A parabolic acceleration time integration method for structural dynamics using quartic B-spline functions

Abstract In this paper, an explicit time integration method is proposed for structural dynamics using periodic quartic B-spline interpolation polynomial functions. In this way, at first, by use of quartic B-splines, the authors have proceeded to solve the differential equation of motion governing SDOF systems and later the proposed method has been generalized for MDOF systems. In the proposed approach, a straightforward formulation was derived in a fluent manner from the approximation of response of the system with B-spline basis. Because of using a quartic function, the system acceleration is approximated with a parabolic function. For the aforesaid method, a simple step-by-step algorithm was implemented and presented to calculate dynamic response of MDOF systems. The proposed method has appropriate convergence, accuracy and low time consumption. Accuracy and stability analyses have been done perfectly in this paper. The proposed method benefits from an extraordinary accuracy compared to the existing methods such as central difference, Runge–Kutta and even Duhamel integration method. The validity and effectiveness of the proposed method is demonstrated with four examples and the results of this method are compared with those from some of the existent numerical methods. The high accuracy and less time consumption are only two advantages of this method.

[1]  T. Belytschko,et al.  Computational Methods for Transient Analysis , 1985 .

[2]  Hikmet Caglar,et al.  B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems , 2006, Appl. Math. Comput..

[3]  Hikmet Caglar,et al.  B-spline method for solving linear system of second-order boundary value problems , 2009, Comput. Math. Appl..

[4]  Hikmet Caglar,et al.  The numerical solution of the one-dimensional heat equation by using third degree B-spline functions , 2008 .

[5]  John C. Houbolt,et al.  A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft , 1950 .

[6]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[7]  K. Park.  An Improved Stiffly Stable Method for Direct Integration of Nonlinear Structural Dynamic Equations , 1975 .

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

[9]  Shuenn-Yih Chang,et al.  Explicit Pseudodynamic Algorithm with Unconditional Stability , 2002 .

[10]  D. F. Rogers,et al.  An Introduction to NURBS: With Historical Perspective , 2011 .

[11]  Hikmet Caglar,et al.  B-spline solution of singular boundary value problems , 2006, Appl. Math. Comput..

[12]  Shuenn-Yih Chang,et al.  A new family of explicit methods for linear structural dynamics , 2010 .

[13]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[14]  Ted Belytschko,et al.  An analysis of an unconditionally stable explicit method , 1983 .

[15]  Robert M. Ebeling,et al.  Accuracy of Response of Single-Degree-of-Freedom Systems to Ground Motion , 1997 .

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

[17]  Anil K. Chopra,et al.  Dynamics of Structures: Theory and Applications to Earthquake Engineering , 1995 .

[18]  Gérard Rio,et al.  Comparative study of numerical explicit time integration algorithms , 2002, Adv. Eng. Softw..

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

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

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

[22]  Shuenn-Yih Chang.  Improved numerical dissipation for explicit methods in pseudodynamic tests , 1997 .

[24]  J. L. Liu,et al.  Solution of dynamic response of SDOF system using piecewise Lagrange polynomial , 2001 .

[25]  SOLUTION OF DYNAMIC RESPONSE OF FRAMED STRUCTURE USING PIECEWISE BIRKHOFF POLYNOMIAL , 2002 .

[26]  Hikmet Caglar,et al.  Fifth-degree B-spline solution for a fourth-order parabolic partial differential equations , 2008, Appl. Math. Comput..