Abstract
In this article, third and fourthorder accurate explicit time integration methods are developed for effective analyses of various linear and nonlinear dynamic problems stated by secondorder ordinary differential equations in time. Two sets of the new methods are developed by employing the collocation approach in the time domain. To remedy some shortcomings of using the explicit RungeKutta methods for secondorder ordinary differential equations in time, the new methods are designed to introduce small period and damping errors in the important lowfrequency range. For linear cases, the explicitness of the new methods is not affected by the presence of nondiagonal damping matrix. For nonlinear cases, the new methods can handle velocity dependent problems explicitly without decreasing order of accuracy. The new methods do not have any undetermined algorithmic parameters. Improved numerical solutions are obtained when they are applied to various linear and nonlinear problems.
Keywords:
Linear and nonlinear dynamics; explicit time integration method; higherorder method
1 Introduction
Stepbystep direct time integrations are dominantly used for transient analyses of linear and nonlinear dynamic problems described by secondorder ordinary differential equations in time. As more sophisticated spatial finite element models [^{1}[1] J. N. Reddy. An Introduction to the Finite Element Method. McGrawHill New York, 3rd edition, 2006.
[2] O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method for Solid and Structural Mechanics. ButterworthHeinemann, 2005.
[3] D. Schillinger, J. A. Evans, F. Frischmann, R. R. Hiemstra, M. Hsu, and T. J. R. Hughes. A collocated c0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics. International Journal for Numerical Methods in Engineering, 102(34):576631, 2015.^{4}[4] W. Kim and J. N. Reddy. Novel mixed finite element models for nonlinear analysis of plates. Latin American Journal of Solids and Structures, 7(2):201226, 2010.] are developed constantly, demands for more accurate time integration methods are also increasing to take full advantage of improved spatial models in transient analyses. Recently, numerous implicit [^{5}[5] W. Kim and S. Y. Choi. An improved implicit time integration algorithm: The generalized composite time integration algorithm. Computers and Structures, 196:341354, 2018.
[6] W. Kim and J. N. Reddy. An improved time integration algorithm: A collocation time finite element approach. International Journal of Structural Stability and Dynamics, 17(02):1750024, 2017.
[7] W. Kim and J. N. Reddy. A new family of higherorder time integration algorithms for the analysis of structural dynamics. Journal of Applied Mechanics, 84(7):071008, 2017.
[8] W. Kim and J. N. Reddy. Effective higherorder time integration algorithms for the analysis of linear structural dynamics. Journal of Applied Mechanics, 84(7):071009, 2017.^{9}[9] W. Kim, S. Park, and J. N. Reddy. A cross weightedresidual time integration scheme for structural dynamics. International Journal of Structural Stability and Dynamics, 14(06):1450023, 2014.] and explicit [^{10}[10] J. Chung and J. M. Lee. A new family of explicit time integration methods for linear and nonlinear structural dynamics. International Journal for Numerical Methods in Engineering, 37(23):39613976, 1994.
[11] G. M. Hulbert and J. Chung. Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Computer Methods in Applied Mechanics and Engineering, 137(2):175188, 1996.
[12] D. Soares. An explicit family of time marching procedures with adaptive dissipation control. International Journal for Numerical Methods in Engineering, 100(3):165182, 2014.
[13] D. Soares. A novel family of explicit time marching techniques for structural dynamics and wave propagation models. Computer Methods in Applied Mechanics and Engineering, 311:838855, 2016.
[14] W. Kim and J. H. Lee. An improved explicit time integration method for linear and nonlinear structural dynamics. Computers and Structures, 206:4253, 2018.
[15] W. Kim. A simple explicit single step time integration algorithm for structural dynamics. International Journal for Numerical Methods in Engineering, pages 0000, 2019.^{16}[16] W. Kim. An accurate twostage explicit time integration scheme for structural dynamics and various dynamic problems. International Journal for Numerical Methods in Engineering, pages 0000, 2019.] time integration methods have been introduced to effectively analyze challenging dynamic problems.
Direct time integration methods are often categorized into implicit and explicit methods. Usually, implicit methods are unconditionally stable for linear analyses, but they require factorization of nondiagonal system matrices. For large systems, factorizing nondiagonal system matrices requires huge computational efforts. On the other hand, explicit methods are conditionally stable, but matrix factorization can be avoided if the mass matrix is diagonal. Accordingly, computational costs per time step of explicit methods are much lower than implicit methods. Due to this reason, explicit methods are more frequently used in analyses of large systems, such as the wave propagation and impact problems, where sizes of optimal time steps are slightly smaller than critical time steps of explicit methods.
According to the past studies [^{10}[10] J. Chung and J. M. Lee. A new family of explicit time integration methods for linear and nonlinear structural dynamics. International Journal for Numerical Methods in Engineering, 37(23):39613976, 1994.,^{17}[17] M. A. Dokainish and K. Subbaraj. A survey of direct timeintegration methods in computational structural dynamics i. explicit methods. Computers and Structures, 32(6):13711386, 1989.], preferable attributes of explicit time integration methods can be summarized as (a) explicit methods should not require iterative solution finding procedures for velocity dependent nonlinear problems; (b) at least secondorder accuracy should be ensured; (c) number of unspecified algorithmic parameters should be minimized; (d) amplification matrices obtained by applying methods to the singledegreeoffreedom problem should not present spurious eigenvalue, or it should be minimized; (e) explicit methods should not involve matrix factorization when the mass matrix is diagonal; (f) explicit methods should not involve differential or integral type calculus; (g) explicit methods should be applicable to nonlinear problems without any modifications.
The central difference (CD) method is one of the most widely used singlestage explicit time integration methods. The CD method is secondorder accurate and nondissipative. Due to its simplicity and good accuracy, the CD method is the standard explicit method of the wellknown software packages [^{18}[18] T. J. R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover, 2012.,^{19}[19] K. J. Bathe. Finite Element Procedures. KlausJurgen Bathe, 2006.]. However, matrix factorization is required in the CD method if the damping matrix is not diagonal. The velocity can be treated explicitly to remedy this, but additional computation is required to retain secondorder accuracy [^{15}[15] W. Kim. A simple explicit single step time integration algorithm for structural dynamics. International Journal for Numerical Methods in Engineering, pages 0000, 2019.]. The CD method cannot satisfy (a) and (b).
Chung and Lee developed a family of secondorder accurate explicit method [^{10}[10] J. Chung and J. M. Lee. A new family of explicit time integration methods for linear and nonlinear structural dynamics. International Journal for Numerical Methods in Engineering, 37(23):39613976, 1994.] with dissipation control capability. The Chung and Lee (CL) method can maintain explicitness in the presence of the nondiagonal damping matrix. However, the amplification matrix of the CL method has a spurious eigenvalue that influences the quality of solutions when large time steps are used. Hulbert and Chung also developed a family of secondorder accurate explicit method [^{11}[11] G. M. Hulbert and J. Chung. Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Computer Methods in Applied Mechanics and Engineering, 137(2):175188, 1996.]. However, the amplification matrix of the Hulbert and Chung (HC) method also has a spurious eigenvalue. The HC method can satisfy all the attributes except (d). The HC method can include a full range of dissipative cases, but the nondissipative case becomes unconditionally unstable for any choices of time steps in the presence of the damping matrix.
Soares developed an explicit time integration method based on the weighted residual method [^{13}[13] D. Soares. A novel family of explicit time marching techniques for structural dynamics and wave propagation models. Computer Methods in Applied Mechanics and Engineering, 311:838855, 2016.]. The Soares method can be used for the analysis of wave propagation problems, but it becomes an only firstorder accurate implicit method in the presence of nondiagonal damping matrix. In addition, it is not easy to use the Soares method for nonlinear analyses, because it has been developed by directly integrating the equations of linear structural dynamics in a weighted residual sense. The Soares method cannot satisfy (a), (b), (d), (e), (f) and (g).
Kim and Lee also proposed a family of secondorder accurate explicit methods [^{14}[14] W. Kim and J. H. Lee. An improved explicit time integration method for linear and nonlinear structural dynamics. Computers and Structures, 206:4253, 2018.]. The Kim and Lee (KL) method can include a full range of dissipative cases and remain as a secondorder explicit method in the presence of a nondiagonal damping matrix. The KL method is very effective, but (c) is not satisfied. Recently, a very accurate twostage explicit method was introduced by Kim [^{16}[16] W. Kim. An accurate twostage explicit time integration scheme for structural dynamics and various dynamic problems. International Journal for Numerical Methods in Engineering, pages 0000, 2019.] to more effectively tackle challenging nonlinear problems.
For more precise and reliable analyses of challenging dynamic problems [^{20}[20] W. Kim and J. Lee. A comparative study of two families of higherorder accurate time integration algorithms. International Journal of Computational Mechanics, accepted.], higherorder accurate methods can be considered as a good option. Many of the standard time integration methods used in structural dynamics, such as the trapezoidal rule and the central difference method, are secondorder accurate, and methods of third or higherorder accuracy are often called higherorder methods [^{7}[7] W. Kim and J. N. Reddy. A new family of higherorder time integration algorithms for the analysis of structural dynamics. Journal of Applied Mechanics, 84(7):071008, 2017.,^{21}[21] Gregory M Hulbert. A unified set of singlestep asymptotic annihilation algorithms for structural dynamics. Computer Methods in Applied Mechanics and Engineering, 113(12):19, 1994.
[22] T.C. Fung. Unconditionally stable higherorder accurate hermitian time finite elements. International journal for numerical methods in engineering, 39(20):34753495, 1996.
[23] T.C. Fung. Weighting parameters for unconditionally stable higherorder accurate time step integration algorithms. part 2secondorder equations. International journal for numerical methods in engineering, 45(8):9711006, 1999.^{24}[24] T.C Fung. Unconditionally stable higherorder accurate collocation timestep integration algorithms for firstorder equations. Computer methods in applied mechanics and engineering, 190(1314):16511662, 2000.]. Computational cost per time step of higherorder methods is higher than secondorder methods because higherorder methods have more stages, while many of secondorder methods use only one stage. However, more accurate predictions can be obtained with large time steps in higherorder methods, and they may become more efficient in getting the same prediction when compared with secondmethods. In addition, some challenging nonlinear problems can only be effectively solved by using proper higherorder methods [^{7}[7] W. Kim and J. N. Reddy. A new family of higherorder time integration algorithms for the analysis of structural dynamics. Journal of Applied Mechanics, 84(7):071008, 2017.,^{8}[8] W. Kim and J. N. Reddy. Effective higherorder time integration algorithms for the analysis of linear structural dynamics. Journal of Applied Mechanics, 84(7):071009, 2017.].
One of the most broadly used explicit higherorder methods is the RungeKutta (RK) methods. The RK methods have been used in numerous engineering areas. Traditionally, the RK methods have been used to solve firstorder ordinary differential equations numerically. Recently, on the other hand, the RK methods have also been used to obtain higherorder accurate transient solutions in structural problems [^{25}[25] J. A. Evans, R. R. Hiemstra, T. J. R. Hughes, and A. Reali. Explicit higherorder accurate isogeometric collocation methods for structural dynamics. Computer Methods in Applied Mechanics and Engineering, 338:208240, 2018.].
For secondorder initial value problems, the RK methods can provide more accurate numerical solutions than secondorder methods if considerably small time steps are used. For example, the convergence rate of the fourstage RK method is fourthorder, and numerical solutions of the fourstage RK method converge to the exact solution with the rate of
In this work, simple and effective third and fourthorder accurate explicit time integration methods are presented to remedy the shortcomings of the RK methods. For a fair comparison, the threestage thirdorder accurate RK method and the fourstage fourthorder accurate RK method are considered. Higherorder accuracies of the new methods are achieved without increasing computational efforts when compared with the equivalent third and fourthorder RK methods. In the development, the displacement vector is approximated over a time domain by using the known interpolation functions in time and the weighted sums of acceleration vectors. Then, the semidiscrete equations of structural dynamics are discretized in time by adopting the collocation method. After determining optimal collocation points in time, simple difference relations are obtained.
2 New explicit method
The governing equations of many dynamic problems can be expressed as
and the initial conditions are given by
where
where
Occasionally, Eq.(3) can also be obtained by spatially discretizing original governing partial differential equations in space and time. Then, Eq.(3) is also called the semidiscrete equation.
In this section, two sets of new higherorder explicit methods are developed to discretize Eqs.(1) and (3) in time. The singledegreeoffreedom case of the linear structural dynamics equation given in Eq.(3) is used in the procedures of the development.
2.1 New thirdorder accurate explicit method
For the development of the new thirdorder accurate explicit method, three sets of interpolation functions in time are used to approximate the displacement and velocity vectors over the time domain
where
The displacement and velocity vectors at
where
Finally, the displacement and velocity vectors at
where
To achieve thirdorder accuracy, extended stability, and preferable spectral properties, permissible
By using the parameters given in Eq.(13) and rearranging the equations, the new thirdorder method are summarized as follow: To start the new thirdorder explicit method, compute the acceleration vector at
where
By using
By using Eq.(17), the displacement and velocity vectors at
By using
Finally, the displacement and velocity vectors at
As shown in Eqs.(14)(22),
2.2 New fourthorder accurate explicit method
The new fourthorder method is developed by considering an additional stage. In the new fourthorder accurate method, the first and second approximations of the displacement and velocity vectors take the same forms as the approximations used in the thirdorder method given in Eqs.(4)(8). The third approximations of the fourthorder method at
where
where
To achieve fourthorder accuracy, extended stability, and preferable spectral properties, permissible
With the parameters given in Eq.(28), the new fourthorder method is summarized as follows: To start the new fourthorder explicit method, compute the acceleration vector at
where
By using
By using Eq.(32), the displacement and velocity vectors at
By using
The interim displacement and velocity vectors at
By using
Finally, the displacement and velocity vectors at
After computing
3 Review of the RungeKutta methods
The RK Methods are frequently used to analyze various linear and nonlinear dynamic systems described by firstorder differential equations. Traditionally, the RK Methods have been considered unsuitable for the analysis of dynamic problems stated by secondorder differential equations due to excessive numerical damping. Recently, however, the RK Methods have also been employed to develop a more accurate numerical procedure which can be used for the analysis of structural dynamics [^{25}[25] J. A. Evans, R. R. Hiemstra, T. J. R. Hughes, and A. Reali. Explicit higherorder accurate isogeometric collocation methods for structural dynamics. Computer Methods in Applied Mechanics and Engineering, 338:208240, 2018.]. In this section, the third and fourthorder RungeKutta methods are reviewed briefly.
3.1 Thirdorder RungeKutta Method
In the thirdorder RK method,
The displacement and velocity vectors at
The interim displacement and velocity vectors at
Finally, the displacement and velocity vectors at
As shown in Eqs.(41)(49), computational structures of the new thirdorder method and the thirdorder RK method are very similar. It can also be observed that each method contains three major computations (i.e.,
3.2 Fourthorder RungeKutta Method
In the fourthorder RK method,
The displacement and velocity vectors at
The displacement and velocity vectors at
The interim displacement and velocity vectors at
Finally, the displacement and velocity vectors at
As shown in Eqs.(50)(61), computational structures of the new fourthorder method and the fourthorder RK method are very similar. It can also be observed that each method contains four major computations (i.e.,
4 Analysis of the new methods
In this section, numerical characteristics of the new methods are analyzed. To investigate numerical characteristics of time integration methods, the linear singledegreeoffreedom problem [^{27}[27] T. J. R. Hughes. The finite element method: linear static and dynamic finite element analysis. Courier Corporation, 2012.
[28] T. J. R. Hughes. Analysis of transient algorithms with particular reference to stability behavior. Computational methods for transient analysis(A 8429160 1264). Amsterdam, NorthHolland, 1983, pages 67155, 1983.^{29}[29] H. M. Hilber. Analysis and design of numerical integration methods in structural dynamics. PhD thesis, University of California Berkeley, 1976.] is frequently used. The linear singledegreeoffreedom problem is given as
with the initial conditions
where
where
where
4.1 Order of accuracy
To investigate the order of accuracy of the new methods for linear problems, the characteristic polynomial of the amplification matrices is used. For a time integration method, the characteristic polynomial of the amplification matrix
where
where
where
The new thirdorder method gives
According to Eq.(69), the new threestage method is thirdorder accurate for the damped case (
and it is fourthorder accurate for the damped and undamped cases according to Eq.(71).
4.2 Spectral radius and stability limit
The spectral radius is frequently used to investigate linear stability of time integration method. The spectral radius is defined as
where
4.3 Period and damping error
The order of accuracy is one of the most important measurements for accuracy. However, the period elongation and the damping ratio are more practical measurements, because they can directly describe important characteristics of time integration algorithms. The relative period error is defined as
The new third and fourthorder explicit methods have much smaller damping and period errors when compared with the third and fourthorder RK methods as shown in Figs.2 and 3. Based on the results presented in Fig.2, it can be expected that the new methods will give very small period error in longterm analyses. This will be verified by using various numerical examples in the next section. Fig.3 is showing that the new explicit methods do not introduce excessive numerical (or algorithmic) damping into the important lowfrequency range, while considerable numerical damping is introduced in the RK methods.
5 Numerical examples
In this section, six illustrative examples are considered to demonstrate improved performances of the new methods. To keep the simplicity of the numerical analysis, only dimensionless problems are considered. Among the examples, twodegreeoffreedom nonlinear springpendulum and double pendulum problems are used to verify the performance of the proposed method for multidegreeoffreedom problems. It should also be noted that the precision of 16 significant digits is used for evaluations of variables in the computer code.
As shown in Table 1, the computation times of the four different higherorder explicit methods are almost identical when the number of degreeoffreedoms is less than hundred. As degreeoffreedoms increase, on the other hand, the computation times of the thirdorder methods become approximately 75% of the fourthorder methods. As expected, computational times of the new explicit methods and the RK methods are almost the same when the same numbers of stages are assumed.
Computation times (in seconds) taken to obtain nonlinear numerical solutions of 10,000th time step for varying sizes of system (i.e., degreeoffreedoms).
5.1 Singledegreeoffreedom problem
Eqs.(62) and (63) are solved by using the new methods and the RK methods. For undamped cases,
Comparison of displacements the single degree of freedom system
Comparison of displacements the single degree of freedom system
Comparison of displacements the single degree of freedom system
Comparison of displacements the single degree of freedom system
5.2 Elastic hardening spring problems
The hardening elastic spring problem is considered. The motion of a hardening elastic spring [^{31}[31] W. L. Wood and M. E. Oduor. Stability properties of some algorithms for the solution of nonlinear dynamic vibration equations. International Journal for Numerical Methods in Biomedical Engineering, 4(2):205212, 1988.
[32] Y. M. Xie. An assessment of time integration schemes for nonlinear dynamic equations. Journal of Sound and Vibration, 192(1):321331, 1996.^{33}[33] J. Liu and X. Wang. An assessment of the differential quadrature time integration scheme for nonlinear dynamic equations. Journal of Sound and Vibration, 314(1):246253, 2008.] is expressed as
where
Phase portrait of the hardening spring problem
Fifty cycles of displacement and velocity solutions are observed. Since this problem is conservative, the displacementvelocity portrait should form a closed cycle as depicted in Fig.8(a) and (b). The results presented in Fig.8 show that the phase portrait obtained by using the new methods is very accurate. Numerical solutions are forming thirtytwo groups on the exact portrait when
Displacement solution of the hardening spring problem
The results presented in Fig.9 support the fact that the small period error of the new methods can improve the quality of solutions in longterm analyses.
5.3 Elastic softening spring problems
The softening equation [^{31}[31] W. L. Wood and M. E. Oduor. Stability properties of some algorithms for the solution of nonlinear dynamic vibration equations. International Journal for Numerical Methods in Biomedical Engineering, 4(2):205212, 1988.
[32] Y. M. Xie. An assessment of time integration schemes for nonlinear dynamic equations. Journal of Sound and Vibration, 192(1):321331, 1996.^{33}[33] J. Liu and X. Wang. An assessment of the differential quadrature time integration scheme for nonlinear dynamic equations. Journal of Sound and Vibration, 314(1):246253, 2008.] is also solved by using the new methods. The motion of the softening elastic spring is expressed as
where
The results presented in Fig.11 also support the fact that the small period and damping errors of the new methods can improve the quality of numerical solutions in longterm analyses.
Phase portrait of the hardening spring problem
Displacement solution of the hardening spring problem
5.4 Nonlinear single pendulum problem
Description of nonlinear single pendulum [^{5}[5] W. Kim and S. Y. Choi. An improved implicit time integration algorithm: The generalized composite time integration algorithm. Computers and Structures, 196:341354, 2018.]
The nonlinear oscillation of single pendulum described in Fig.12 is numerically solved. The motion of the single pendulum is described
with initial conditions
where
This simple nonlinear pendulum problem is suitable for the test of time integration methods, because important information regarding motions of the pendulum (such as the period and the maximum angle) can be exactly computed by using the initial conditions given in Eqs.(76a) and (76b). For details, please see Refs.[^{5}[5] W. Kim and S. Y. Choi. An improved implicit time integration algorithm: The generalized composite time integration algorithm. Computers and Structures, 196:341354, 2018.], [^{34}[34] T. C. Fung. Solving initial value problems by differential quadrature methodpart 2: second and higherorder equations. International Journal for Numerical Methods in Engineering, 50(6):14291454, 2001.], and [^{35}[35] T. C. Fung. On the equivalence of the time domain differential quadrature method and the dissipative rungekutta collocation method. International Journal for Numerical Methods in Engineering, 53(2):409431, 2002.].
For the test of the new methods, two sets of initial conditions are used. First, the initial conditions that have been used in Refs.[^{5}[5] W. Kim and S. Y. Choi. An improved implicit time integration algorithm: The generalized composite time integration algorithm. Computers and Structures, 196:341354, 2018.,^{7}[7] W. Kim and J. N. Reddy. A new family of higherorder time integration algorithms for the analysis of structural dynamics. Journal of Applied Mechanics, 84(7):071008, 2017.,^{14}[14] W. Kim and J. H. Lee. An improved explicit time integration method for linear and nonlinear structural dynamics. Computers and Structures, 206:4253, 2018.] are also used in this study to synthesize a highly nonlinear case where the pendulum oscillates continuously between two peak points not making complete turns. Second, the initial conditions are chosen as
Angles of the oscillating nonlinear single pendulum problem described in Fig.12(a) for varying values of
Angles of the oscillating nonlinear single pendulum problem described in Fig.12(a) for varying values of
Angles of the rotating nonlinear single pendulum problem described in Fig.12(b) for varying values of
Angles of the rotating nonlinear single pendulum problem described in Fig.12(b) for varying values of
Comparison of relative error
For the first set of the initial conditions, the numerical solution of the new thirdorder method is almost superposing the exact solution (the dotted red line) as shown in Fig.13. However, the thirdorder RK method is presenting noticeable period error as shown in Fig.13. The numerical solutions of the new fourthorder methods are completely superposing the exact solution (the dotted red line) as shown in Fig.14. On the other hand, the fourthorder RK method is showing better results when compared with the thirdorder RK method, but the period error is still noticeable as shown in Fig.14.
For the second set of the initial conditions, the numerical solution of the new thirdorder method is almost superposing the exact solution (the dotted red line) as shown in Fig.15. However, the thirdorder RK method is presenting huge period error and giving a completely incorrect prediction as shown in Fig.15. The result of the thirdorder RK method indicates that the pendulum is oscillating between the two peak points instead of making complete turns. The numerical solution of the new fourthorder method is completely superposing the exact solution (the dotted red line) as shown in Fig.16, while the fourthorder RK method is showing noticeable period error.
To investigate convergence rate of the nonlinear solutions obtained from various methods, the first set of the initial conditions is used. Errors
5.5 Springpendulum problem
The twodegreeoffreedom springpendulum problem [^{10}[10] J. Chung and J. M. Lee. A new family of explicit time integration methods for linear and nonlinear structural dynamics. International Journal for Numerical Methods in Engineering, 37(23):39613976, 1994.] is used for the test of the new methods. The configuration of the twodegreeoffreedom springpendulum problem is described in Fig.18. The governing equations are given by
and the initial conditions are
where
Description of spring pendulum [^{5}[5] W. Kim and S. Y. Choi. An improved implicit time integration algorithm: The generalized composite time integration algorithm. Computers and Structures, 196:341354, 2018.].
Comparison of relative error
In this particular problem, the new fourthorder method is presenting accurate solutions while the new thirdorder method and the third and fourth order RK methods are not as presented in Figs.1920. Interestingly, the convergence rate of the new thirdorder explicit method for the springpendulum is thirdorder as shown in Fig.21. This is due to the velocity dependent nonlinearity included in the governing equations. The results presented in Fig.21 are in good agreement with the order of accuracy computed by using the linear singledegreeoffreedom problem given in Eqs.(70) and (71).
5.6 Double pendulum problem
A double pendulum problem described in Fig.22 is solve to test the new methods. This double pendulum problem can also be used to investigate chaos dynamics [^{36}[36] S. Maiti, J. Roy, A. K. Mallik, and J. K. Bhattacharjee. Nonlinear dynamics of a rotating double pendulum. Physics Letters A, 380(3):408412, 2016.].
The governing equations of the double pendulum [^{37}[37] T. Stachowiak and T. Okada. A numerical analysis of chaos in the double pendulum. Chaos, Solitons & Fractals, 29(2):417422, 2006.] is given by
where
Dimensionless properties
Comparison of relative error
6 Concluding remarks
The new third and fourthorder explicit methods presented in this paper could provide accurate numerical solutions when applied to various linear and nonlinear test problems. Simple but meaningful examples were selected from the existing studies where various methods have been developed and tested. The selected illustrative examples were also used to verify improved performances of the new methods. Due to the computational structures of the new methods which are similar to those of the equivalent RK methods, various nonlinear problems of structural dynamics were successfully solved. The new explicit methods could provide more accurate numerical solutions when compared with the wellknown third and fourthorder RK methods.
The advantages of the new explicit methods can be summarized as:

(a) The new methods could be applied to linear and nonlinear problems in a consistent and unified manner.

(b) The final forms of the new methods did not have any undetermined parameters, which reduced the efforts of a user.

(c) For velocity independent nonlinear problems and undamped linear problems, the new thirdorder method could provide improved solutions with less computational efforts when compared with the fourthorder accurate RK method.
Acknowledgments
The author truly appreciates the support and love from Donghee Son. Thank you.
References

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^{[2]}O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method for Solid and Structural Mechanics. ButterworthHeinemann, 2005.

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^{[4]}W. Kim and J. N. Reddy. Novel mixed finite element models for nonlinear analysis of plates. Latin American Journal of Solids and Structures, 7(2):201226, 2010.

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^{[6]}W. Kim and J. N. Reddy. An improved time integration algorithm: A collocation time finite element approach. International Journal of Structural Stability and Dynamics, 17(02):1750024, 2017.

^{[7]}W. Kim and J. N. Reddy. A new family of higherorder time integration algorithms for the analysis of structural dynamics. Journal of Applied Mechanics, 84(7):071008, 2017.

^{[8]}W. Kim and J. N. Reddy. Effective higherorder time integration algorithms for the analysis of linear structural dynamics. Journal of Applied Mechanics, 84(7):071009, 2017.

^{[9]}W. Kim, S. Park, and J. N. Reddy. A cross weightedresidual time integration scheme for structural dynamics. International Journal of Structural Stability and Dynamics, 14(06):1450023, 2014.

^{[10]}J. Chung and J. M. Lee. A new family of explicit time integration methods for linear and nonlinear structural dynamics. International Journal for Numerical Methods in Engineering, 37(23):39613976, 1994.

^{[11]}G. M. Hulbert and J. Chung. Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Computer Methods in Applied Mechanics and Engineering, 137(2):175188, 1996.

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^{[13]}D. Soares. A novel family of explicit time marching techniques for structural dynamics and wave propagation models. Computer Methods in Applied Mechanics and Engineering, 311:838855, 2016.

^{[14]}W. Kim and J. H. Lee. An improved explicit time integration method for linear and nonlinear structural dynamics. Computers and Structures, 206:4253, 2018.

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^{[16]}W. Kim. An accurate twostage explicit time integration scheme for structural dynamics and various dynamic problems. International Journal for Numerical Methods in Engineering, pages 0000, 2019.

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^{[21]}Gregory M Hulbert. A unified set of singlestep asymptotic annihilation algorithms for structural dynamics. Computer Methods in Applied Mechanics and Engineering, 113(12):19, 1994.

^{[22]}T.C. Fung. Unconditionally stable higherorder accurate hermitian time finite elements. International journal for numerical methods in engineering, 39(20):34753495, 1996.

^{[23]}T.C. Fung. Weighting parameters for unconditionally stable higherorder accurate time step integration algorithms. part 2secondorder equations. International journal for numerical methods in engineering, 45(8):9711006, 1999.

^{[24]}T.C Fung. Unconditionally stable higherorder accurate collocation timestep integration algorithms for firstorder equations. Computer methods in applied mechanics and engineering, 190(1314):16511662, 2000.

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Publication Dates

Publication in this collection
22 July 2019 
Date of issue
2019
History

Received
29 Apr 2019 
Reviewed
18 May 2019 
Accepted
22 May 2019 
Published
03 June 2019