International Journal of Structural Stability and Dynamics Vol. 16 (2016) 1550009 (22 pages) # .c World Scienti¯c Publishing Company DOI: 10.1142/S0219455415500091
Time Integration Method Based on Discrete Transfer Function
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M. Rezaiee-Pajand* and M. Hashemian Department of Civil Engineering Ferdowsi University of Mashhad, Iran *
[email protected] Received 28 July 2014 Accepted 23 February 2015 Published 16 April 2015 Complex structural dynamic problems are normally analyzed by ¯nite element and numerical integration techniques. An explicit time integration algorithm with second-order accuracy and unconditional stability is presented for dynamic analysis. Utilizing weighted factors, the current displacement and velocity relations are de¯ned in terms of the accelerations of two previous time steps. The concept of discrete transfer function and the pole mapping rule from the control theory are exploited to develop the new algorithm. Several linear and nonlinear dynamic analyses are performed to verify the e±ciency of the method compared with the well-known Newmark method. Keywords: Discrete transfer function; explicit time integration; unconditional stability; accuracy; structural dynamics.
Nomenclature A: C: GðsÞ: GðzÞ: K: L: M: R: s: T: : :: X; X; X:
Ampli¯cation matrix Damping matrix Continuous transfer function Discrete transfer function Sti®ness matrix Load operator Mass matrix Load vector Complex argument in Laplace domain Period of the system Displacement, velocity and acceleration vectors
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z: t; dt: : ; ; ; ’; : !: :
Complex argument in z-domain Time step value Spectral radius Weighted factors Angular frequency Damping ratio
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1. Introduction The dynamic analysis of linear and nonlinear problems in structural mechanics requires solving the equation of motion in a stable and accurate manner. The linear dynamic equilibrium equation of motion is written in the following form1: ::
:
MX þ CX þ KX ¼ R;
ð1Þ
where M, C and K are the mass, damping and sti®ness matrices, respectively, R is the vector of external forces, X is the nodal displacement vector, and each super dot indicates di®erentiation with respect to the time. In general, there is no closed-form solution for Eq. (1), and in most cases, direct time integration methods are used for ¯nding the approximate solution. The main objective of these numerical approaches is to achieve a function X ¼ XðtÞ, satisfying the equation of motion at discrete time intervals t apart, and also, the next initial conditions: :
Xð0Þ ¼ X0 ;
:
:
Xð0Þ ¼ X0 ;
ð2Þ
where X0 and X0 are known displacement and velocity vectors at t0 , respectively. It should be added that a nonlinear analysis will be required, if the mass, damping or internal force of the system is a nonlinear function of the displacement. All time integration schemes can fundamentally be classi¯ed as either explicit or implicit. The procedure is called explicit, if the unknown solutions in each time step can be directly derived using the known solutions of the previous time steps. On the other hand, in implicit approaches, a system of equations consists of the known solutions of previous time steps plus the unknown solutions of the current time step must be solved simultaneously to obtain the unknown solutions. Each way de¯nitely has its merits and demerits. The most striking trait of implicit formulations is their large stability ranges. However, since the convergence of these methods usually requires iteration procedure at each time step, they are often complicated and time consuming. Great e®orts have been made so far to obtain less computationally expensive methods.2–5 The well-known Newmark processes,6 HHT-algorithm,7 WBZ scheme,8 generalized- method,9 the family of IHOA tactics,10 the technique presented by Rezaiee Pajand and Sarafrazi in Ref. 11 and the family of MIHOA algorithms12 are some examples of existing implicit methods. 1550009-2
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Time Integration Method Based on Discrete Transfer Function
On the other hand, there are explicit schemes. The most important feature of these techniques is their independence of any matrix operation or iterative procedure at each time step. If the mass and damping matrices are diagonal, an explicit scheme will be accomplished by vector operators.13,14 Consequently, compared to an implicit procedure, an explicit scheme contains less computational e®ort. This advantage makes this technique a suitable method for problems with dominant high frequency modes.15 The implementation of explicit methods is also easier in parallel processing techniques, which leads to reduced computational time. Despite being economical, the conditional stability of explicit methods has created di±culties in their applications to structural dynamics. Apart from a few methods presented mostly in the last decade,15–22 other existing explicit methods are conditionally stable.23–26 In order to overcome this problem, numerous studies have been carried out.11,13,18,27 In many of the proposed algorithms, only the results of the last time step are used in the equations and the known solutions of previous time steps are ignored. To enhance the e±ciency of numerical time integration, some researchers have employed these data in the main equations. In 2008, Rezaiee-Pajand and Alamatian o®ered a predictor– corrector algorithm using the acceleration of several time steps in the displacement and velocity formulations.10 This technique has also been used by Keierleber et al. in an implicit algorithm.28 In 1996, Zhai o®ered a simple explicit scheme that uses the accelerations of two previous time steps.26 In a latter paper he also developed a predictor–corrector integration algorithm employing his proposed explicit method as a predictor and the Newmark implicit method as a corrector. According to Zhai, in spite of the existence of advanced techniques, the second order accurate central di®erence method (CDM) is still the most popular explicit algorithm. He also mentioned that the reason of this popularity is the acceptable stability range of CDM, while the other methods have limitations in their stability ranges. Consequently in the latter paper, he tried to develop a method with a stability limit at least equal to the CDM one. The stability analysis of an algorithm for structural analysis is performed by investigating its spectral radius. In control engineering, to describe the dynamic behavior of a linear time-invariant (LTI) system, speci¯cally its stability conditions, the concept of transfer function is usually used. For a LTI system, the transfer function transfers the input of that system to its output. These functions are classi¯ed into two categories: continuous and discrete. The continuous transfer functions are de¯ned in the Laplace domain, while discrete transfer functions are described in the Z-domain.29 The stability criteria according to the control theory are the poles of the transfer function. As it was mentioned by Chen and Ricles, the poles of a discrete transfer function are equal to the eigenvalues of the ampli¯cation matrix of a time integration algorithm.17 By utilizing this fact in the present paper, a new technique is proposed. Extensive numerical experiences demonstrate that the proposed method possesses some notable properties. It is explicit, second-order accurate, and at the same time unconditionally stable.
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2. Stability in Control Theory In structural engineering, the most common technique for analyzing the stability of a time integration algorithm is to investigate the eigenvalues of the ampli¯cation matrix of the system. To this end, usually a linear SDOF system is considered. The discretized form of equation of motion for a SDOF system can be given as ::
:
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mxiþ1 þ cxiþ1 þ kxiþ1 ¼ riþ1 ;
ð3Þ
where m, c and k are the mass, damping and sti®ness, respectively, riþ1 is the external force at time tiþ1 , xiþ1 is the nodal displacement at this instant, and each super dot indicates di®erentiation with respect to time. Using the relations of displacement and velocity as well as Eq. (3), we can write the following matrix equation: 8 :: 9 8 :: 9 < xiþ1 = < xi = : : ¼ A þ fLgð tþ rÞ; ð4Þ x x : iþ1 ; : i; xiþ1 xi where matrix A and vector L are called the integration approximation (or ampli¯cation matrix) and load operator, respectively.12 However, if the accelerations of previous time steps are included in the calculation, they will be added to Eq. (4)28: 9 9 8 :: 8 :: xi > xiþ1 > > > > > > > > > > > > > . > . > > > > > > > = = < .. > < .. > :: ¼ A :: þ fLgð tþ rÞ: ð5Þ ximþ2 > ximþ1 > > > > > > > > > > > > > > > > x: iþ1 > > x: i > > > > > ; ; : : xiþ1 xi The spectral radius of an algorithm is de¯ned by ¼ maxði Þ, where i is the ith eigenvalue of A. For linear problems, an integration approach is unconditionally stable if 1 for all t=T 2 ½0; 1Þ.9 Moreover, it is conditionally stable if 1 for any time step t, when t=T is smaller than or equal to a certain value, usually called the stability limit. In control engineering, to analyze the stability of a system, the poles of the transfer function of the system are the most common criteria that must be examined. For a linear, time-invariant, di®erential equation system, the transfer function represents the relationship between the input and output. For continuous-time input xðtÞ and output yðtÞ, the transfer function is the ratio of the Laplace-transform of the output to the Laplace-transform of the input, when all the initial conditions are taken as zero.30 For a linear time-invariant system, the di®erential equation that includes the nth derivative of the output yðtÞ and the mth derivative of the input xðtÞ is as follows: :
:
a0 y ðnÞ þ a1 y ðn1Þ þ þ an1 y þ an y ¼ b0 x ðmÞ þ b1 x ðm1Þ þ þ bm1 x þ bm x: ð6Þ 1550009-4
Time Integration Method Based on Discrete Transfer Function
The continuous transfer function has the form below:
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GðsÞ ¼
L½output Y ðsÞ b0 s m þ b1 s m1 þ þ bm1 s þ bm ¼ ¼ L½input XðsÞ a0 s m þ a1 s m1 þ þ an1 s þ an
ð7Þ
where GðsÞ is the continuous transfer function, s is a complex argument in the Laplace domain, Y ðsÞ and XðsÞ are the Laplace transforms of yðtÞ and xðtÞ, respectively and bi and ai are real coe±cients of the numerator and denominator determined by the characteristics of the system. For discrete-time systems, the transfer function is the ratio of the z-transform of the output to the z-transform of the input, when all initial conditions are taken as zero. For a LTI system, the di®erential equation that includes the nth time step data of the output y and the mth time step data of the input x, the next formula is held: a0 yn þ a1 yn1 þ þ an1 y1 þ an y0 ¼ b0 xm þ b1 xm1 þ þ bm1 x1 þ bm x0 : ð8Þ The discrete transfer function can be written in the form below: GðzÞ ¼
Z½output Y ðzÞ b0 z m þ b1 z m1 þ þ bm1 z þ bm ¼ ¼ ; Z½input XðzÞ a0 z m þ a1 z m1 þ þ an1 z þ an
ð9Þ
where GðzÞ is the discrete transfer function, z is a complex argument in the z-domain, Y ðzÞ and XðzÞ are the z-transforms of y and x, respectively and bi and ai are real coe±cients of the numerator and denominator determined by the characteristics of the system. For both continuous and discrete transfer functions, the roots of the denominator polynomial are called the poles, while the roots of the numerator polynomial are called the zeros of the transfer function.30 In control theory, a continuous system is stable if all the poles of the transfer function lie on the imaginary axis of the Laplace domain or on the left-hand side of this axis, and a discrete system is stable when all the poles of the transfer function are inside or on the unit circle of the z-domain. According to Chen and Ricles,17 the characteristic polynomial of a dynamical system is identical to the denominator of the transfer function of the discrete system and consequently, the eigenvalues of the ampli¯cation matrix, which are the roots of the characteristic polynomial of the system, are exactly equal to the poles of the discrete transfer function. This fact is the foundation of the new algorithm proposed in the present paper. Based on the control theory, discrete transfer function of a linear time-invariant system can be obtained through conversion of the continuous transfer function of the system using a discretization method. The most commonly used scheme for converting a continuous system into a discrete one is to use the bilinear transformation. This procedure is also called the trapezoid approximation or Tustin transformation, which leads to the following relationship31: 2 z1 s¼ : ð10Þ t z þ 1
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3. Proposed Method To solve the equation of motion using numerical procedures, two key steps must be taken. The ¯rst step is discretizing the di®erential equation of motion in the time domain. It is noteworthy that these techniques do not satisfy the motion equation in all time t, but only at time intervals t. The next step is to assume speci¯c equations for both displacement and velocity, within each time interval. The discretized form of Eq. (1) can be written in the form below: ::
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:
::
:
MXiþ1 þ CXiþ1 þ KXiþ1 ¼ Riþ1 ;
ð11Þ
where Xiþ1 , Xiþ1 and Xiþ1 are the displacement, velocity and acceleration vectors at time tiþ1 , respectively, and Riþ1 is the vector of external loads at the discrete time point, i þ 1. 3.1. New integration formulation In 1996, enlightened by the Newmark process and by using the information of two previous time steps, Zhai proposed a simple explicit two-step algorithm with the following relationships: : :: :: 1 Xiþ1 ¼ Xi þ tXi þ þ t 2 Xi ð Þt 2 Xi1 ; ð12Þ 2 :
:
::
::
Xiþ1 ¼ Xi þ ð1 þ ’ÞtXi ð’ÞtXi1 :
ð13Þ
::
In these formulas, Xi1 is the acceleration vector at ði 1Þth time station and must be stored in computer memory. Furthermore, and ’ are weighted factors, which control the stability and numerical dissipation of procedure. As mentioned before, his e®orts for ¯nding the stability limit, which at least must be identical to the stability limit of CDM, led to the conditions of Table 1.26 According to Zhai, by changing the values of ’ and , in compliance with the conditions of Table 1, the stability limit of his explicit scheme is similar to or slightly better than the stability limit of CDM. By applying the Taylor formula, Zhai also
Table 1. Conditions of stability of Zhai explicit method. ’ ’>
1 2
1 2
’
’¼
1 2
0
