3D elastic wave equation forward modeling based on the precise integration method

The Finite Difference （FD） method is an important method for seismic numerical simulations. It helps us understand regular patterns in seismic wave propagation, analyze seismic attributes, and interpret seismic data. However, because of its discretization, the FD method is only stable under certain conditions. The Arbitrary Difference Precise Integration （ADPI） method is based on the FD method and adopts an integration scheme in the time domain and an arbitrary difference scheme in the space domain. Therefore, the ADPI method is a semi-analytical method. In this paper, we deduce the formula for the ADPI method based on the 3D elastic equation and improve its stability. In forward modeling cases, the ADPI method was implemented in 2D and 3D elastic wave equation forward modeling. Results show that the travel time of the reflected seismic wave is accurate. Compared with the acoustic wave field, the elastic wave field contains more wave types, including PS- and PP- reflected waves, transmitted waves, and diffracted waves, which is important to interpretation of seismic data. The method can be easily applied to elastic wave equation numerical simulations for eoloical models.

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.

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

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.

Response of train-bridge system under intensive seismic excitation by random vibration method

Earthquake is a kind of sudden and destructive random excitation in nature.It is significant to determine the probability distribution characteristics of the corresponding dynamic indicators to ensure the safety and the stability of structures when the intensive seismic excitation,the intensity of which is larger than 7,acts in train-bridge system.Firstly,the motion equations of a two-dimensional train-bridge system under the vertical random excitation of track irregularity and the vertical seismic acceleration are established,where the train subsystem is composed of 8 mutually independent vehicle elements with 48 degrees of freedom,while the single-span simple supported bridge subsystem is composed of 102D beam elements with 20 degrees of freedom on beam and 2 large mass degrees of freedom at the support.Secondly,Monte Carlo method and pseudo excitation method are adopted to analyze the statistical parameters of the system.The power spectrum density of random excitation is used to define a series of non-stationary pseudo excitation in pseudo excitation method and the trigonometric series of random vibration history samples in Monte Carlo method,respectively solved by precise integral method and Newmark-βmethod through the inter-system iterative procedure.Finally,the results are compared with the case under the weak seismic excitation,and show that the samples of vertical acceleration response of bridge and the offload factor of train obeys the normal distribution.In a high probability,the intensive earthquakes pose a greater threat to the safety and stability of bridges and trains than the weak ones.

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.

High order symplectic conservative perturbation method for time-varying Hamiltonian system

This paper presents a high order symplectic con- servative perturbation method for linear time-varying Hamil- tonian system. Firstly, the dynamic equation of Hamilto- nian system is gradually changed into a high order pertur- bation equation, which is solved approximately by resolv- ing the Hamiltonian coefficient matrix into a ＂major compo- nent＂ and a ＂high order small quantity＂ and using perturba- tion transformation technique, then the solution to the orig- inal equation of Hamiltonian system is determined through a series of inverse transform. Because the transfer matrix determined by the method in this paper is the product of a series of exponential matrixes, the transfer matrix is a sym- plectic matrix; furthermore, the exponential matrices can be calculated accurately by the precise time integration method, so the method presented in this paper has fine accuracy, ef- ficiency and stability. The examples show that the proposed method can also give good results even though a large time step is selected, and with the increase of the perturbation or- der, the perturbation solutions tend to exact solutions rapidly.

Analysis of VIV for Barrel-Shaped Mast with Covering Ice by HPD-S Method

The vortex-induced vibrations （VIV） of barrel-shaped mast are calculated by three numerical metods, i.e. New- mark- β, HPD-L （High Precision Direct integration scheme-Linear form）, and HPD-S （High Precision Direct integration scheme-Sinusoidal form）. According to the measured value, the accuracy curves are given to show the advantages of HPD-S method over others. Based on the comparison above, HPD-S method is used to calculate the influence of ice covering on the mast to VIV responses. It has been proved that the vortex-induced responses of barrel-shaped mast are changed along with ice thicknesses and types.

Radial integral boundary element method for simulating phase change problem with mushy zone

A radial integral boundary element method(BEM)is used to simulate the phase change problem with a mushy zone in this paper.Three phases,including the solid phase,the liquid phase,and the mushy zone,are considered in the phase change problem.First,according to the continuity conditions of temperature and its gradient on the liquid-mushy interface,the mushy zone and the liquid phase in the simulation can be considered as a whole part,namely,the non-solid phase,and the change of latent heat is approximated by heat source which is dependent on temperature.Then,the precise integration BEM is used to obtain the differential equations in the solid phase zone and the non-solid phase zone,respectively.Moreover,an iterative predictor-corrector precise integration method(PIM)is needed to solve the differential equations and obtain the temperature field and the heat flux on the boundary.According to an energy balance equation and the velocity of the interface between the solid phase and the mushy zone,the front-tracking method is used to track the move of the interface.The interface between the liquid phase and the mushy zone is obtained by interpolation of the temperature field.Finally,four numerical examples are provided to assess the performance of the proposed numerical method.

New matrix method for response analysis of circumferentially stiffened non-circular cylindrical shells under harmonic pressure

Based on the governing equation of vibration of a kind of cylindrical shells written in a matrix differential equation of the first order, a new matrix method is presented for steady-state vibration analysis of a noncircular cylindrical shell simply sup- ported at two ends and circumferentially stiffened by rings under harmonic pressure. Its difference from the existing works by Yamada and Irie is that the matrix differential equation is solved by using the extended homogeneous capacity precision integration＇ approach other than the Runge-Kutta-Gill integration method. The transfer matrix can easily be determined by a high precision integration scheme. In addition, besides the normal interacting forces, which were commonly adopted by researchers earlier, the tangential interacting forces between the cylindrical shell and the rings are considered at the same time by means of the Dirac-δ function. The effects of the exciting frequencies on displacements and stresses responses have been investigated. Numerical results show that the proposed method is more efficient than the aforementioned method.

An Integral Method for Solving Dynamic Equations with Fluid–Solid Coupling

In this work,a new methodology is presented to mainly solve the fluid–solid interaction(FSI)equation.This methodology combines the advantages of the Newmark precise integral method(NPIM)and the dual neural network(DNN)method.The NPIM is employed to modify the exponential matrix and loading vector based on the DNN integral method.This involves incorporating the basic assumption of the Newmark-βmethod into the dynamic equation and eliminating the acceleration term from the dynamic equilibrium equation.As a result,the equation is reduced to a first-order linear equation system.Subsequently,the PIM is applied to integrate the system step by step within the NPIM.The DNN method is adopted to solve the inhomogeneous term through fitting the integrand and the original function with a pair of neural networks,and the integral term is solved using the Newton–Leibniz formula.Numerical examples demonstrate that the proposed methodology significantly improves computing efficiency and provides sufficient precision compared to the DNN method.This is particularly evident when analyzing large-scale structures under blast loading conditions.

The Calculation of Parameters for DNA Kinetic Structure Based on Monte-Carlo Multiple Integrals

Based on protein-DNA complex crystal structural data in up-to-date Nucleic Acid Database,the related parameters of DNA Kinetic Structure were investigated by Monte-Carlo Multiple Integrals on the base of modified DNA structure statistical mechanical model,and time complexity and precision were analyzed on the calculated results.

利用压印法制造直齿锥齿轮精锻模

简要叙述了用压印法制造直齿锥齿轮精锻模的问题对于母模齿轮的设计，从理论和实际两方面进行了分析和讨论．

AN ADAPTIVE ALGORITHM OF PRECISE INTEGRATION FOR TRANSIENT ANALYSIS

This paper presents an improved precise integration algorithm fortransient analysis of heat transfer and some other problems. Theoriginal precise integration method is improved by means of the inve-rse accuracy analysis so that the parameter N, which has been takenas a constant and an independent pa- rameter without consideration ofthe problems in the original method, can be generated automaticallyby the algorithm itself.

AN IMPLICIT SERIES PRECISE INTEGRATION ALGORITHM FOR STRUCTURAL NONLINEAR DYNAMIC EQUATIONS

Nonlinear dynamic equations can be solved accurately using a precise integration method. Some algorithms exist, but the inversion of a matrix must be calculated for these al- gorithms. If the inversion of the matrix doesn’t exist or isn’t stable, the precision and stability of the algorithms will be afected. An explicit series solution of the state equation has been pre- sented. The solution avoids calculating the inversion of a matrix and its precision can be easily controlled. In this paper, an implicit series solution of nonlinear dynamic equations is presented. The algorithm is more precise and stable than the explicit series solution and isn’t sensitive to the time-step. Finally, a numerical example is presented to demonstrate the efectiveness of the algorithm.

Nonlinear vibration of embedded single-walled carbon nanotube with geometrical imperfection under harmonic load based on nonlocal Timoshenko beam theory

Based on the nonlocal continuum theory, the nonlinear vibration of an embedded single-walled carbon nanotube （SWCNT） subjected to a harmonic load is in- vestigated. In the present study, the SWCNT is assumed to be a curved beam, which is unlike previous similar work. Firstly, the governing equations of motion are derived by the Hamilton principle, meanwhile, the Galerkin approach is carried out to convert the nonlinear integral-differential equation into a second-order nonlinear ordinary differ- ential equation. Then, the precise integration method based on the local linearzation is appropriately designed for solving the above dynamic equations. Besides, the numerical example is presented, the effects of the nonlocal parameters, the elastic medium constants, the waviness ratios, and the material lengths on the dynamic response are analyzed. The results show that the above mentioned effects have influences on the dynamic behavior of the SWCNT.

Study on Location of Wuhan Steel Logistics Distribution Center

This paper studies the location of Wuhan steel logistics distribution center. First of all, according to Wuhan Iron and Steel Plant sales in Hunan Province and the relative position of the city, the transport costs are calculated from Wuhan Iron and Steel Plant to the demand point. We further analyze the necessity of establishing steel logistics distribution center, using the precise center of gravity to determine the actual location of the distribution center. After the establishment of distribution center, the total freight is reduced by 15.46 million yuan from Wuhan Iron and Steel Plant to each city in Hunan province via distribution center each year. The results of this paper can provide theoretical basis for the logistics node planning of related enterprises.