Precise Time-step Integration Research Articles

New unconditionally stable and high-accuracy finite element procedures for high-frequency solid-fluid coupled dynamic systems

To achieve high-precision computation of the dynamic response of saturated poroelastic media under high-frequency loads, this paper develops new, unconditionally stable and high-accuracy finite element procedures based on the u-w and u-U solid-fluid coupled dynamic systems. In the procedures the solution domain is spatially discretized using the standard finite element method, while a precise time step integration method is introduced for temporal domain computations. A matrix-based spectral analysis is conducted to validate the stability of the proposed procedures. Subsequently, an error estimation and convergence analysis of the procedures are performed. Four classical examples involving different load types with varying time dependences are used to validate the numerical performance of the proposed procedures. Through rigorous comparison with analytical/reference solutions, the proposed procedures have been convincingly demonstrated to accurately capture the relative motion between the solid skeleton and the pore fluid under high-frequency loads. Furthermore, the procedures have remarkable advantages in both accuracy and efficiency compared to the commonly employed Newmark schemes.

An alternate state-space algorithm for dynamic solution, sensitivity analysis and parameter identification of dry friction systems

Dry friction is one of the most complicated nonlinear phenomena in mechanical systems, due to its discontinuous stick–slip property. This property certainly brings difficulty to the solution and identification of dry friction systems. To overcome the difficulty, an alternate state-space algorithm is proposed in this paper to transform the dry friction system into a piecewise linear system. In doing so, the stick–slip process is firstly modeled by a zero–one representation, with which the stick–slip transition conditions are deduced. Then, the transition points are detected by the transition conditions to break the system into several pure stick or slip phases; each phase corresponds to a linear system. Dynamic solution of the piecewise linear system is quickly acquired by precise time step integration over an arbitrary pure stick or slip phase and the sensitivity analysis, pertaining also to a piecewise linear equation, can proceed exactly in a similar way. As is noteworthy, the sensitivity may admit jumps at the transition points due to the discontinuity of the dry friction system and therefore, the affine relation between the sensitivity before and after a transition point is elaborately derived. Eventually, based on the acquired sensitivity and the Tikhonov regularization, a sensitivity approach is developed to inversely identify the parameters of the dry friction system. Numerical examples and an experimental case are studied to demonstrate the accuracy and efficiency of the proposed alternate state-space algorithm.

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 semianalytical solution for one‐dimensional transient wave propagation in a saturated single‐layer porous medium with a fluid surface layer

SummaryOne‐dimensional transient wave propagation in a saturated single‐layer porous medium with a fluid surface layer is studied in this paper. An analytical solution for a special case with a dynamic permeability coefficient kf → ∞ and a semianalytical solution for a general case with an arbitrary dynamic permeability coefficient are presented. The eigenfunction expansion and precise time step integration methods are employed. The solution is presented in series form, and thus, the long‐term dynamic responses of saturated porous media with small permeability coefficients can be easily computed. We first transform the nonhomogeneous boundary conditions into homogeneous boundary conditions, and then we obtain the eigenvalues and orthogonal eigenfunctions of the fluid–solid system. Finally, the solutions in the time domain are developed. As the model is one dimensional, geometric attenuation is absent, and only the attenuation in the saturated porous medium is considered. We can apply this model to analyse the influences of different seabed types on the propagation of acoustic waves in the fluid layer, which is very important in ocean acoustics and ocean seismic. This solution can also be employed to validate the accuracies of various numerical methods.

Improved Krylov Precise Time Integration Algorithm for Structural Dynamic Equations

An improved Krylov precise time-step integration algorithm is proposed to solve the second-order differential equations with arbitrary excitation directly and efficiently. This method could tackle the complex excitations and improve the computing efficiency by determining the bound of iterative subspace () accurately. The upper bound of could be obtained by computational efficiency analysis, whereas the error estimation is employed to compute the lower bound of . Hence, the application of the improved Krylov precise time-step integration algorithm could be extended widely by determining the bound of . Two numerical examples are also presented to demonstrate the practicability and the applicability of the proposed method.

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 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

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.

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.

Reduced-order method for computing critical eigenvalues in ultra large-scale power systems

This study presents a novel reduced-order method to find critical eigenvalues of ultra large-scale power system. First, the numerical solution of matrix exponential is computed by precise time-step integration. Secondly, based on the numerical solution, the numerical curve of the trace of matrix exponential is formed. Thirdly, the candidates of critical eigenvalues are extracted from the numerical curve of the trace by Prony analysis, and finally, a set of weight coefficients is calculated to confirm critical eigenvalues from candidate eigenvalues. Since the trace contains all eigenvalues, no critical eigenvalue can be lost in analysing the numerical curve of the trace. In the later period of time integration, the effect of all non-critical eigenvalues to the trace is decayed and oppositely the effect of all critical eigenvalues is amplified. Thus, several rightmost eigenvalues can only be extracted from the numerical curve of the trace. Case studies for 16 and 9004 order system have validated that the proposed method can be used to find all critical eigenvalues of ultra large-scale power system.

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.