Precise Time Step Integration Method Research Articles

A new two-dimensional transient forecast model of wellbore temperature based on precise time step integration method

Dramatic temperature changes in the wellbore can lead to the severe deformation and breakage of the wellbore string, packer seal malfunction, and other operational accidents. A new two-dimensional (2D) transient forecast model of the wellbore temperature is established to provide accurate temperature parameters for the precision mechanical analysis of the wellbore string and the safety evaluation of well integrity. Furthermore, a precise time step integration scheme of the forecast model using the precise time step integration method of a one-point subdomain is proposed. A solution method for the 2D transient forecast model of the wellbore temperature is proposed by combining the precise time step integration method. Finally, a case study focused on a high-temperature deep gas well with a transient wellbore temperature is presented by comparing the calculation results of the different models and solution methods. The sensitivity analysis of the wellbore temperature is performed as well. The results indicate that the forecast model in this paper has higher accuracy than the existing two-dimensional transient model. By contrast analysis of the finite difference method and the solution method that combines the precise time step integration method, the solution method has the advantages of better convergence and faster calculation speed. The major sensitive parameters that affect the stability time of wellbore temperature are ranked as follows: gas production rate, tubing size, specific heat, temperature gradient, gas relative density, and thermal conductivity. The work presented can provide a theoretical foundation and technological basis for the transient forecast of the wellbore temperature, which is critical for well completion design and wellbore safety evaluation.

Numerical simulation of dynamic fractures in 2D FGMs using the numerical manifold method

In this study, the numerical manifold method is used with an explicit time integration scheme to implement dynamic fracture analysis of 2D functionally graded materials. The framework of the numerical manifold method is used to discretize the dynamic equilibrium equation in the spatial domain. Instead of the commonly used Newmark method, the precise time step integration method is used for temporal discretization, and the detailed implementations of the proposed technique are described in detail in this paper. The dynamic stress intensity factors that can describe the inertia effect are extracted by the domain form of the interaction integral with nonequilibrium formulation. For verification purposes, four numerical examples with increasing complexity are investigated, in which different material gradients, crack configurations and loading forms are considered. Simulation results show that the proposed method achieves high reliability and precision for dynamic problems of 2D cracked functionally graded materials.

Implementation of a high-accuracy manifold element modelling scheme for dynamic fracture analysis under thermal-mechanical shock

In this study, a high-accuracy numerical manifold modelling scheme is proposed for simulating dynamic thermoelastic fracture problems of two-dimensional stationary cracks in linear elastic solids. The advantages of the numerical manifold method in discretizing the physical model with multiple discontinuities, and those of the precise time step integration method in time integral are integrated together in the proposed numerical scheme. The implementation details of the proposed modelling scheme are well presented. Dynamic stress intensity factor considering the inertial effect is computed. Through three dynamic fracture examples with pure thermal shock, heat flux shock and mechanical shock, respectively, the validity of the present scheme is first verified. Then the proposed scheme is applied for dynamic fracture analysis under mixed thermal-mechanical shock, in which different loading forms, such as step loading, linear loading and harmonic loading, are considered. The results show that the present modelling scheme is extremely stable and convergence. The advantages of the present scheme on accuracy, efficiency, time-step-independence over commonly used Newmark method are also fully demonstrated.

Dynamic fracture analysis using a high-accuracy manifold element modelling scheme

In this paper, a high-accuracy manifold element modelling scheme for dynamic fracture analysis of two-dimensional elastic cracked bodies is presented. The displacement discontinuity across crack surfaces is naturally simulated attributing to the dual cover systems in the numerical manifold method. The singularity of field gradients at crack tips is appropriately captured through the use of asymptotic basis functions. The discrete equations for dynamic analysis are solved by a new fully explicit direct time integration algorithm based on precise time step integration method, in which Taylor expansion theorem and downscaling time step integration method are used to calculate the exponential transfer matrix with high accuracy. To eliminate the sensitivity of accuracy of results to time-step size, Fourier expansion with trigonometric basis functions is utilized to obtain the approximate expression for different time-dependent loadings, the exponential transfer matrix is also further utilized to calculate the response matrix of inhomogeneous term. A matrix-based spectral analysis is used to prove the stability and convergence of proposed scheme, the results demonstrate that the present scheme is unconditionally stable and convergent. Finally, the applicability and accuracy of the proposed scheme for calculating dynamic stress intensity factors are examined by two numerical examples with different crack configurations and time-dependent loadings. By comparing with the analytical/reference solutions, it is proved that the present scheme has significant advantages in accuracy and efficiency than Newmark scheme for dynamic fracture analysis.

A multi-temporal series high-accuracy numerical manifold method for transient thermoelastic fracture problems

This paper presents a multi-temporal series high-accuracy numerical manifold method (NMM) for fracture analysis of solid materials under highly time-dependent thermal loadings. In the present method, the framework of NMM is used to perform space discretization of the thermo-mechanical coupling system, due to its advantages in accurately calculating field variables and describing discontinuity. An explicit direct integration scheme based on the precise time step integration method (or PTSIM scheme for short) is constructed for discretization in the time domain. In the present PTSIM scheme, the time-dependent thermal loadings are expressed by means of the basic function in a polynomial form. Highly accurate calculation of the response matrix for the load item is achieved by introducing the exponential transfer matrix. The results of stability, error and convergence analyses indicate that the proposed method is unconditionally stable and convergent. The superior advantages of present method are firstly verified by one example for transient thermal analysis, then four numerical examples with different time-dependent thermal boundary conditions and crack configurations are used for fracture analysis. The results show that the present method is unconditionally stable and time-independent, and high-accuracy can be achieved even for highly time-dependent thermal loadings and large time-step sizes.

Transient wave propagation in a multi‐layered soil with a fluid surface layer: 1D analytical/semi‐analytical solutions

AbstractThis paper studies one‐dimensional transient wave propagation in a saturated multi‐layered soil with a fluid surface layer. Based on Biot theory, the eigenfunction expansion method, the transfer matrix method, the state‐space method, and the precise time step integration method are employed to develop analytical/semi‐analytical solutions. For a special case (large dynamic permeability coefficient), we obtain an analytical solution. For a general case (arbitrary dynamic permeability coefficient), due to the influence of the damping term, we develop a semi‐analytical solution that can be used to evaluate sediment with small dynamic permeability coefficients. The newly derived solutions are verified with solutions obtained by the Laplace transformation and numerical inverse Laplace transformation, which are also presented here. The newly derived solutions are presented in series form in the time domain and are straightforward and explicit so that the solutions are easy to implement. Through numerical examples, we analyse the influences of the dynamic fluid permeability coefficients on the transient response of the model, which is important in marine seismic and ocean acoustics applications.

An unconditionally stable and high-accuracy finite element scheme for dynamic analysis of saturated poroelastic media

This paper presents a finite element scheme for analyzing the dynamic response of the saturated poroelastic media. The proposed scheme employs the standard finite element method for space domain discretization of the dynamic u-p equation system. We introduce the precise time step integration method as an explicit, direct integration algorithm for handling the time derivatives. The spectral radius analysis, the error and convergence analysis show that the proposed scheme is unconditionally stable as well as convergent with numerical solutions that are of any order accuracy to exact solutions. Compared with the Newmark scheme, the advantages of the proposed scheme are demonstrated by simulating five examples in both one- and two-dimensions. The proposed scheme yields high-precision results and exhibits good adaptability under large time step size, which remarkably outperforms the Newmark scheme.

A novel finite element two-step solution scheme for fully coupled hydro-mechanical processes in poroelastic media

A novel two-step solution scheme (TSSS) is described for fully coupled hydro-mechanical (FCHM) analysis of saturated poroelastic media. The TSSS is based on the pressure formulation in two-step. In step one, the pressure field is obtained directly by solving the sub-problems with a reduced scale of displacement variables. This process is fully decoupled. In step two, the displacement field is calculated by staggered iteration of pressure variables. The finite element method (FEM) is used for discretization of the FCHM differential equations in the space domain. The precise time step integration method is performed for the time derivatives. The stability and convergence of the TSSS are proved using a matrix-based spectral analysis in the time domain. It is demonstrated that the TSSS is unconditionally stable, fully explicit and highly precise. The algorithmic error estimation results indicate that the numerical performance in the time domain can match the computer precision. Theoretically, the algorithmic error is caused only by mesh discretization. The stability and accuracy of the TSSS are verified and calibrated by numerical examples. By comparing with the analytical or reference solutions, it is shown that the TSSS result is highly precise, and it is remarkably better than the standard FEM in terms of precision. In addition, the numerical results are stable and independent of the time step size. The numerical experiments also demonstrate the stability, convergence and precision for the pressure formulation of TSSS. The proposed scheme has great potential in engineering applications for long timescale problems, especially the problems focusing on the evolution of pressure field.

An unconditionally stable explicit and precise multiple timescale finite element modeling scheme for the fully coupled hydro-mechanical analysis of saturated poroelastic media

An unconditionally stable, fully explicit and highly precise multiple timescale finite element modeling scheme is described for a fully coupled hydro-mechanical (FCHM) analysis of saturated poroelastic media. The finite element method (FEM) is used for the discretization of the FCHM differential equation in the space domain. Direct integration is performed based on the precise time step integration method (PTSIM) for the time derivatives. Two configurations for the proposed scheme are constructed (abbreviated as PTSIM-f1 and -f2, respectively). The stability and convergence of the PTSIM-f1 and -f2 are proved using a matrix-based spectral analysis in the time domain. It is demonstrated that the explicit scheme proposed in this paper is unconditionally stable and independent of the time-step size. The algorithmic error estimation results indicate that the numerical modeling performed using PTSIM-f1 and -f2 in the time domain match the computer precision. Theoretically, the algorithmic error is caused by only the mesh discretization. Therefore, the proposed modeling scheme is a semi-analytical scheme. The applicability and accuracy of the proposed scheme are examined using sample calculations. By comparing with the analytical solutions, it is indicated that the modeling results have significant advantages over the standard FEM in terms of precision and computational efficiency for large timescales.

Variational Multiscale Element-Free Galerkin Method and Precise Time Step Integration Method for Convection–Diffusion Problems

To simulate the convection–diffusion problems and improve the computational efficiency of Galerkin meshless methods, an explicit algorithm that combines variational multiscale element-free Gakerkin method with precise time step integration is presented. Variational multiscale element-free Gakerkin method is used to overcome the numerical spurious oscillations for convection-dominated problems; meanwhile in order to save the computational time, the precise time step integration method is adopted for the solution of the ordinary differential equations. Two numerical examples are considered to demonstrate the stability, accuracy, and computational efficiency of the method. The numerical results illustrate that the presented method not only has high calculation accuracy, but also brings a significant computational time saving for solving the unsteady convection–diffusion problems.

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

An improved element free Galerkin method coupled the precise time-step integration method in the time domain is proposed for solving transient heat conduction problem with spatially varying conductivity in the paper. Firstly the nodal influence domain of element free Galerkin methods is extended to arbitrary convex polygon rather than rectangle and circle. When the dimensionless size of the nodal influence domain is 1.01, the shape function almost possesses interpolation property, thus essential boundary conditions can be implemented without any difficulties for the meshless method. Secondly, the precise time-step integration method is selected for the time discretization in order to improve the computational efficiency. Lastly, one numerical example is given to illustrate the highly accurate and efficient algorithm.

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

Sensitivity and Hessian Matrix Analysis of Evolutionary PSD Functions for Nonstationary Random Seismic Responses

AbstractThis paper describes a numerical method for calculation of the sensitivity and Hessian matrix of the response evolutionary power spectral density (PSD) functions of structures subjected to uniformly modulated evolutionary random seismic excitation. The method is formulated based on the pseudoexcitation method and the Gauss precise time step integration method. The evolutionary nonstationary random response analysis is converted into step-by-step integration computations using the pseudoexcitation method. The formulas of the pseudoresponses and their first and second derivatives with respect to the structural design variables are derived based on the Gauss precise time step integration method. The evolutionary PSD functions, their sensitivity, and the Hessian matrix are calculated using the pseudoresponses and their first and second derivatives, respectively. Finally, the evolutionary PSD functions’ sensitivity and Hessian matrix analysis of a three-story, two-bay planar frame subjected to uniforml...

Transient heat conduction analysis using the MLPG method and modified precise time step integration method

The meshless local Petrov–Galerkin (MLPG) method in conjunction with the modified precise time step integration method in the time domain is proposed for transient heat conduction analysis in this paper. The MLPG method is often referred to as a truly meshless method because it requires no elements or background cells for either field interpolation or background integration. Local weak forms are developed using weighted residual method locally from the partial differential equation of transient heat conduction. In order to simplify the treatment of essential boundary conditions, the natural neighbour interpolation (NNI) is employed for the construction of trial functions. Moreover, the three-node triangular FEM shape functions are taken as test functions to reduce the order of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with modified precise time step integration method in the time domain. The availability and accuracy of the present method for transient heat conduction analysis are tested through numerical examples.

A numerical method of calculating first and second derivatives of dynamic response based on Gauss precise time step integration method

This paper describes an accurate and efficient method for calculating the first and second derivatives of dynamic response with respect to design variables for linear structural systems subjected to transient loads. An efficient algorithm to calculate the dynamic responses and their first and second derivatives is formulated based on Gauss precise time step integration method. The algorithm is achieved by direct differentiation and only a single dynamic analysis is required. Several numerical examples are comparatively demonstrated using the new developed method, analytical method, and central difference method. The results show that the new method is highly accurate compared with the analytical approach and is more efficient than the central difference method.

On the precise integration methods based on Padé approximations

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can give precise numerical results approaching exact solutions at the integration points, but it is conditionally stable. By combining Padé approximation and the generalized Padé approximation of the matrix exponential function in precise integration, and by using three different types of quadrature formulae, a new generalized family of precise time step integration methods is developed to achieve unconditional stability and arbitrary order of accuracy. Numerical studies indicate that they are unconditionally stable algorithms with controllable numerical dissipation. They also demonstrate the validity and efficiency of these algorithms.

Precise Time-Step Integration Algorithms Using Response Matrices with Expanded Dimension

In this paper, the precise time-step integration method by step-response and impulsive-response matrices is further developed by expanding the dimension of the matrices so as to avoid computing the particular solutions separately. Two new precise time-step integration algorithms with excitations described by second-order and first-order differential equations are proposed. The first method is a direct extension of the existing algorithms. However, the extended system matrices are not symmetrical. In the second method, the Duhamel integrals are used as the particular solutions. As a result, the responses can be expressed in terms of the given initial conditions and the step-response matrix, the impulsive-response matrix and a newly derived Duhamel-response matrix. The symmetry property of the system matrices can be used in the computation. However, it will first require a calculation of the Duhamel-response matrix and its derivative. To reduce the computational effort, the relation between the Duhamel-response matrix and its derivative is established. A special computational procedure for periodic excitation is also discussed. Numerical examples are given to illustrate the present highly efficient algorithms.

A Fast Architecture-Level Thermal Analysis Method for Runtime Thermal Regulation

As power consumption and the corresponding heat dissipated on a die grow rapidly, efficient on-chip temperature regulation becomes imperative for today's high performance microprocessors. Temperature tracking based on the on-chip thermal sensors is not sufficient as the temperature hot spots keep changing with the load. One way to mitigate this problem is by means of software sensors, where temperature of any location is computed based on realtime power information and calibrated with the physical sensors. In this paper, we present a very efficient numerical thermal analysis method, which is suitable for fast temperature tracking and runtime thermal regulation. The proposed method, called FEKIS, combines two existing numerical techniques: extended Krylov subspace reduction technique to reduce the thermal circuit complexity and large-step integration method to exploit the sampled-based power input traces, which is typical in the power traces at the architectural and operation system levels. Experimental results show that FEKIS archives 10x speedup over recently proposed state-of-the art thermal moment matching technique 1,2 and the precise time-step integration method only, and three orders of magnitude faster than the traditional numerical integration method with high accuracy.

A Precise Time-Step Integration Method for Transient Analysis of Lossy Nonuniform Transmission Lines

This paper presents a novel time-domain integration method for transient analysis of nonuniform multiconductor transmission lines (MTLs). It can solve the time response of various kinds of transmission lines with arbitrary coupling status. The spatial discretization in this method is the same as the finite-difference time-domain (FDTD) algorithm. However, in order to eliminate the Courant-Friedrich-Levy condition constraint, a precise time-step integration method is utilized in time-domain calculation. It gives an analytical solution in the time domain for the spatial discretized Telegrapher's equations with linear boundary conditions. Large time steps can be adopted in the integration process to achieve accurate results efficiently. In the analysis of transmission lines with frequency-dependent parameters, a passive equivalent model is introduced, which leads to the similar semidiscrete model as that for the frequency-independent case. In addition, a rigorous proof of the passivity of the model is provided. Numerical examples are presented to demonstrate the accuracy and stability of the proposed method.

A Novel Time-domain Integration Method for Transient Analysis of Nonuniform Transmission Lines

This paper presents a precise time-step integration method for transient analysis of lossy nonuniform transmission lines. The spatial discretization in this method is the same to the finite difference time domain (FDTD) algorithm. However, in order to eliminate the Courant condition constraint, the precise integration method is utilized in the time-domain calculation. It gives quasi-exact solutions in time domain for the spatial discretized Telegrapher’s equations. Since the exponential matrix is calculated precisely, large time step can be adopted in the integration process to achieve accurate results efficiently. A numerical example is presented to demonstrate the accuracy and stability of the proposed method. DOI: 10.2529/PIERS060904030908