Dynamic relaxation with concentrated damping

Author
S. R. Sarafrazi
University of Birjand
Abstract
Solving a system of linear or non-linear equations is required to analyze any kind of structures. There are many ways to solve a system of equations. They can be classified as implicit and explicit techniques. The explicit methods eliminate round-off errors and use less memory. The dynamic relaxation method (DRM) is one of the powerful and simple explicit processes. The important point is that the DRM does not require to storage the global stiffness matrix. It just uses the residual loads vector.
Utilizing the virtual masses, damping and time steps, the DRM convert a system of static equations to dynamic ones. The process is started by assuming an initial solution. The next steps are done in such a way that the residual forces are decreased. The proper value of fictitious mass and time step guarantees the convergence of the proposed DR procedure. On the other hand, the convergence rate is dependent on value of damping factor, which is calculated using the lowest eigenvalue of artificial dynamic system in the common dynamic relaxation method. It is evidence; the dynamic system oscillates when damping is zero. The convergence of DRM with zero damping factors is achieved utilizing kinetic damping or -damping. In the kinetic dynamic relaxation process, the velocities of the joints are set to zero when a fall in the level of total kinetic energy of the structure occurs. However, it is difficult to calculate the extreme point of kinetic energy. Topping suggested assuming the peak point at the mid-point of the previous time-step, when a fall down in kinetic energy is occurred. The factor  in the -damping method is time step ratio of two sequence steps. The time-step ratio can be calculated in such a way that the responses converge to exact solutions.
In this paper, a comprehensive review of dynamic relaxation algorithms is presented. Of these, the popular and kinetic damping DR methods are described in detail. Then, the new dynamic relaxation algorithm is proposed. In this procedure, the artificial mass and time steps are similar to the DR methods that have been recently introduced. However, the damping factor is different with these methods. Damping factor is calculated in some specified steps. In other words, damping is zero in the most step of DR algorithm. Therefore, the total number of calculations is reduced. The concentrated damping is imposed, when the value of total kinetic energy of system is at its peak point. Utilizing the proper values of concentrated damping factors, the kinetic energy converges to zero. The presented formulation shows the relation between common and kinetic dynamic relaxation processes, too. It should be noted; the procedures of minimizing the kinetic energy of proposed method and Topping algorithm are different. The kinetic technique is required more calculations. Finally, some benchmark problems of truss and frame structures are selected. The linear and geometric nonlinear analyses are performed. The numerical results also show that the convergence rate of the new DRM increases in the majority of cases with respect to kinetic damping and also popular damping.

Keywords

[1]        Sarafrazi S.R. Numerical integration for structural dynamic analysis. Dissertation submitted in partial fulfillment of the requirement for the degree of doctor of philosophy, Ferdowsi University, Mashhad, Iran; 2010 (In Persian).
[2]           Frankel, S. P. (1950), “Convergence rates of iterative treatments of partial differential equations”, Math. Tables Aids Comput , vol. 4, no. 30, pp. 65–75.
 [3]          Otter, J. R. H. (1966), “Dynamic relaxation compared with other iterative finite difference methods”, Nucl. Eng. Des., vol. 3, no. 1, pp. 183–185.
[4]           Rezaiee-Pajand, M. Sarafrazi, S. R. and Rezaiee, H. (2012), “Efficiency of dynamic relaxation methods in nonlinear analysis of truss and frame structures”, Comput. Struct., vol. 112–113, pp. 295–310.
[5]           Rushton, K. R. (1969), “Dynamic-relaxation solution for the large deflection of plates with specified boundary stresses”, J. Strain Anal. Eng. Des ., vol. 4, no. 2, pp. 75–80.
[6] Brew, J. S. and Brotton, D. M. (1971), “Non-linear structural analysis by dynamic relaxation”, Int. J. Numer. Methods Eng ., vol. 3, no. 4, pp. 463–483.
[7]           Wood, W. L. (1971), “Note on dynamic relaxation”, Int. J. Numer. Methods Eng. , vol. 3, no. 1, pp. 145–147.
[8] Bunce, J. W.  (1972), “A note on the estimation of critical damping in dynamic relaxation”, Int. J. Numer. Methods Eng. , vol. 4, no. 2, pp. 301–303.
[9]           Cundall, P. A. (1976), “Explicit finite-difference method in geomechanics”, presented at the Numerical Methods in Geomechanics, Blacksburg, 1976, pp. 132–150.
[10]         Topping, B. H. V. and Ivanyi, P. (2008), Computer aided design of cable membrane structures. Kippen, Stirlingshire, Scotland: Saxe-Coburg Publications.
[11]         Cassell, A. C. and Hobbs, R. E. (1976), “Numerical stability of dynamic relaxation analysis of non-linear structures”, Int. J. Numer. Methods Eng. , vol. 10, no. 6, pp. 1407–1410.
[12]         Frieze, P. A. Hobbs, R. E. and Dowling, P. J. (1978), “Application of dynamic relaxation to the large deflection elasto-plastic analysis of plates”, Comput. Struct. , vol. 8, no. 2, pp. 301–310.
[13]         Papadrakakis, M. (1981) “A method for the automatic evaluation of the dynamic relaxation parameters”, Comput. Methods Appl. Mech. Eng., vol. 25, no. 1, pp. 35–48.
[14]         Qiang, S. (1988), “An adaptive dynamic relaxation method for nonlinear problems”, Comput. Struct. , vol. 30, no. 4, pp. 855–859.
[15]         Al-Shawi, F. A. N. and Mardirosian, A. H. (1987), “An improved dynamic relaxation method for the analysis of plate bending problems”, Comput. Struct., vol. 27, no. 2, pp. 237–240.
[16]         Zhang, L. G. and Yu, T. X. (1989), “Modified adaptive dynamic relaxation method and its application to elastic-plastic bending and wrinkling of circular plates”, Comput. Struct. , vol. 33, no. 2, pp. 609–614.
[17]         Zhang, L. C. Kadkhodayan, M.and Mai, Y.-W. (1994), “ Development of the maDR method”, Comput. Struct ., vol. 52, no. 1, pp. 1–8.
[18]         Munjiza, A. Owen, D. R. J. and Crook, A. J. L. (1998), “An M(M−1K)m proportional damping in explicit integration of dynamic structural systems”, Int. J. Numer. Methods Eng. , vol. 41, no. 7, pp. 1277–1296.
[19]         Rezaiee Pajand, M. and Taghavian Hakkak, M. (2006), “Nonlinear analysis of truss structures using dynamic relaxation (research note)”, Int. J. Eng. - Trans. B Appl., vol. 19, no. 1, p. 11.
[20]         Kadkhodayan, M. Alamatian, J.and Turvey, G. J. (2008), “A new fictitious time for the dynamic relaxation (DXDR) method”, Int. J. Numer. Methods Eng., vol. 74, no. 6, pp. 996–1018.
[21]         Rezaiee Pajand, M. and Alamatian, J. (2005), “Best time step for geometric nonlinear analysis using dynamic relaxation”, Modarres Eng., vol. 19, No. 17, pp. 61-74 (In Persian).
 [22]        Rezaiee Pajand, M. and Alamatian, J. (2009), “Dynamic relaxation method for tracing the statical path of truss structures”, J. Model. Eng. , vol. 3, No. 17, pp. 27-39 (In Persian).
 [23]        Rezaiee Pajand, M. and Alamatian, J. (2010), “The dynamic relaxation method using new formulation for fictitious mass and damping”, Struct. Eng. Mech. , vol. 34, No. 1, pp. 109-133.
[24]         Rezaiee-Pajand, M. and Sarafrazi, S. R. (2011), “Nonlinear dynamic structural analysis using dynamic relaxation with zero damping”, Comput Struct, vol. 89, no. 13–14, pp. 1274–1285.
[25]         Rezaiee-Pajand, M. and Rezaiee, H. (2011), “A technique for improving dynamic relaxation method”, J. Model. Eng., vol. 9, no. 24, pp. 39–52 (In Persian).
 [26]        M. Rezaiee-Pajand and S. R. Sarafrazi, (2010), “Nonlinear structural analysis using dynamic relaxation method with improved convergence rate”, Int. J. Comput. Methods, vol. 07, no. 04, pp. 627–654
Modares Civil Engineering journal
Volume 17, Issue 3
September and October 2017
Pages 146-156