A DELAY-DEPENDENT STABILITY CRITERION FOR NONLINEAR STOCHASTIC DELAY-INTEGRO-DIFFERENTIAL EQUATIONS

A type of complex systems under both random influence and memory effects is considered. The systems are modeled by a class of nonlinear stochastic delay-integrodifferential equations. A delay-dependent stability criterion for such equations is derived under the condition that the time lags are small enough. Numerical simulations are presented to illustrate the theoretical result.

Dissipativity of Multistep Runge-Kutta Methods for Nonlinear Neutral Delay-Integro-Differential Equations with Constrained Grid

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear neutral delay-integro-differential equations.We investigate the dissipativity properties of-algebraically stable multistep Runge-Kutta methods with constrained grid.The finite-dimensional and infinite-dimensional dissipativity results of-algebraically stable multistep Runge-Kutta methods are obtained.

General Modified Split-Step Balanced Methods for Stiff Stochastic Differential Equations

A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already reported split-step balanced methods, the drift increment function of the methods can be taken from any chosen ane-step ordinary differential equations （ODEs） solver. The schemes is proved to be strong convergent with order one. For the mean-square stability analysis, the investigation is confined to two cases. Some numerical experiments are reported to testify the performance and the effectiveness of the methods.

An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models

A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters.

Three-stage Stiffly Accurate Runge-Kutta Methods for Stiff Stochastic Differential Equations

In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations（SDEs）.Two methods,a three-stage stiffly accurate semi-implicit（SASI3） method and a three-stage stiffly accurate diagonally implicit （SADI3） method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.

Numerical Solution for Stiff Dynamic Equations of Flexible Multibody System

A nonlinear numerical integration method, based on forward and backward Euler integration methods, is proposed for solving the stiff dynamic equations of a flexible multibody system, which are transformed from the second order to the first order by adopting state variables. This method is of A0 stability and infinity stability. The numerical solutions violating the constraint equations are corrected by Blajer＇s modification approach. Simulation results of a slider-crank mechanism by the proposed method are in good agreement with ones from other literature.

Modeling Fast Diffusion Processes in Time Integration of Stiff Stochastic Differential Equations

Numerical algorithms for stiff stochastic differential equations are developed using lin-ear approximations of the fast diffusion processes,under the assumption of decoupling between fast and slow processes.Three numerical schemes are proposed,all of which are based on the linearized formulation albeit with different degrees of approximation.The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems.Convergence analysis is conducted for one of the schemes,that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1.Approximations arriving at the other two schemes are discussed.Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems.

Numerical Study of the Vibrations of Beams with Variable Stiffness under Impulsive or Harmonic Loading

The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.

Development of component stiffness equations for thread-fixed one-side bolt connections to an enclosed rectangular hollow section column under tension

The derivation and validation of analytical equations for predicting the tensile initial stiffness of threadfixed one-side bolts(TOBs),connected to enclosed rectangular hollow section(RHS)columns,is presented in this paper.Two unknown stiffness components are considered:the TOBs connection and the enclosed RHS face.First,the trapezoidal thread of TOB,as an equivalent cantilevered beam subjected to uniformly distributed loads,is analyzed to determine the associated deformations.Based on the findings,the thread-shank serial-parallel stiffness model of TOB connection is proposed.For analysis of the tensile stiffness of the enclosed RHS face due to two bolt forces,the four sidewalls are treated as rotation constraints,thus reducing the problem to a two-dimensional plate analysis.According to the load superposition method,the deflection of the face plate is resolved into three components under various boundary and load conditions.Referring to the plate deflection theory of Timoshenko,the analytical solutions for the three deflections are derived in terms of the variables of bolt spacing,RHS thickness,height to width ratio,etc.Finally,the validity of the above stiffness equations is verified by a series of finite element(FE)models of T-stub substructures.The proposed component stiffness equations are an effective supplement to the component-based method.

Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method

In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.

A ONE-STEP EXPLICIT FORMULA FOR THE NUMERICAL SOLUTION OF STIFF ORDINARY DIFFERENTIAL EQUATION

In this paper, a new one-step explicit method of fourth order is derived. The new method is proved to be A-stable and L-stable, and it gives exact results when applied to the test equation y’=λy with Re(λ)【0, Also several numerical examples are included.

ARC-LENGTH METHOD FOR DIFFERENTIAL EQUATIONS

An arc_length method is presented to solve the ordinary differential equations (ODEs) with certain types of singularity such as stiff property or discontinuity on continuum problem. By introducing one or two arc_length parameters as variables, the differential equations with singularity are transformed into non_singularity equations, which can be solved by usual methods. The method is also applicable for partial differential equations (PDEs), because they may be changed into systems of ODEs by discretization. Two examples are given to show the accuracy, efficiency and application.

STIFFNESS EQUATION OF FINITE SEGMENT FOR FLEXIBLE BEAM-FORMED STRUCTURAL ELEMENTS

The finite segment modelling for the flexible beam-formed structural elements is presented, in which the discretization views of the finite segment method and the difference from the finite element method are introduced. In terms of the nodal model, the joint properties are described easily by the model of the finite segment method, and according to the element properties, the assumption of the small strain is only met in the finite segment method, i. e., the geometric nonlinear deformation of the flexible bodies is allowable. Consequently,the finite segment method is very suited to the flexible multibody structure. The finite segment model is used and the are differentiation is adopted for the differential beam segments. The stiffness equation is derived by the use of the principle of virtual work. The new modelling method shows its normalization, clear physical and geometric meanings and simple computational process.

A micromechanical model based on hypersingular integro-differential equations for analyzing micro-crazed interfaces between dissimilar elastic materials

The current work models a weak(soft) interface between two elastic materials as containing a periodic array of micro-crazes. The boundary conditions on the interfacial micro-crazes are formulated in terms of a system of hypersingular integro-differential equations with unknown functions given by the displacement jumps across opposite faces of the micro-crazes. Once the displacement jumps are obtained by approximately solving the integro-differential equations, the effective stiffness of the micro-crazed interface can be readily computed. The effective stiffness is an important quantity needed for expressing the interfacial conditions in the spring-like macro-model of soft interfaces. Specific case studies are conducted to gain physical insights into how the effective stiffness of the interface may be influenced by the details of the interfacial micro-crazes.

解Stiff常微分方程的精确指数拟合法

提出解Ｓｔｉｆ方程的精确指数拟合法。一个数值方法称为精确指数拟合法，如果将它用于试验方程ｙ＝λｙ，Ｒｅ（λ）＜０所得到的数值解对一切λ（Ｒｅλ＜０）都是精确的。显然，精确指数拟合法的要求比Ｌｉｎｉｇｅｒ和Ｗｉｌｏｕｇｈｂｙ［１］提出的指数拟合法更强。此外，我们建立解Ｓｔｉｆ方程的Ｅ稳定概念：一个单步方法说成是Ｅ稳定的，如果将它用于试验方程ｙ＝λｙ，Ｒｅ（λ）＜０时的数值解满足ｙｎ＋１／ｙｎ＝ｅλｈ，可见精确指数拟合法一定是Ｅ稳定的。又由Ｅ稳定可推出Ｌ稳定性和Ａ稳定性。从而精确指数拟合法一定是Ｅ稳定的Ｌ稳定的和Ａ稳定的。根据Ｓｔｉｆ方程解的特征，我们在积分区间的每个部分区间〔ｔｎ，ｔｎ＋１〕上局部地用形如Ｉ（ｔ）＝ＡｅＢｔ或Ｉ（ｔ）＝Ａ＋ＢｅＣｔ的指数型函数来逼近微分方程的解ｙ（ｔ）。

一类求解Stiff方程高精度L─稳定的显示单步方法

求文在参考文献［１］、［２］的基础上，构造了一类求解Ｓｔｉｆｆ方程组Ｌ－稳定的高精度显式单步法．本文方法是对参考文献［１］、［２］中的方法的改进与推广．与现有的隐式方法相比，具有稳定性好，精度高计算简便，适用范围广泛等特点．对于某些类型的Ｓｔｉｆｆ问题是很有效的．

A GENERAL PROCEDURE TO CAPTURE THE "DYNAMIC STIFFNESS

A general procedure to capture the 'dynanmic Stiffness' is presented in this paper. The governing equations of motion are formulated for an arbitrary flexible body in large overall motion based on Kane's equations . The linearization is performed peroperly by means of geometrically nonlinear straindisplacement relations and the nonlinear expression of angular velocity so that the 'dynamical stiffness' terms can be captured naturally in a general tcase. The concept and formulations of partial velocity and angular velocity arrays of Huston's method are extended to the flexible body and form the basis of the analysis. The validity and generality of the procedure presented in the paper are verified by numerical results of its application in both the beam and plate models.

液压系统Stiff隐式状态方程的数值解法

本文论述了液压系统模块式建模法中隐式状态方程的Stiff问题和数值解法,计算了含有稳定参数s的四阶Runge-Kutta型公式的绝对稳定区间,通过实例检验表明直接代数解法具有稳定可靠、精度高等优点。采用含稳定参数s的Runge-Kutta法的直接代数解法是解决液压仿真中Stiff问题的可行途径。

基于滑动元法的连续拉索滑移分析

实际工程中,拉索在索撑节点处滑移十分普遍。考虑滑移的影响,索撑节点不能简单地用铰接模拟。现有的拉索滑移分析方法往往需要迭代,计算过程复杂且收敛性无法保证。本文提出一种拉索滑移分析新方法——滑动元法,以预应力鱼腹梁为例进行理论推导,介绍了滑动元法的基本原理和计算步骤。结合有限元软件,利用滑动元法对算例进行计算分析,结果表明,滑动元法计算快速无需迭代,可以较为准确地分析拉索滑移。

解常微分方程初值问题的K步k+1阶线性多步公式集及其stiff稳定性

本文利用算子方法导出了一般的k步k+1阶线性多步公式集其中的系数β_i及误差系数C_(k+2)可以表示为α_i的函数(i=0,1,2,…,k):从而可以方便地构造出满足稳定性要求的任意k步k+1阶线性多步公式,并同时给出它的误差系数。是否存在k步k+1阶stiff稳定的线性多步公式?,对于k=1,2,3的情形,本文作出了论证,答案是否定的。