A new simple method of implicit time integration for dynamic problems of engineering structures

Transcription

1 Acta Mech Sin :91 99 DOI /s RESEARCH PAPER A new simple method of implicit time integration for dynamic problems of engineering structures Jun Zhou Youhe Zhou Received: 17 April 2006 / Revised: 10 July 2006 / Accepted: 6 September 2006 / Published online: 16 January 2007 Springer-Verlag 2007 Abstract This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai s approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems. Keywords Initial-value problems Time integration Implicit method Higher accuracy Time step and stability The project supported by the National Key Basic Research and Development Foundation of the Ministry of Science and Technology of China G , 2003CB716707, the National Science Fund for Distinguished Young Scholars , the National Natural Science Foundation of China Key Program , the Research Fund for Oversea Chinese The English text was polished by Keren Wang. J. Zhou Y. Zhou B Department of Mechanics, Lanzhou University, Lanzhou , China zhouyh@lzu.edu.cn 1 Introduction The simulation of a complex dynamic system requires a high efficient algorithm of time integration, with high accuracy and limited amount of computation. This requirement has attracted many researchers 1,2]. Many approaches have been established to solve different dynamic problems, such as structure, fluid flowing, and multi-rigid-body systems 3 ]. They are mainly classified into two categories: explicit and implicit algorithms. The advantage of an explicit approach is that it avoids solving algebraic equations at each time step and leads to less computation. Recently, some comparative studies of explicit time integration methods were conducted 13, 1]. These methods were improved for solving linear or nonlinear dynamic problems 15, 16], and some of them were applied to large-scale dynamic systems such as railway ballast vibration systems 16, 17]. However, the inherent deficiency of an explicit approach is that it is conditionally stable and may require a tradeoff between the accuracy, stability and the computation time in simulating dynamics of large-scale structures. The trade-off sometimes depends on the researchers experience. In order to overcome this deficiency, some implicit approaches are proposed to achieve unconditional stability for large time-step sizes, e.g., Wilson-θ method 18], Newmark-β method 19] and Runge Kutta method. Of course, implicit methods may also need a balance between the accuracy and the computation time in practice when the stability is to be assured. In order to get higher order accuracy most studies were based on the Newmark method for different applications in dynamic systems 20 26]. To solve nonlinear dynamic systems with energy conservation, for example, the basic Newmark-β method was modified by keeping

2 92 J. Zhou, Y. Zhou the total energy conservation at the cost of reducing accuracy 2]. Thus the accuracy of the modified algorithm is degenerated to order one for some circumstances and a small time step is required to ensure the desired accuracy. In order to get an accuracy of arbitrary order, a complex-time-step Newmark method was established by operating complex matrices at each time step, with a number of parameters chosen 20]. It is obvious that this method may render emerge computations comparable with other ones when it is applied to a dynamic system with large number of degrees of freedom. In Ref. 25], a multi-time-step technique was developed on the basis of the basic formulae of the Newmark method when the formulae are re-written in the matrix form, and the accuracy and stability remain unchanged. In this paper, a new simple two-parameter implicit time integration method is developed. The new implicit scheme possesses satisfactory integration properties and may serve as a possible basis for further modifications to achieve better performances on accuracy, stability and numerical dissipation and dispersion. It has a second-order accuracy in general, and reaches a thirdorder accuracy as its control parameters are set to their optimal values. Its stability is analysized and its amplitude decay rate AD and the relative period error TD are lower than those of the Newmark-β method and the Zhai-method 16]. In the following section, the time integration algorithm is briefly introduced, while the analysis of accuracy and stability of the algorithm are carried out in Sect. 3. After that, the evaluations of amplitude decay rate AD and relative error of period TD insimulating a harmonic system are conducted in Sect., and comparisons with other efficient methods are also presented. Finally, some concluding remarks are made in Sect Essence of the new implicit time integration method The dynamic systems described by second order ordinary differential equations can be written in a compact form of matrix Mẍ + Cẋ + Kx = F, 1 or MA + CV + KX = F, 2 where M, C, and K stand for the mass, damping, and stiffness matrices, respectively; x, V= ẋ, and A= ẍ are the vectors of displacement, velocity, and acceleration, respectively; and F represents the external force. The symbol dot over a quantity means differentiation of the quantity with respect to time variable t. For the initial-value problem of a dynamic system, the initial conditions are formulated as X t=0 = X 0 and V t=0 = V 0, 3 in which X 0 and V 0 are specific vectors. The goal of a numerical integration of Eqs. 1 or2 associated with the initial conditions of Eq. 3 is to find an efficient algorithm when the matrices in the equations are very large. Applying the Taylor expansion of Vt n + t up to order 2 and Xt n + t up to the order 3 at the time step t n = n t, we have Vt n + t Vt n + At n t + 1 2!Ȧt n t 2, Xt n + t Xt n + Vt n t + 1 2! At n t !Ȧt n t 3, 5 where A n, V n, and X n are, respectively, the acceleration, velocity, and displacement at time step t n n = 0, 1, 2,..., i.e., A n = A t=tn, V n = V t=tn, and X n = X t=tn. Further, the central difference is employed to the calculation of Ȧt n, i.e., Ȧt n = A n+1 A n t Substituting Eq. 6 into Eqs. and 5, we obtain the functions of V n+1 and X n+1 in terms of variables of A n 1, A n, and A n+1 except for V n and X n. Once we introduce six weighted factors to the former three variables in the resulting two formulae, we have V n+1 = V n + ξ 1 A n t + 1 δ 1A n+1 t 1 ζ 1A n 1 t, 7 X n+1 = X n + V n t ξ 2A n t δ 2A n+1 t 2 1 ζ 2A n 1 t 2. 8 For A n 1 and A n+1 in the above equations, we take their Taylor expansions at A n up to order 3 and have A n 1 A n Ȧ n t Än t 1 A n t 3, 6 9 A n+1 A n + Ȧ n t Än t + 1 A n t

3 A new simple method of implicit time integration 93 Then, the substitution of Eqs. 9 and 10 into Eqs. 7 and 8 leads to V n+1 = V n + ξ 1 + δ 1 ζ 1 A n t + δ 1 + ζ 1 Ȧ n t 2 + δ 1 ζ 1 Ä n t δ 1 + ζ 1 A n t + O t 5, 11 2 ξ2 X n+1 = X n + V n t δ 2 ζ 2 A n t 2 + δ 2 + ζ 2 Ȧ n t 3 v δ 2 ζ 2 Ä n t 2 + δ 2 + ζ 2 A n t 5 + O t Comparing Eqs. 11 and witheqs. and 5, respectively, the accuracy of order 2 of the scheme in fact, the velocity has an accuracy of order 2 and the displacement has a accuracy of order 3 may be ensured by using the following relations ξ 1 = ζ 1, δ 1 = 2 ζ 1, ξ 2 = ζ 2, δ 2 = 2 ζ Thus two parameters in the scheme are independent of each other. Substituting the relations in Eq. 13 into Eqs. 11 and, we get the implicit algorithm of time integration with two control parameters in the form 1 V n+1 = V n ζ ζ 1 2 X n+1 = X n + V n t ζ 2 A n t A n+1 t 1 ζ 1A n 1 t, ζ 2 A n t 2 6 A n+1 t 2 1 ζ 2A n 1 t Substituting Eqs. 1 and 15 into Eq. 2 at time step t n+1,or MA n+1 + CV n+1 + KX n+1 = F n+1, 16 we obtain A n+1 = M 1 F n+1, 17 here, M = M + ζ C t + ζ K t 2, 18 6 F n+1 = F n+1 KX n C + K tv n + ζ1 C t + ζ ζ1 C t + ζ 2 K t ] K t 2 A n 3 ] A n From Eqs. 1 19, one sees that the calculation of time integration for Eqs. 1 or2 may start once the initial conditions of Eq. 2 are known and the values of acceleration A at the initial instant and at one time step before that time are specified. Without loss of generality, the initial acceleration may be obtained by substituting the initial conditions of Eq. 2 into the differential Eq. 2, and assuming A 1 = 0. FromEq.18, we find that the revised mass matrix is a constant matrix once the parameters ζ 1 and ζ 2 are specified and the equal time step t is taken, so the calculations of inversion matrix M 1 can be avoided at all time steps except for the first time step. Thus a large amount of CPU time may be saved when it is applied to complex dynamic systems with emerge degrees of freedom. Suppose X R m and a fixed time step size is used in the computation, then the computation of inverse matrix M 1 needs to be done only once. And subsequently, one needs about O2m 2 operations to compute vector F n+1, and another Om 2 operations to calculate vector A n+1 in each time step, as can be seen in Eqs. 17 and 19. Thus, the total number of operations is about O3m 2 in each time step, which is about the same as the Newmark method and the Zhai s method. Of course, the new method needs a bit more computation for vectors V n+1 and X n+1 in each time step, as compared with the Newmark method and the Zhai s method. 3 Analysis of accuracy and stability As shown in the previous section, the implicit method of time integration with Eqs has an accuracy of order 2 when the relations in Eq. 13 are satisfied. In this section, we present some conditions for higher accuracy of the scheme and stability of the calculations.

4 9 J. Zhou, Y. Zhou 3.1 Higher order accuracy The substitution of the expansions of A n 1 and A n+1 expressed in Eqs. 9 and 10 into Eqs. 1 and 15 leads to the following formulae V n+1 = V n + A n t + 1 2Ȧn t ζ 1Ä n t A n t +O t 5, 20 X n+1 = X n + V n t A n t Ȧn t ζ 2Ä n t + 1 A n t 5 +O t It is obvious that the accuracy of the scheme depends on the two independent parameters ζ 1 and ζ 2. Subtracting the Taylor expansions of V n+1 and X n+1 of one order higher than the formulae of Eqs. and 5 from Eqs. 20 and 21, respectively, we get the local truncation errors of the form EV n+1 = ζ 1 Ä n t 3 + O t, 22 EX n+1 = ζ 2 Ä n t + O t When ζ 1 = 1/3, 2 we have EV n+1 = O t and EX n+1 = O t, which tells us that the scheme has an accuracy of order 3 higher than that of simple methods of the Newmark method and the Wilson-θ method. 3.2 Stability analysis For simplicity, here, we present a stability analysis of the proposed scheme of implicit time integration with two parameters for the linear dynamic systems. In this case, the system of linear dynamic equations formulated by Eq. 2 may be always decoupled into a set of independent single-degree-of-freedom ordinary differential equations by using the modal decomposition technique. Thus, the stability analysis is usually performed on the basis of a dynamic system of a single-degree-of-freedom without damping in the dimensionless form, i.e., ẍ + ω 2 x = 0, 25 or equivalently in the difference form, a n+1 + ω 2 x n+1 = 0, 26 where ω= k/m is a natural frequency of the system; a = ẍ, and m and k are the mass and the stiffness, respectively. By means of Eqs. 1 and 15 for the case of the single-degree-of-freedom system, and substituting the formulas into Eq. 26, then eliminating variables V n and V n+1, we get a difference equation of Eq. 2 as follows ζ ] p ζ 2 ζ 1 2 x n+2 + ] p ζ 2 ζ ] 1 p x n+1 ζ2 x n + ζ ] 1 p x n 1 =0, 27 here, p = ω t 2 and n 1. Let x n+1 = λx n, then one obtains an eigenvalue equation of Eq. 27 in terms of λ as follows ζ 2p + 1 ] ζ 2 1 ] ζ 1 p λ 2 ] p λ+ p = 0. λ ζ ζ 1 1 ζ 2 1 ζ 1 28 Denote the roots of Eq. 28byλ i i = 1, 2, 3. Following the theory of difference 27], we know that the general solution of Eq. 27 can be expressed as x n = 3 c i λ n i, n = 1, 2, 3,..., 29 i=1 in which the coefficients c i i = 1, 2, 3 are determined by the initial conditions of Eq. 3. The stability of the scheme requires that the general solution x n should be bounded as n, which means that the roots should all have a magnitude less than or equal to 1, i.e., λ i 1, i ={1, 2, 3}, 30 which determine the regions of parameters. In order to get the formulae of stability conditions, we introduce the complex variable transformation of the form λ = 1 + z/1 z. 31 From the theory of complex variables, we know that the conditions in Eq. 30 are equivalent to Rez <0. Substituting Eq. 31 into28, we get ã 3 z 3 + ã 2 z 2 + ã 1 + ã 0 = 0, 32 where ã 0 = ã 1 = 3p, ã 2 = 3ζ 1 + 1p, and ã 3 = 2ζ 2 3ζ 1 + 1p. Thus, the stability conditions become Rez i <0, i = 1, 2, 3, 33

5 A new simple method of implicit time integration 95 in which z i i = 1, 2, 3 are the root of Eq. 32. According to the Routh Hurwitz criterion 28] of stability for Eq. 32, the stability conditions correspond to the case of no sign change of the first column in the Routh criterion table as follows λ 3 λ 2 ã 3 ã 1 ã 2 ã 0 λ 1, 3 b 2 0 λ 0 c 2 0 where b 2 = ã 3 ã 0 ã 2 ã 1 /ã 2 and c 2 = ã 0.Duetothe fact that p > 0or c 2 < 0, the stability conditions of the scheme reduce to the form 2ζ 2 3ζ 1 + 1p < 0, 3ζ 1 + 1p < 0, 35 and ζ 2 3ζ 1 p 2 > 0. From the inequalities in Eq. 35, we see that the unconditional stability of the scheme is obtained when the parameters satisfy the following inequalities 2ζ 2 3ζ , 3ζ , 36 and ζ 2 3ζ 1 > 0, or explicitly ζ 1 1/3, and 3ζ 1 <ζ 2 3ζ 1 1/2. 37 If the parameters are in the regions of 2ζ 2 3ζ > 0 and 3ζ > 0, the conditions of conditional stability of the scheme are obtained, ζ 1 > 1/3, ζ 2 > 3ζ 1, 38 and p < /2ζ 2 3ζ It is evident that the optimal values of ζ 1 expressed in Eq. 2 do not satisfy the first two conditions of Eq. 37 but 38, which implies that the stability of the new scheme with accuracy of order 3 is conditional. The time step should satisfy p = ω t 2 < 6/ζ 2 or t < 2.9/ω ζ 2 ζ 2 > 1. Usually, the time step is not greater than 0.2T, orf t 0.2, where T = 2π/ω and f = 1/T represent a period of the dynamic system and corresponding frequency, respectively. Corresponding to the criterion value of f t max = 0.2, we get q 0 = 625/π , where q = 2ζ 2 3ζ 1 +1 = q 0. For the case of q 625/π 2, the following inequality is obtained by the last inequality of Eq. 38 2ζ 2 3ζ /π Fig. 1 The conditional and unconditional stability regions of parameters ζ 1 and ζ 2 in the implicit scheme of time integration for the dynamic systems. Here, q = 2ζ 2 3ζ 1 + 1andq 0 = Then, selecting different value of q q 0, we can draw the lines of 2ζ 2 3ζ = q in the plane of ζ 1 ζ 2. According to the conditions of Eqs. 37 and 38, the regions of unconditional and conditional stability of the scheme are plotted in Fig. 1 for the parameters ζ 1 and ζ 2 of the scheme in the plane of ζ 1 ζ 2. When the parameters are in these two regions, the implicit method of time integration proposed in this paper for a dynamic system has an accuracy of at least order 2, and it has a third order accuracy when the parameters are chosen on the line of ζ 1 = 1/3 in the conditional stability region. When the parameters are on the line of 2ζ 2 3ζ = q, when q q 0, in the conditional stability region, the third inequality of Eq. 38 may be rewritten as f t /q /q 0, which gives a restriction of the time step with respect to the frequency of the dynamic system concerned. Figure 2 shows the curves of t versus frequency for different parameter q q 0.Asq increases, the curve drops down, and the time step should be in the region under and/or on the line marked by the value of q. For the parameters in the unconditional stability region and those under line of 2ζ 2 3ζ = q 0 in the conditional stability region, the time step should be selected in the region under the curve marked by q = q 0 in Fig. 2. Evaluation of dissipation and dispersion In order to further show the efficiency of the scheme, we need to evaluate the dissipation and dispersion of the new scheme. That is, we should evaluate the amplitude and period for a periodical dynamics. From the algebraic

6 96 J. Zhou, Y. Zhou Fig. 2 The diagram of log log curves of time step t versus frequency f Fig. 3 Curves of relative error of period TD versus the dimensionless time step t/t for the case of AD = 0 theory, we know that Eq. 28 has at least one real root and may have two conjugate complex roots. Denote the complex roots by λ 1 and λ 2, and the real root by λ 3.Let the conjugate roots be λ 1,2 = α ± iβ, where α and β are real and i = 1. When ζ 2 = 3ζ 1, we get λ 3 = 0. In the stability regions of parameters near the line of ζ 2 = 3ζ 1, we know that the conjugate complex roots are principal, or λ 3 0. In this case, the general solution of Eq. 27 may be rewritten as x n = eᾱt n c 1 cos ω t n + c 2 sin ωt n ], 0 in which ᾱ = lnα2 + β 2 2 t, ω = tan 1 β/α. 1 t After that, the measures of dissipation and dispersion are, respectively, defined by the amplitude decay function AD and the relative error of period TD of the form AD := 1 e 2πᾱ/ω, 2 TD := T T = 1, 3 T ω ω where T = 2π/ω and T = 2π/ ω. For the case of ζ 2 = 3ζ 1, it is shown that λ 1 λ 2 = 1, or α 2 + β 2 = 1, and { 2 λ 1,2 = ] 2 ζ 1 p / ±i } ] 2 3ζ 1p, here, = p p 2 /3 p 2 ζ 1. The complex roots of λ 1 and λ 2 satisfy the condition of >0 which means the second inequality of Eq. 35. In addition, the result α 2 + β 2 = 1 leads to AD = 0 from the first formula of Eq. 1 and Eq. 2. These results show that the line of ζ 2 = 3ζ 1 in the plane of parameters shown in Fig. 1 is in the stability regions, and the harmonic response may be well obtained by the new algorithm proposed in this paper for a harmonic dynamic system. Comparing this method with others in literature, e.g., the Newmarkβ method and the Zhai s method 16], one sees that AD = 0 is held only for a set of parameters in them, for example, γ = 1/ and β = 1/2 in the Newmark-β method, and ψ = φ = 1/2 in the Zhai s method. Our method, however, gives the parameters ζ 1 and ζ 2 in the line ζ 2 = 3ζ 1 with ζ 1 > 1/3. It is evident that the selected parameters for AD = 0 in the method of this paper are much more convenient than those in other methods. Considering the relative error of period TD for the case of AD = 0, we find that the relative error depends on both the independent parameter ζ 1 and the time step t. Figure 3 plots the curves of the relative error of period versus the ratio of dimensional time step for different parameter ζ 1 in the region of 1/8, 1/2]. From this figure, one sees that all relative errors for the case in the regions mentioned above are within 6%, and they are always less than those of the Newmark-β method and the Zhai s method, where the maximum relative errors are about 16 and 10%, respectively. When the new scheme has accuracy of order 3 and AD = 0, i.e., ζ 1 = 1/3 and ζ 2 = 1, the curve indicates that the TD is only within 1.3%. Figure shows the curves of amplitude dissipation ratio versus the dimensional time step for different parameters ζ 1 and ζ 2 in the vicinity of the stability regions near the line of ζ 2 = 3ζ 1. When ζ 1 is in the region of 1/8, 1/2], for example, the value of ζ 2 is chosen 20% higher than 3ζ 1,orζ 2 = 3.6ζ 1. This departure of parameters from those when AD = 0 in this paper is greater than those selected in other methods. For the curves of the Newmark method and the Zhai s method, there are 5%

7 A new simple method of implicit time integration 97 Fig. Characteristic curves of amplitude dissipation ratio AD versus dimensionless time step for different parameters ζ 1 and ζ 2 in the stability region Fig. 5 Comparison of the simulation results with analytical solution for the displacement response of a harmonic motion. The curves marked by values of ζ 1 and ζ 2 are obtained by the new scheme of time integration proposed in this paper departures from their perfect parameters. From Fig., we find that the amplitude dissipation ratio depends on the selection of the parameters and the time step, and most of the relative errors of the new method are less than those of other methods. When t/t < 0.15, the amplitude dissipation ratio of this method is always less than those of others, and the ratio generated from this new method is insensitive to the ratio of time step to period when the dimensionless time step is not greater than 0.1, but the dissipation ratio generated from other methods shown in this figure is strongly sensitive to the dimensionless time step even if the dimensionless time step is small, or close to zero. 5 Case study In this section, two numerical examples of applying the new scheme of time integration are displayed to show the efficiency of the scheme. The first one is the harmonic motion of Eq. 25 with the initial conditions of x0 = 0.1 and ẋ0 = 0, and with frequency ω = 10 rad/s. To this example, there is no difficulty to obtain its analytical solution xt = 0.1 cos10t. In the numerical simulation of this dynamic problem, we choose the time step t = T/. Figure 5 shows the simulation curves of displacement response of the problem during 5 10 periods. By comparing them with those obtained by the Newmark method and the Zhai s approach, we also plot their simulation results when their parameters in their schemes are taken as the perfect ones for the case of AD = 0. From this figure, one sees that the simulation results of this new scheme with accuracy of order 3, when either AD = 0see the curve marked by ζ 1 = 1/3 and ζ 2 = 3ζ 1 or AD = 0 see the curve marked by ζ 1 = 1/3 and ζ 2 = 1.1 are consistent with the analytical solution, while the curve with parameters ζ 1 = 1/6 and ζ 2 = 3ζ 1 when the scheme has accuracy of order 2 with AD = 0 is better than the curves of other methods with AD = 0 as compared to the analytic solution. The other example is a multi-degree-of-freedom dynamic system, a similar system of discrete axial vibration of one-dimensional straight bar, without damping and excitation, M 1 ẍ 1 + k 1 + k 2 x 1 k 2 x 2 = 0, M i ẍ i k i x i 1 + k i + k i+1 x i k i+1 x i+1 = 0, M n ẍ n k n x n 1 + k n x n = 0, 2 i n 1, 5 where M i and k i are the mass and spring stiffness, respectively; and the integer n 1 denotes the number of degrees of freedom. Here, x i represent the coordinates of mass M i relative to their equilibrium position. In this case, the parameters are taken as M i = kg and k i = N/m corresponding to a practical bar. In order to get a response consisting of all order vibrations, here, the initial conditions of displacement disturbance are chosen in the form x i 0 = 1 10 m, ẋ i 0 = 0, i = 1, 2,..., n. 6 From the modal analysis, it is not difficult for us to get the natural frequencies or periods of the dynamic system. Denote the maximum and minimum periods by T max and T min, respectively. In the simulation of this dynamic system, we take ζ 1 = 1/3 and ζ 2 = 1 in the new

8 98 J. Zhou, Y. Zhou 6 Conclusions A new simple algorithm of implicit time integration of dynamic systems with multi-degrees-of-freedom is proposed in this paper. It is found that this new algorithm with two control parameters has accuracy of at least order 2 when the parameters are in the stability regions, and of order 3 when the parameters are optimally taken as ζ 1 = 1/3 and ζ 2 ζ 1. The stability regions of the parameters and the time step are obtained by analysis of stability behavior of the scheme calculations and amplitude dissipation and dispersion. The numerical examples show that the proposed new algorithm can achieve satisfactory accuracy. Fig. 6 Numerical solutions of time response of the 11th mass in the duration of 30T max, which are obtained by the Newmark method, Zhai s method and the new scheme, respectively Fig. 7 Higher order components of the numerical solutions of the 11th mass scheme with third order accuracy and t = T min /6, and the simulation of response is conducted in the duration of 30T max. To make comparison, the explicit Runge Kutta method of order 29,30] is employed as the third-party method to yield a solution of the system with sufficient accuracy by setting a small time-step size, then numerical solutions are obtained by the Newmark method, Zhai-approach and the new method, respectively. Figure 6 shows the displacement response of the 11th mass. As can be seen in Fig. 6, the solution obtained by the new scheme is in good agreement with the ones obtained by the Newmark-method, Zhai s method and the Runge Kutta method. In addition, the new scheme can also achieve satisfactory accuracy for higher order components of the solution, which is shown in Fig. 7 by zooming out the figure of the displacement response. References 1. Belytschko, T., Hughes, T.J.R.: Computational Methods for Transient Analysis Computational Methods in Mechanics vol. 1. NorthHooland Elsevier, Amsterdam Wang, X.C., Shao, M.: The Fundamentals of the Finite Element Method and Numerical Methods, 2nd edn. Tsinghua University Press, Beijing Hilber, H.M., Hughes, T.J., Taylor, R.L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Eng. Struct. Dyn. 5, Stewart, D.E., Trinkle, J.C.: An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction. Int. J. Numer. Methods Eng. 39, Villasenor, R.: A fully coupled, implicit, numerical scheme for laminar and turbulent parabolic flows. Int. J. Numer. Methods Eng. 0, Chen, S., Hansen, J.M., Tortorelli, D.A.: Unconditionally energy stable implicit time integration: application to multibody system analysis and design. Int. J. Numer. Methods Eng. 8, Modak, S., Sotelino, E.D.: The iterative group implicit algorithm for parallel transient finite element analysis. Int. J. Numer. Methods Eng. 7, Mugan, A., Hulbert, G.M.: Frequency-domain analysis of time-integration methods for semidiscrete finite element equations part I: Parabolic problems. Int. J. Numer. Methods Eng. 51, Mugan, A., Hulbert, G.M.: Frequency-domain analysis of time-integration methods for semidiscrete finite element equations part II: Hyperbolic and parabolic-hyperbolic problems. Int. J. Numer. Methods Eng. 51, Greyvenstein, G.P.: An implicit method for the analysis of transient flows in pipe networks. Int. J. Numer. Methods Eng. 53, Schreyer, H.L.: On time integration of viscoplastic constitutive models suitable for creep. Int. J. Numer. Methods Eng. 53, Mugan, A.: Discrete equivalent time integration methods for transient analysis. Int. J. Numer. Methods Eng. 57, Rio, G., Soive, A., Grolleau, V.: Comparative study of numerical explicit time integration algorithms. Adv. Eng. Softw. 36,

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