Precise Integration Method Research Articles

An Improved Element Free Galerkin Method and Precise Time-Step Integration Method for Solving Transient Heat Conduction Problems

An Improved Element Free Galerkin Method and Precise Time-Step Integration Method for Solving Transient Heat Conduction Problems

A modal precise integration method for the calculation of footbridge vibration response

This paper presents a computationally efficient methodology for determining footbridge vibration response when subject to pedestrian excitation suitable for use in a Monte Carlo framework. A general modal precise integration method is developed for moving force problems. A closed-form solution for the transition matrix for the important case of a simply-supported beam subject to harmonic moving loads is given. The proposed method is efficient and accurately estimates important statistics of response in pedestrian excitation problems. This method should therefore find value in assessing new or existing footbridges when subject to random pedestrian excitation.

Study on the propagation properties of one-dimensional photonic crystal with oblique incidence by the precise integration method

Propagation properties of one-dimensional photonic crystal with oblique incidence are simulated and analyzed using precise integration method in the paper. The variational principle based on the single variable in conditions of the oblique incidence is derived. Then, the dual variables are introduced and the equations are derived to the Hamiltonian system formulation. A precise integration method based on the Riccati equations is used to generate the export stiffness matrices. The problem can be solved by merging the export stiffness matrices and using the boundary conditions. Numerical examples demonstrate the high accuracy and efficiency of this method for solving problems of transmission property of one-dimensional photonic crystal.

Time-Varying LQG Control for Vibration of Coupled Vehicle-Bridge System

Time-varying linear quadratic Gaussian (LQG) control for vibration of coupled vehicle-bridge system is studied. The vehicle is modeled as a moving mass model with three degrees of freedom, which consists of vehicle body, bogie and wheel. Active suspensions are adopted for the primary and secondary ones, and the control forces are produced by two actuators placed between the bogie and wheel, and between the vehicle body and the bogie, respectively. Vehicle-bridge coupling systems are time-dependent, which lead to the time-varying Riccati differential equation and the time-varying Kalman-Bucy filter equation in the LQG controller design. However, both of them are solved precisely via precise integration method and symplectic conservative perturbation method. In the example, the time history responses of the bridge and the vehicle were calculated respectively for the vehicle with passive suspensions or with active suspensions. Numerical results show that with active suspensions adopted, ride comfort can be improved when the vehicles passing through the bridge.

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.

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.

A precise integration approach for dynamic impedance of rigid strip footing on arbitrary anisotropic layered half-space

The precise integration method (PIM) is proposed for the dynamic response analysis of rigid strip footing resting on arbitrary anisotropic multi-layered half-space. In the frequency domain, the governing equation of wave motion is converted into dual vector form of first-order ordinary differential equations which is solved by PIM. Each layer is divided into a large number (say, 2N) of mini-layers of equal thickness, within which characteristic matrices are assumed to vary following the Taylor series expansion to the fourth order. As a result, any desired accuracy of the displacements and stresses can be achieved by PIM. In addition, dual vector form equation makes it quite easily to combine two adjacent mini-layers into a new one. Each pass of combination reduces the total number of mini-layers by a half. The computational effort for the evaluation of the dynamic impedance of rigid strip footing can be reduced to a great extent. Numerical examples are provided to validate the efficiency and accuracy of the proposed approach.

An efficient approach for dynamic impedance of surface footing on layered half-space

A semi-analytical model for the evaluation of dynamic impedance of rigid surface footing bonded to multi-layered subsoil is proposed. The technique is based on the dual vector form of wave motion equation and Green's influence function of subdisk for horizontally layered half-space. The multi-layered half-space is divided into a quite large number of mini-layers and the precise integration method (PIM) is introduced for the numerical implementation. The PIM is highly accurate for solving sets of first-order ordinary differential equations with specified two-end boundary conditions. It can produce numerical results of Green's influence functions up to the precision of computer used. The dual vector form of wave motion equation makes the combination of two adjacent mini-layers/layers very easy. As a result, the computational effort for the evaluation of Green's influence function of the multi-layered half-space is reduced to a great extent. In order to satisfy the mixed boundary condition at the surface, the footing–soil interface is discretized into a number of uniformly spaced subdisk-elements. Comparisons illustrating the efficiency and accuracy of the proposed approach are made with a number of solutions available in the literature.

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 geological models.

A Wavelet Collocation Precise Integration Method for Laminated Rectangular Plates with Four Sides Clamped

By using Quasi-Shannon wavelet, we develop a wavelet collocation method for the Hamilton canonical equation of elastic bodies, and get the Hamilton canonical equation which is discrete in the plane of the rectangular laminated plate and continuum along its thickness. The approximate solution expressed by Quasi-Shannon wavelet satisfies the clamped boundary conditions of the plate easily. Hence, it is convenient to establish the wavelet collocation precise integration method for the static problem of the rectangular laminated plate with four sides clamped. Numerical results show that the current approach offers excellent predictive capability for rectangular composite laminated plate with clamped boundary conditions.

Numerical simulation of temperature field in heterogeneous material with the XFEM

The purpose of this paper is to focus on modeling temperature field in heterogeneous materials. The heat conductivity is adopted as the basic parameter for calculation. The extended finite element method (XFEM) is applied for simulation of temperature field. For one element that contains no material interface, the temperature function will be degenerated into that of the conventional finite element. For the element containing material interfaces, the standard temperature based approximation is enriched by incorporating level-set-based enrichment functions which model the interfaces. For unsteady temperature field, the improved precise integration method is adopted for the solution of the ordinary differential equations. The mesh generation can be considerably simplified and high-quality meshes are obtained; meanwhile the solution of good precision and stability can be achieved.

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 investigated. 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 differential 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.

Precise Integration Method for Solving Noncooperative LQ Differential Game

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

Application of precise integration in waveguide discontinuities with anisotropic dielectric

Waveguide discontinuities with anisotropic dielectric are simulated and analyzed by the precise integration method. The discrete coefficient matrices for the cross-section of the waveguide, which contains anisotropic dielectric, are deduced from the variational principle based on single variable corresponding to the vector wave equation. Introducing the dual-variables, the stiff matrices are calculated by using precise integration method in a Hamiltonion system. Then the problem is solved by assembling the finite elements. Numerical results show accuracy and good efficiency of the method. The influence of the components of permittivity and permeability on the waveguide transmission characteristic is also discussed.

Adaptive Wavelet Precise Integration Method for Nonlinear Black-Scholes Model Based on Variational Iteration Method

An adaptive wavelet precise integration method (WPIM) based on the variational iteration method (VIM) for Black-Scholes model is proposed. Black-Scholes model is a very useful tool on pricing options. First, an adaptive wavelet interpolation operator is constructed which can transform the nonlinear partial differential equations into a matrix ordinary differential equations. Next, VIM is developed to solve the nonlinear matrix differential equation, which is a new asymptotic analytical method for the nonlinear differential equations. Third, an adaptive precise integration method (PIM) for the system of ordinary differential equations is constructed, with which the almost exact numerical solution can be obtained. At last, the famous Black-Scholes model is taken as an example to test this new method. The numerical result shows the method's higher numerical stability and precision.

HAM-Based Adaptive Multiscale Meshless Method for Burgers Equation

Based on the multilevel interpolation theory, we constructed a meshless adaptive multiscale interpolation operator (MAMIO) with the radial basis function. Using this operator, any nonlinear partial differential equations such as Burgers equation can be discretized adaptively in physical spaces as a nonlinear matrix ordinary differential equation. In order to obtain the analytical solution of the system of ODEs, the homotopy analysis method (HAM) proposed by Shijun Liao was developed to solve the system of ODEs by combining the precise integration method (PIM) which can be employed to get the analytical solution of linear system of ODEs. The numerical experiences show that HAM is not sensitive to the time step, and so the arithmetic error is mainly derived from the discrete in physical space.

Wavelet-Based Homotopy Analysis Method for Nonlinear Matrix System and Its Application in Burgers Equation

To generalize the homotopy analysis method (HAM) to multidegree-of-freedom nonlinear system, the adaptive precise integration method (APIM) is introduced into the HAM, with which the almost exact value of the exponential matrix can be obtained. Combining the interval interpolation wavelet collocation method, HAM-based APIM can be employed to solve the nonlinear PDEs. As an example, Burgers equation is spatially discretized by the interval quasi-Shannon wavelet collocation method and solved by the proposed method to illustrate the effectiveness and great potential of the homotopy analysis method in nonlinear problems.

An efficient algorithm for computing the dynamic responses of one-dimensional periodic structures and periodic structures with defects

This paper proposes an efficient algorithm for computing the dynamic responses of one-dimensional periodic structures and periodic structures with defects. It uses the symmetric property of the periodic structure and the energy propagation feature of the dynamic system to analyze the algebraic structure of the matrix exponential corresponding to one-dimensional periodic structures and periodic structures with defects. By using the special algebraic structure of this matrix exponential and the precise integration method, an efficient and accurate algorithm is proposed for computing the matrix exponential corresponding to one-dimensional periodic structures or periodic structures with defects. Hence an efficient method is presented for computing the dynamic responses of one-dimensional periodic structures and periodic structures with defects. It is accurate, efficient and saves memory.

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 show that the IPIM is stable and gives the machine accuracy solutions.

Research on Dynamic Response of Bridge Subjected to Kinds of Load

A vehicle-bridge system is simplified as a model of Euler-Bernoulli beam subjected to kinds of load. Dynamic behavior of the beam is researched in this paper. Several equations of the beam traversed by static or moving load and mass are given. Some of these equations are time-varying nonlinear. In order to avoid the problem of instability solving the equations, precise time integration method is used. Results show that the transversal deflection of the mid-span of the beam is greater when a harmonic force or a mass is traveling on the bridge at a lower velocity or by accelerating.