Stability Analysis and Performance Evaluation of Additive Mixed-Precision Runge-Kutta Methods

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.

Research and experimental analysis of precision degradation method based on the ball screw mechanism

As a key transmission component in computer numerical control(CNC) machine tools,the ball screw mechanism(BSM) is usually investigated under working load conditions. Its accuracy degradation process is relatively long,which is not conducive to the design and development of new products. In this paper,the normal wear depth of the BSM nut raceway is calculated under the variable speed operation condition using the fractal wear analysis method and the BSM’s accelerated degradation proportional wear model. Parameters of the acceleration degradation model of the double-nut preloaded ball screw pair are calculated based on the physical simulation results. The accelerated degradation test platform of the BSM is designed and manufactured to calculate the raceway wear model when the lubrication condition is broken under the variable-speed inertial load and the boundary lubrication condition under the uniform speed state. Three load forces and two samples are selected for the accelerated degradation test of the BSM. The measured friction torque of the BSM is employed as the evaluation index of the accuracy degradation test. In addition,the life cycle of the accuracy retention is accurately calculated by employing the parameters of the physical simulation model of the BSM. The calculations mentioned above can be used to estimate BSM’s accuracy performance degradation law under normal operating conditions. The application of the proposed model provides a new research method for researching the precision retention of the BSM.

Accuracy and Precision of Energy Dispersive X-Ray Fluorescence (EDXRF) Analysis of Trace and Major Elements in Rock Standard Reference Materials Using Fine Powder

In this work the performance of a screening analytical method for Energy Dispersive X-Ray Fluorescence (EDXRF) analysis in terms of accuracy and precision was evaluated through analysis of rock standard reference materials. The method allowed the division of elements into four groups taking into account the excitation energies and measurement conditions of the sample. Two standard reference materials were used and 15 sample replicates were prepared and analyzed, then statistics were applied to assess the precision and accuracy of analytical results. The obtained results show that major compounds or elements (SiO2, P2O5, K2O, CaO, Fe2O3, Ti) can be determined in fine powder sample with a deviation lower than 15%, and a relative standard deviation in the range (1 - 10)%. The deviation was found to be lower than 5% for major compounds such as K2O, and CaO, which suggest that the EDXRF is accurate in evaluating major elemental concentrations in rock samples. It was also found that the method seems to be more accurate and precise for major elements than for trace element investigation. This screening analytical method can be used for routine analysis with acceptable results, even though the method should be optimized to increase its precision and accuracy.

High-precision solution to the moving load problem using an improved spectral element method

In this paper, the spectral element method(SEM)is improved to solve the moving load problem. In this method, a structure with uniform geometry and material properties is considered as a spectral element, which means that the element number and the degree of freedom can be reduced significantly. Based on the variational method and the Laplace transform theory, the spectral stiffness matrix and the equivalent nodal force of the beam-column element are established. The static Green function is employed to deduce the improved function. The proposed method is applied to two typical engineering practices—the one-span bridge and the horizontal jib of the tower crane. The results have revealed the following. First, the new method can yield extremely high-precision results of the dynamic deflection, the bending moment and the shear force in the moving load problem.In most cases, the relative errors are smaller than 1%. Second, by comparing with the finite element method, one can obtain the highly accurate results using the improved SEM with smaller element numbers. Moreover, the method can be widely used for statically determinate as well as statically indeterminate structures. Third, the dynamic deflection of the twin-lift jib decreases with the increase in the moving load speed, whereas the curvature of the deflection increases.Finally, the dynamic deflection, the bending moment and the shear force of the jib will all increase as the magnitude of the moving load increases.

A HIGH CONVERGENT PRECISION EXACT ANALYTIC METHOD FOR DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS

The exact analytic method was given by [1] . It can be used for arbitrary variable coefficient differential equations and the solution obtained can have the second order convergent precision. In this paper, a new high precision algorithm is given based on [1], through a bending problem of variable cross-section beams. It can have the fourth convergent precision without increasing computation work. The present computation method is not only simple but also fast. The numerical examples are given at the end of this paper which indicate that the high convergent precision can be obtained using only a few elements. The correctness of the theory in this paper is confirmed.

Eddy Current Analyses by Domain Decomposition Method Using Double-Double Precision

A matrix equation solved in an eddy current analysis,??-??method based on a domain decomposition method becomes a complex symmetric system.In general,iterative method is used as the solver.Convergence of iterative method in an interface problem is improved by increasing an accuracy of a solution of an iterative method of a subdomain problem.However,it is difficult to improve the convergence by using a small convergence criterion in the subdomain problem.Therefore,authors propose a method to introduce double-double precision into the interface problem and the subdomain problem.This proposed method improves the convergence of the interface problem.In this paper,first,we describe proposed method.Second,we confirm validity of the method by using Team Workshop Problem 7,standard model for eddy current analysis.Finally,we show effectiveness of the method from two numerical results.

Preliminary Application of Combinatorial Measurement and Regression Analysis Method to High Precision Instrumental Analysis

With determination micro-Fe by 1, 10-phenanthroline spectrophotometry for example, they are systematically introduced the combinatorial measurement and regression analysis method application about metheodic principle, operation step and data processing in the instrumental analysis, including: calibration curve best linear equation is set up, measurand best linear equation is set up, and calculation of best value of a concentration. The results showed that mean of thrice determination , s = 0 μg/mL, RSD = 0. Results of preliminary application are simply introduced in the basic instrumental analysis for atomic absorption spectrophotometry, ion-selective electrodes, coulometry and polarographic analysis and are contrasted to results of normal measurements.

MEASURING PRECISION OF INDUCED POLARIZATION METHOD

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A Comprehensive Method for Evaluating Precision of Transfer Alignment on a Moving Base

In this study, we propose the use of the Degree of Alignment(DOA) in engineering applications for evaluating the precision of and identifying the transfer alignment on a moving base. First, we derive the statistical formula on the basis of estimations. Next, we design a scheme for evaluating the transfer alignment on a moving base, for which the attitude error cannot be directly measured. Then, we build a mathematic estimation model and discuss Fixed Point Smoothing(FPS), Returns to Scale(RTS), Inverted Sequence Recursive Estimation(ISRE), and Kalman filter estimation methods, which can be used when evaluating alignment accuracy. Our theoretical calculations and simulated analyses show that the DOA reflects not only the alignment time and accuracy but also differences in the maneuver schemes, and is suitable for use as an integrated evaluation index. Furthermore, all four of these algorithms can be used to identify the transfer alignment and evaluate its accuracy. We recommend RTS in particular for engineering applications. Generalized DOAs should be calculated according to the tactical requirements.

A Precision-Positioning Method for a High-Acceleration Low-Load Mechanism Based on Optimal Spatial and Temporal Distribution of Inertial Energy

High-speed and precision positioning are fund a mental requirements for high-acceleration low-load mechanisms in integrated circuit(IC) packaging equipment. In this paper, we derive the transient nonlinear dynamicresponse equations of high-acceleration mechanisms, which reveal that stiff ness, frequency, damping, and driving frequency are the primary factors. Therefore, we propose a new structural optimization and velocity-planning method for the precision positioning of a high-acceleration mechanism based on optimal spatial and temporal distribution of inertial energy. For structural optimization, we first reviewed the commonly flexible multibody dynamic optimization using equivalent static loads method(ESLM), and then we selected the modifled ESLM for optimal spatial distribution of inertial energy; hence, not only the stiff ness but also the inertia and frequency of the real modal shapes are considered. For velocity planning, we developed a new velocity-planning method based on nonlinear dynamic-response optimization with varying motion conditions. Our method was verifled on a high-acceleration die bonder. The amplitude of residual vibration could be decreased by more than 20% via structural optimization and the positioning time could be reduced by more than 40% via asymmetric variable veloci ty planning. This method provides an effective theoretical support for the precision positioning of high-acceleration low-load mechanisms.

Fast precise integration method for hyperbolic heat conduction problems

A fast precise integration method is developed for the time integral of the hyperbolic heat conduction problem. The wave nature of heat transfer is used to analyze the structure of the matrix exponential, leading to the fact that the matrix exponential is sparse. The presented method employs the sparsity of the matrix exponential to improve the original precise integration method. The merits are that the proposed method is suitable for large hyperbolic heat equations and inherits the accuracy of the original version and the good computational efficiency, which are verified by two numerical examples.

A COMBINED PARAMETRIC QUADRATIC PROGRAMMING AND PRECISE INTEGRATION METHOD BASED DYNAMIC ANALYSIS OF ELASTIC-PLASTIC HARDENING/SOFTENING PROBLEMS

The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems.

CHARACTERISTIC GALERKIN METHOD FOR CONVECTION-DIFFUSION EQUATIONS AND IMPLICIT ALGORITHM USING PRECISE INTEGRATION

This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.

ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations ( ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.

Improved precise integration method for differential Riccati equation

An improved precise integration method （IPIM） for solving the differential Riccati equation （DRE） is presented. The solution to the DRE is connected with the exponential of a Hamiltonian matrix, and the precise integration method （PIM） for solving the DRE is connected with the scaling and squaring method for computing the exponential of a matrix. The error analysis of the scaling and squaring method for the exponential of a matrix is applied to the PIM of the DRE. Based ,on the error analysis, the criterion for choosing two parameters of the PIM is given. Three kinds of IPIMs for solving the DRE are proposed. The numerical examples machine accuracy solutions. show that the IPIM is stable and gives the

Effects of Process Parameters on the Temperature Field in Ti-6Al-4V Alloy Blade Precision Forging Process

Blade precision forging is a high temperature and large plastic deformation process. Process parameters have a great effect on temperature distribution in billet, so in this paper, by taking a Ti-6Al-4V alloy blade with a tenon as an object, the influence of process parameters on the temperature distribution in precision forging process was investigated using 3D coupled thermo-mechanical FEM （finite element method） code developed by the authors. The results obtained illustrate that： （1） the gradient of temperature distribution increases with increasing the deformation degree; （2） with increasing the initial temperature of the billet, the zones of high temperature become larger, and the gradient of temperature distribution hardly has any increase; （3） friction factors have little effect on the distribution of temperature field; （4） with increasing upper die velocity, temperature of the billet increases while the temperature gradient in billet decreases. The results are helpful to the design and optimization of the process parameters in precision forging process of Ti-alloy blade.

Optimization of a precise integration method for seismic modeling based on graphic processing unit

General purpose graphic processing unit （GPU） calculation technology is gradually widely used in various fields. Its mode of single instruction, multiple threads is capable of seismic numerical simulation which has a huge quantity of data and calculation steps. In this study, we introduce a GPU-based parallel calculation method of a precise integration method （PIM） for seismic forward modeling. Compared with CPU single-core calculation, GPU parallel calculating perfectly keeps the features of PIM, which has small bandwidth, high accuracy and capability of modeling complex substructures, and GPU calculation brings high computational efficiency, which means that high-performing GPU parallel calculation can make seismic forward modeling closer to real seismic records.

Precise integration method for a class of singular two-point boundary value problems

In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.

Precise integration method for solving singular perturbation problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.