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Various parameter values are provided in the form of data tables, where data keys are ordered and unevenly spaced in general, for real‐time simulation of dynamic systems. However, most parameter values required for simulation do not explicitly exist in data tables. Thus, unit intervals, including parameter values, are searched rather than the data keys. Since real‐time constraint enforces use of a fixed step size in integration of system differential equations because of the inherent nature of input from and output to real hardware, the worst case of iterated probes in searching algorithms is the core measure for comparison. The worst case is expressed as Big O. In this study, conventional bisection, interpolation, and fast searches are analyzed and compared in Big O as well as the newly developed searching algorithms: modified fast search and modified regular falsi search. If the criterion is actual execution time required for searching, most numerical tests in this paper show that bisection search is superior to the others. Interpolation search and its variations show better performance in the case of linear or near linear data distribution than bisection search. The numerical tests show that modified regular falsi search is faster than the other interpolation searches in either expected time or worst cases. Given parameter tables should be carefully examined for their data distribution in order to determine the most appropriate searching algorithm for the application.

The purpose of the method is to develop a numerical method for the solution of nonlinear partial differential equations.

A new numerical approach based on Barycentric Rational interpolation has been used to solve partial differential equations.

A numerical technique based on barycentric rational interpolation has been developed to investigate numerical simulation of the Burgers’ and Fisher’s equations. Barycentric interpolation is basically a variant of well-known Lagrange polynomial interpolation which is very fast and stable. Using semi-discretization for unknown variable and its derivatives in spatial direction by barycentric rational interpolation, we get a system of ordinary differential equations. This system of ordinary differential equation’s has been solved by applying SSP-RK43 method. To check the efficiency of the method, computed numerical results have been compared with those obtained by existing methods. Barycentric method is able to capture solution behavior at small values of kinematic viscosity for Burgers’ equation.

To the best of the authors’ knowledge, the method is developed for the first time and validity is checked by stability and error analysis.

This paper aims to predict and evaluate the wear rate of TiAl-2 Wt.% MoO₃ tabular crystals (TMCs) using the Newton interpolation methods.

The friction and wear behaviors of TMC were examined using pin-on-disc apparatus at different times, namely, 1,200, 2,400, 3,600, 4,800 and 6,000 s. The wear rates of five different times as interpolation nodes were measured and calculated by electron probe microanalysis (EMPA) and field emission electron microscope (FESEM). Then, the prediction formula of wear rate was constructed using the Newton interpolation method. The accuracy of the prediction formula and the relationship with friction layer and worn surface are verified for evaluating the reliability of the prediction formula.

The prediction formula shows a similar variation trend of TMC as the experimental results, indicating that the prediction formula can forecast the wear rate and working condition of TMC. Moreover, the microstructures of friction layer and worn surface also have a strong impact on the prediction formulas.

The prediction formulas of the Newton interpolation polynomial can be adopted to predict working longevity in the mechanical components, which can guide the practical engineering application in industrial fields.

We introduce a second‐order accurate time‐marching pressure‐correction algorithm to accommodate weakly‐compressible highly‐viscous liquid flows at low Mach number. As the incompressible limit is approached (Ma ≈ 0), the consistency of the compressible scheme is highlighted in recovering equivalent incompressible solutions. In the viscous‐dominated regime of low Reynolds number (zone of interest), the algorithm treats the viscous part of the equations in a semi‐implicit form. Two discrete representations are proposed to interpolate density: a piecewise‐constant form with gradient recovery and a linear interpolation form, akin to that on pressure. Numerical performance is considered on a number of classical benchmark problems for highly viscous liquid flows to highlight consistency, accuracy and stability properties. Validation bears out the high quality of performance of both compressible flow implementations, at low to vanishing Mach number. Neither linear nor constant density interpolations schemes degrade the second‐order accuracy of the original incompressible fractional‐staged pressure‐correction scheme. The piecewise‐constant interpolation scheme is advocated as a viable method of choice, with its advantages of order retention, yet efficiency in implementation.

The problems of shear flexible beams are analyzed by the MLPG method based on a locking‐free weak formulation. In order for the weak formulation to be locking‐free, the numerical characteristics of the variational functional for a shear flexible beam, in the thin beam limit, are discussed. Based on these discussions a locking‐free local symmetric weak form is derived by changing the set of two dependent variables in governing equations from that of transverse displacement and total rotation to the set of transverse displacement and transverse shear strain. For the interpolation of the chosen set of dependent variables (i.e. transverse displacement and transverse shear strain) in the locking‐free local symmetric weak form, the recently proposed generalized moving least squares (GMLS) interpolation scheme is utilized, in order to introduce the derivative of the transverse displacement as an additional nodal degree of freedom, independent of the nodal transverse displacement. Through numerical examples, convergence tests are performed. To identify the locking‐free nature of the proposed method, problems of shear flexible beams in the thick beam limit and in the thin beam limit are analyzed, and the numerical results are compared with analytical solutions. The potential of using the truly meshless local Petrov‐Galerkin (MLPG) method is established as a new paradigm in totally locking‐free computational analyses of shear flexible plates and shells.

The purposes of this research are to study the theory and method of multi-attribute index system design and establish a set of systematic, standardized, scientific index systems for the design optimization and inspection process. The research may form the basis for a rational, comprehensive evaluation and provide the most effective way of improving the quality of management decision-making. It is of practical significance to improve the rationality and reliability of the index system and provide standardized, scientific reference standards and theoretical guidance for the design and construction of the index system.

Using modern methods such as complex networks and machine learning, a system for the quality diagnosis of index data and the classification and stratification of index systems is designed. This guarantees the quality of the index data, realizes the scientific classification and stratification of the index system, reduces the subjectivity and randomness of the design of the index system, enhances its objectivity and rationality and lays a solid foundation for the optimal design of the index system.

Based on the ideas of statistics, system theory, machine learning and data mining, the focus in the present research is on “data quality diagnosis” and “index classification and stratification” and clarifying the classification standards and data quality characteristics of index data; a data-quality diagnosis system of “data review – data cleaning – data conversion – data inspection” is established. Using a decision tree, explanatory structural model, cluster analysis, K-means clustering and other methods, classification and hierarchical method system of indicators is designed to reduce the redundancy of indicator data and improve the quality of the data used. Finally, the scientific and standardized classification and hierarchical design of the index system can be realized.

The innovative contributions and research value of the paper are reflected in three aspects. First, a method system for index data quality diagnosis is designed, and multi-source data fusion technology is adopted to ensure the quality of multi-source, heterogeneous and mixed-frequency data of the index system. The second is to design a systematic quality-inspection process for missing data based on the systematic thinking of the whole and the individual. Aiming at the accuracy, reliability, and feasibility of the patched data, a quality-inspection method of patched data based on inversion thought and a unified representation method of data fusion based on a tensor model are proposed. The third is to use the modern method of unsupervised learning to classify and stratify the index system, which reduces the subjectivity and randomness of the design of the index system and enhances its objectivity and rationality.

The purpose of this paper is to propose an automatic interpolation algorithm for robot spraying trajectories based on cubic Non-Uniform Rational B-Splines (NURBS) curves, to solve the problem of sparse and incomplete trajectory points of the head and heel of the shoe sole when extracting robot motion trajectories using structured-light 3D cameras and to ensure the robot joints move smoothly, so as to achieve a good effect of automatic spraying of the shoe sole with a 7-degree-of-freedom (DOF) robot.

Firstly, the original shoe sole edge trajectory position points acquired by the 3D camera are fitted with NURBS curves. Then, the velocity constraint at the local maximum of the trajectory curvature is used as the reference for curve segmentation and S-shaped acceleration and deceleration planning. Immediately, real-time interpolation is performed in the time domain to obtain the position and orientation of each point of the robot motion trajectory. Finally, the inverse kinematics of the anthropomorphic motion of the 7-DOF robot arm is used to obtain the joint motion trajectory.

The simulation and experiment prove that the shoe sole spraying trajectory is complete, the spraying effect is good and the robot joint movement is smooth, which show that the algorithm is feasible.

This study is of good practical value for improving the quality of automated shoe sole spraying, and it has wide applicability for different shoe sole shapes.

The goal is to derive a numerical method for computing parametric reduced-order models (PROMs) from finite-element (FE) models of microwave structures that feature geometrical parameters.

First, a parameter-dependent FE mesh is constructed by a topology-preserving mesh-morphing algorithm. Then, multivariate polynomial interpolation is employed to achieve explicit geometrical parameterization of all FE matrices. Finally, a PROM based on parameter-dependent projection matrices is constructed by means of interpolation and state transformation techniques.

The resulting PROMs are of low dimension and fast to evaluate. Moreover, the method features high rates of convergence, and the number of FE solutions required for constructing the PROM is small. The accuracy of the PROM is only limited by that of the underlying FE model and can be controlled by varying the PROM dimension.

Since the method uses topology-preserving mesh-morphing algorithms to instantiate FE models at a number of interpolation points in geometrical parameter space, there are limitations to the amount of deformation that can be handled.

PROM evaluations are computationally cheap. In many cases they can be evaluated hundreds or even thousands of times per second. Therefore, PROMs are very well-suited for parametric studies or numerical optimization.

The presented methodology employs a new way of constructing parameter-dependent interpolation matrices, based on interpolation and space transformations. The proposed methodology yields better accuracy and higher rates of convergence than previous approaches.

In a previous chapter (Klimberg, Ratick, & Smith, 2018), we introduced a novel approach in which cluster centroids were used as input data for the predictor variables of a multiple linear regression (MLR) used to forecast fleet maintenance costs. We applied this approach to a real data set and significantly improved the predictive accuracy of the MLR model. In this chapter, we develop a methodology for adjusting moving average forecasts of the future values of fleet service occurrences by interpolating those forecast values using their relative distances from cluster centroids. We illustrate and evaluate the efficacy of this approach with our previously used data set on fleet maintenance.

This work compares the performance of the three boundary element techniques for solving Helmholtz problems: dual reciprocity, multiple reciprocity and direct interpolation. All techniques transform domain integrals into boundary integrals, despite using different principles to reach this purpose.

Comparisons here performed include the solution of eigenvalue and response by frequency scanning, analyzing many features that are not comprehensively discussed in the literature, as follows: the type of boundary conditions, suitable number of degrees of freedom, modal content, number of primitives in the multiple reciprocity method (MRM) and the requirement of internal interpolation points in techniques that use radial basis functions as dual reciprocity and direct interpolation.

Among the other aspects, this work can conclude that the solution of the eigenvalue and response problems confirmed the reasonable accuracy of the dual reciprocity boundary element method (DRBEM) only for the calculation of the first natural frequencies. Concerning the direct interpolation boundary element method (DIBEM), its interpolation characteristic allows more accessibility for solving more elaborate problems. Despite requiring a greater number of interpolating internal points, the DIBEM has presented higher-quality results for the eigenvalue and response problems. The MRM results were satisfactory in terms of accuracy just for the low range of frequencies; however, the neglected higher-order primitives impact the accuracy of the dynamic response as a whole.

There are safe alternatives for solving engineering stationary dynamic problems using the boundary element method (BEM), but there are no suitable comparisons between these different techniques. This paper presents the particularities and detailed comparisons approaching the accuracy of the three important BEM techniques, aiming at response and frequency evaluation, which are not found in the specialized literature.