, particularly due to its flexibility to form complex arithmetic equation. However, the conventional MLS method is suffering to deal with discontinuity of field variables. Varied strategies of overcoming such shortfall are discussed in current work. Although numerous MLS variants were proposed since the introduction of MLS method in numerical/statistical analysis, there is no technical review made on how the methods evolve. The current review is structured according to major strategies on how to improvise MLS method: the modification of weight function, the manipulation of discrete norms, the inclusion of iterative feature for residuals minimising and integration of these strategies for more robust computation. A wide range of advanced MLS variants have been compiled, summarised, and reappraised according to its underlying principle of improvement. In addition, inherent limitation of MLS method and its possible strategy of improvement is discussed too in this article. The current work could render valuable reference to implement and develop advanced MLS schemes, whenever complexity of the specific scientific problems arose.

Improving spatial estimation of hydrologic attributes via optimized moving search strategies

The anisotropy and directional variation of spatial attributes is one of the most important challenges in spatial estimation of environmental features. This problem received a suitable response in the realm of stochastic modeling using directional variography; however, there is not any specific solution in the field of deterministic methods. In this research, Moving Least Square (MLS) and Moving Inverse Distance Weighting (MIDW) have been used to assess a moving domain as the remedy; although, there has been no research on the spatial estimation with the MLS method. The methods are evaluated in three case studies, (I) using anisotropic piezometric head data from the Wolfcamp aquifer in Texas, (II) long-term annual average precipitation measurements in Namak Lake Watershed (NLW) located in the central part of Iran, and (III) estimation of annual precipitation with remote-sensing products, TRMM, in NLW as well. To achieve the best performance of the methods, parameters have been optimized using the modified version of shuffled complex evolutionary method. Assessment of statistical metrics, Taylor Diagram, and numerical results showed that spatial interpolation using MIDW and MLS revealed better results, with an improvement of 45.3% on average in terms of Root Mean Square Error (RMSE), in comparison with original IDW as the benchmark evaluation approach. This shows the power of moving data partitioning on the performance of deterministic estimators. Additionally, MLS was more effective than MIDW with an average of 0.55, 10.53, and 17.19% reduction in RMSE of case I, II, and III, respectively.

Probabilistic analyses of structural dynamic response with modified Kriging-based moving extremum framework

Mechanical system is usually composed of multiple complex structures, which endure the combine action of multi-physical fields (e.g., flow field, thermal field, structural field, and so forth) during operation. Structural response, for instance, deformation, stress, strain, fatigue life, etc., have dynamic and time-varying attributes, their values play an important role in safeguarding function properly, and affects the safety of entire system. Probabilistic analyses (including reliability evaluation and sensitivity analysis) can effectively identify potential risk and improve the reliability level. The traditional way is to use Monte Carlo (MC) simulation to achieve probabilistic analyses of complex structures. However, the MC simulation needs to perform a large amount of calculations to complete the specified task analysis, the computational burden is too heavy for the reliability evaluation and sensitivity analysis of structural dynamic response. To efficiently perform probabilistic analyses of structural dynamic response, we therefore developed a surrogate model method, namely modified Kriging-based moving extremum framework (MKMEF, short for), absorbing extremum thought, moving least squares (MLS) technique, Kriging model and collaborative evolution genetic algorithm (CEGA). For this proposed MKMEF, the extremum thought is used to transform dynamic output response into extremum values within a time domain, and the MLS method is applied to obtain efficient samples to derive Kriging model. Besides, the CEGA is employed to replace gradient descent method to resolve maximum likelihood equation (MLE) and find the optimal hyperparameters of Kriging model. Furthermore, an aeroengine high-pressure turbine blisk is treated as a case study, the probabilistic analyses of radial running deformation is executed to validate the effectiveness of the proposed method by considering the fluid-thermal-solid interaction. The analytical results show that the reliability degree of turbine blisk is 99.78%, when the allowable value of radial running deformation is 2.5071 × 10−3 m (determined by 3 sigma levels), and the highest impact on the turbine blisk radial running deformation is gas temperature, followed by inlet velocity, rational speed, material density and outlet pressure. The comparison of methods including direct simulation, response surface method-based extremum thought (ERSM) and traditional Kriging model, shows that the developed MKMEF holds high-efficiency and high-accuracy for the probabilistic analyses of turbine blisk radial running deformation. The presented efforts provide a useful insight for assessing probabilistic analyses of structural dynamic response and enrich mechanical reliability theory.

Robust high precision multi-frame motion detection for PMLSMs’ mover based on local upsampling moving least square method

This paper studies a multi-frame visual sub-pixel measurement technique to achieve the precise positioning of the permanent magnet linear synchronous motor (PMLSM) rapidly. A new matrix multiplication discrete Fourier transform (DFT) upsampling moving least square (MLS) algorithm is proposed to improve the accuracy and robustness of PMLSM positioning. Firstly, a line-scan vision-based measurement system is designed, and an aperiodic sinusoidal fringe image is placed on the pedestal as target image. Secondly, the local upsampling cross-correlation curve of adjacent one-dimensional signals is calculated to obtain sub-pixel displacement using the matrix-multiplied DFT method. Finally, to further improve the displacement accuracy, the MLS method is applied. Simulation tests verify that the proposed method is superior to other algorithms in measurement accuracy, calculation efficiency and measurement range. The experiment results confirm the feasibility and validity of the proposed method for detecting the mover motion under different working environments.