A fast and accurate dynamic relaxation scheme

Abstract

Dynamic relaxation method (DRM) is one of the suitable numerical procedures for nonlinear structural analysis. Adding the fictitious inertia and damping forces to the static equation, and turning it to the dynamic system, are the basis of this technique. Proper selection of the DRM artificial factors leads to the better convergence rate and efficient solutions. This study aims to increase the numerical stability, and to decrease the analysis time. To fulfil this objective, the reduction rate of analysis error for consecutive iterations is minimized. Based on this formulation, a new time step is found for the viscous dynamic relaxation. After combining this novel relationship with the other DRM factors, various geometrical nonlinear structures, such as trusses, frames, and shells, are analyzed. The obtained results verify the efficiency of authors’ scheme.

Keywords

viscous dynamic relaxation / time step / displacement error / geometric nonlinear analysis

References

[1] Day A S. An introduction to dynamic relaxation. Engineer, 1965, 219: 218–221

[2] Brew J S, Brotton D M. Non-linear structural analysis by dynamic relaxation. International Journal for Numerical Methods in Engineering, 1971, 3(4): 463–483

[3] Bunce J W. A note on the estimation of critical damping in dynamic relaxation. International Journal for Numerical Methods in Engineering, 1972, 4(2): 301–303

[4] Cassell A C, Hobbs R E. Numerical stability of dynamic relaxation analysis of non-linear structures. International Journal for Numerical Methods in Engineering, 1976, 10(6): 1407–1410

[5] Papadrakakis M. A method for the automatic evaluation of the dynamic relaxation parameters. Computer Methods in Applied Mechanics and Engineering, 1981, 25(1): 35–48

[6] Underwood P.Dynamic relaxation. Computational Methods for Transient Analysis, 1983, 245–265

[7] Qiang S. An adaptive dynamic relaxation method for nonlinear problems. Computers & Structures, 1988, 30(4): 855–859

[8] Munjiza A. A K_m proportional damping for dynamic relaxation. International Journal for Engineering Modelling, 1996, 9: 1–9

[9] Munjiza A, Owen D R J, Crook A J L, AnM(M⁻¹K)Mproportional damping in explicit integration of dynamic structural systems. International Journal for Numerical Methods in Engineering, 1998, 41(7): 1277–1296 doi:10.1002/(SICI)1097-0207(19980415)41:7<1277::AID-NME335>3.0.CO;2-9

[10] Zhang L G, Yu T X. Modified adaptive dynamic relaxation method and its application to elastic-plastic bending and wrinkling of circular plates. Computers & Structures, 1989, 33(2): 609–614

[11] Zhang L C, Kadkhodayan M, Mai Y W. Development of the maDR method. Computers & Structure, 1994, 52(1): 1–8

[12] Barnes M. Form finding and analysis of tension structures by dynamic relaxation. International Journal of Space Structures, 1999, 14(2): 89–104

[13] Topping B H V, Ivanyi P. Computer Aided Design of Cable Membrane Structures.Stirling: Saxe-Coburg Publications, 2008

[14] Rezaiee-Pajand M, Taghavian Hakkak M. Nonlinear analysis of truss structures using dynamic relaxation. International Journal of Engineering, 2006, 19(1): 11–22

[15] Kadkhodayan M, Alamatian J, Turvey G J. A new fictitious time for the dynamic relaxation (DXDR) method. International Journal for Numerical Methods in Engineering, 2008, 74(6): 996–1018

[16] Rezaiee-Pajand M, Alamatian J. Nonlinear dynamic analysis by dynamic relaxation method. Structural Engineering and Mechanics, 2008, 28(5): 549–570

[17] Rezaiee-Pajand M, Alamatian J. The dynamic relaxation method using new formulation for fictitious mass and damping. Structural Engineering and Mechanics, 2010, 34(1): 109–133

[18] Rezaiee-Pajand M, Sarafrazi S R. Nonlinear structural analysis using dynamic relaxation method with improved convergence rate. International Journal of Computational Methods, 2010, 7(4): 627–654

[19] Rezaiee-Pajand M, Sarafrazi S R. Nonlinear dynamic structural analysis using dynamic relaxation with zero damping. Computers & Structures, 2011, 89(13–14): 1274–1285

[20] Rezaiee-Pajand M, Sarafrazi S R, Rezaiee H. Efficiency of dynamic relaxation methods in nonlinear analysis of truss and frame structures. Computers & Structures, 2012, 112‒113: 295–310

[21] Alamatian J. Displacement-based methods for calculating the buckling load and tracing the post-buckling regions with dynamic relaxation method. Computers & Structures, 2013, 114–115: 84–97

[22] Rezaiee-Pajand M, Rezaee H. Fictitious time step for the kinetic dynamic relaxation method. Mechanics of Advanced Materials and Structures, 2014, 21(8): 631–644

[23] Rezaiee-Pajand M, Estiri H. Mixing dynamic relaxation method with load factor and displacement increments. Computers & Structures, 2016, 168: 78–91

[24] Rezaiee-Pajand M, Estiri H. Finding equilibrium paths by minimizing external work in dynamic relaxation method. Applied Mathematical Modelling, 2016, 40(23–24): 10300–10322

[25] Rezaiee-Pajand M, Alamatian J, Rezaee H. The state of the art in dynamic relaxation methods for structural mechanics Part 1: Formulations. Iranian Journal of Numerical Analysis and Optimization, 2017, 7(2): 65–86

[26] Rezaiee-Pajand M, Alamatian J, Rezaee H. The state of the art in dynamic relaxation methods for structural mechanics Part 2: Applications. Iranian Journal of Numerical Analysis and Optimization, 2017, 7(2): 87–114

[27] Rezaee H. Nonlinear structural analysis using dynamic relaxation method. Dissertation for the Doctoral Degree. Mashhad: Ferdowsi University, 2012 (in Persian)

[28] Rezaiee-Pajand M, Alamatian J. Automatic DR structural analysis of snap-through and snap-back using optimized load increments. Journal of Structural Engineering, 2011, 137(1): 109–116

[29] Alamatian J. A new formulation for fictitious mass of the dynamic relaxation method with kinetic damping. Computers & Structures, 2012, 90‒91: 42–54

[30] Rezaiee-Pajand M, Estiri H. Computing the structural buckling limit load by using dynamic relaxation method. International Journal of Non-linear Mechanics, 2016, 81: 245–260

[31] Kuo-Mo H. Nonlinear analysis of general shell structures by flat triangular shell element. Computers & Structures, 1987, 25(5): 665–675

[32] Kuznetsov V V, Levyakov S V. Phenomenological invariant-based finite-element model for geometrically nonlinear analysis of thin shells. Computer Methods in Applied Mechanics and Engineering, 196(49): 4952–4964

[33] Boutagouga D, Gouasmia A, Djeghaba K. Geometrically nonlinear analysis of thin shell by a quadrilateral finite element with in-plane rotational degrees of freedom. European Journal of Computational Mechanics, 2012, 19(8): 707–724 doi:10.3166/ejcm.19.707-724

RIGHTS & PERMISSIONS

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature