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.

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.

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.

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.

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.

An Adaptively Filtered Precise Integration Method Considering Perturbation for Stochastic Dynamics Problems

The difficulty in solving stochastic dynamics problems lies in the need for a large number of repeated computations of deterministic dynamic equations,which has been a challenge in stochastic dynamics analysis and was discussed in this study.To efficiently and accurately compute the exponential of the dynamics state matrix and the matrix functions due to external loads,an adaptively filtered precise integration method was proposed,which inherits the high precision of the precise integrationmethod,improves the computational efficiency and saves the memory required.Moreover,the perturbation method was introduced to avoid repeated computations of matrix exponential and terms due to external loads.Based on the filtering and perturbation techniques,an adaptively filtered precise integration method considering perturbation for stochastic dynamics problems was developed.Two numerical experiments,including a model of phononic crystal and a bridge model considering random parameters,were performed to test the performance of the proposed method in terms of accuracy and efficiency.Numerical results show that the accuracy and efficiency of the proposed method are better than those of the existing precise integration method,the Newmark-βmethod and the Wilson-θmethod.

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.

Research of a Response Precise Algorithm on a System with a Pseudo-viscous Frictional Energy Dissipator

The pseudo-viscous frictional energy dissipator（PVFED） is a new energy dissipator. This dissipator can be widely used in engineering for not only the friction is in direct ratio to velocity, but also the problem of viscous energy dissipator mucilage easily leaked has been overcome. The problem of how to get response of the PVFED sys- tem need to be solved before this dissipator can be used widely in engineering. The response calculation methods of the PVFED system on sina load was researched. Wilson-θ,Newmark-β and a precise integration algorithm was used separately to solve the system response and the calculation result in a different time step was compared. It was found from comparison that three calculation results were almost equivalent in a small time step. Calculation precision of Newmark-β and Wilson-θ was reduced and high calculation precision of a precise integration algorithm was kept in a large time step. The results show that it is an effective way to solve the response of a PVFED system by a precise integration method.

精细积分方法的发展与扩展应用

钟万勰院士于1991年首先提出计算矩阵指数的精细积分方法,其要点是2N类算法和增量存储。精细积分方法可给出矩阵指数在计算机意义上的精确解,为常微分方程的数值计算提供了高精度、高稳定性的算法,现已成功应用于结构动力响应、随机振动、热传导以及最优控制等众多领域。本文首先介绍矩阵指数精细积分方法的提出、基本思想和发展;然后依次介绍在时不变/时变线性微分方程、非线性微分方程以及大规模问题求解中发展起来的各种精细积分方法,分析了其优缺点和适用范围;最后介绍了精细积分方法的基本思想在两点边值问题、椭圆函数和病态代数方程等问题的扩展应用,进一步展示了该思想的特色。

A modified highly precise direct integration method for a class of linear time-varying systems

It is always a bottleneck to design an effective algorithm for linear time-varying systems in engineering applications.For a class of systems,whose coefficients matrix is based on time-varying polynomial,a modified highly precise direct integration(VHPD-T method)was presented.Through introducing new variables and expanding dimensions,the system can be transformed into a timeinvariant system,in which the transfer matrix can be computed for once and used forever with a highly precise direct integration method.The method attains higher precision than the common methods(e.g.RK4 and power series)and high efficiency in computation.Some numerical examples demonstrate the validity and efficiency of the method proposed.

大数据画像结合多种教学方法在医学影像技术专业磁共振实习中的应用

磁共振实习一直是医学影像技术专业学生实习的难点,各教学基地对于如何提高磁共振实习质量仍处于探索阶段。文章针对学生实习过程中的困惑、磁共振实习阶段存在问题,提出大数据画像结合多种教学方法,构建医学影像技术专业学生在磁共振实习中的大数据画像模型,详细阐述实施计划及设计流程并对实行此项教学改革的显著效果进行了分析与总结。此项教学改革有效避免了传统实习模式的固化形式,对学生进行因材施教精准培养的同时可提高教师自身专业水平,实现了师生双方的共同成长,为医学影像技术专业学生实习培养提供了新的思路和方法,为培养出更多优秀的毕业生奠定坚实的基础。

Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method

The Non-uniform rational B-spline （NURBS） enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper. The scaled boundary finite element method is a semi-analytical technique, which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction. In this method, only the boundary is discretized in the finite element sense leading to a re- duction of the spatial dimension by one with no fundamental solution required. Neverthe- less, in case of the complex geometry, a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often un- avoidable in the conventional finite element approach, which leads to huge computational efforts and loss of accuracy. NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape. In the proposed methodology, the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions, while the straight part of the boundary is discretized by the conventional Lagrange shape functions. Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analy- sis and the solution is obtained using the modified precise integration method. The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion. Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method. The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.

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.

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.