A composite collocation method with low-period elongation for structural dynamics problems

Abstract This paper presents a novel time integration algorithm for solving linear structural dynamic problems in the framework of the high-order collocation method. When two Gauss points in the integration interval are selected as collocation points, both an A-stable algorithm with third order accuracy and a non-dissipative algorithm with fourth order accuracy can be derived from a second order collocation polynomial. The only difference is that the former obtains a numerical solution at the middle point of the time interval, while the latter has a solution at the end of the interval. A new composite method is established through applying these two algorithms alternately, which combines the advantages of the numerical dissipation property of the third order algorithm and the high-order accuracy of the fourth order algorithm. The usage frequency of the two algorithms during the whole step-by-step integration procedure is an important parameter affecting the numerical dissipation, which is investigated in this study. As the algebraic equations systems solved by the two algorithms are exactly same, no extra computation effort is introduced.

[1]  F. Ubertini,et al.  The Nørsett time integration methodology for finite element transient analysis , 2002 .

[2]  Delfim Soares,et al.  A novel family of explicit time marching techniques for structural dynamics and wave propagation models , 2016 .

[3]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[4]  Gene H. Golub,et al.  Matrix computations , 1983 .

[5]  Jennifer A. Scott,et al.  On Signed Incomplete Cholesky Factorization Preconditioners for Saddle-Point Systems , 2014, SIAM J. Sci. Comput..

[6]  Zeng-Qi Wang,et al.  On parameterized inexact Uzawa methods for generalized saddle point problems , 2008 .

[7]  T. C. Fung,et al.  Solving initial value problems by differential quadrature method—part 1: first‐order equations , 2001 .

[8]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[9]  J. M. Sanz-Serna,et al.  Runge-kutta schemes for Hamiltonian systems , 1988 .

[10]  M. Dokainish,et al.  A survey of direct time-integration methods in computational structural dynamics—I. Explicit methods , 1989 .

[11]  Javad Alamatian,et al.  Implicit Higher-Order Accuracy Method for Numerical Integration in Dynamic Analysis , 2008 .

[12]  Thomas J. R. Hughes,et al.  Improved numerical dissipation for time integration algorithms in structural dynamics , 1977 .

[13]  K. Bathe,et al.  Insight into an implicit time integration scheme for structural dynamics , 2012 .

[14]  K. Bathe.  Finite Element Procedures , 1995 .

[15]  Jintai Chung,et al.  A new family of explicit time integration methods for linear and non‐linear structural dynamics , 1994 .

[16]  T. Fung.  Higher‐order accurate least‐squares methods for first‐order initial value problems , 1999 .

[17]  Webe João Mansur,et al.  A review of automatic time-stepping strategies on numerical time integration for structural dynamics analysis , 2014 .

[18]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

[19]  T. C. Fung,et al.  Weighting parameters for unconditionally stable higher-order accurate time step integration algorithms. Part 2—second-order equations , 1999 .

[20]  K. Bathe,et al.  On a composite implicit time integration procedure for nonlinear dynamics , 2005 .

[21]  T. C. Fung,et al.  Unconditionally stable higher-order accurate collocation time-step integration algorithms for first-order equations , 2000 .

[22]  Jennifer A. Scott,et al.  HSL_MI28 , 2014, ACM Trans. Math. Softw..

[23]  F. Ubertini,et al.  Collocation methods with controllable dissipation for linear dynamics , 2001 .

[24]  Edward L. Wilson,et al.  Nonlinear dynamic analysis of complex structures , 1972 .

[25]  T. C. Fung,et al.  Solving initial value problems by differential quadrature method?part 2: second- and higher-order equations , 2001 .

[26]  T. Fung.  Complex-time-step Newmark methods with controllable numerical dissipation , 1998 .

[27]  Jintai Chung,et al.  Explicit time integration algorithms for structural dynamics with optimal numerical dissipation , 1996 .

[28]  T. C. Fung,et al.  Complex‐time‐step methods for transient analysis , 1999 .

[29]  S. Shojaee,et al.  An unconditionally stable implicit time integration algorithm , 2015 .