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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (1) : 176-189     https://doi.org/10.1007/s11709-018-0486-2
RESEARCH ARTICLE |
A fast and accurate dynamic relaxation scheme
Mohammad REZAIEE-PAJAND(), Mohammad MOHAMMADI-KHATAMI
Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
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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     
Corresponding Authors: Mohammad REZAIEE-PAJAND   
Just Accepted Date: 29 May 2018   Online First Date: 17 July 2018    Issue Date: 04 January 2019
 Cite this article:   
Mohammad REZAIEE-PAJAND,Mohammad MOHAMMADI-KHATAMI. A fast and accurate dynamic relaxation scheme[J]. Front. Struct. Civ. Eng., 2019, 13(1): 176-189.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-018-0486-2
http://journal.hep.com.cn/fsce/EN/Y2019/V13/I1/176
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Mohammad REZAIEE-PAJAND
Mohammad MOHAMMADI-KHATAMI
Fig.1  Arch truss
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
12345
benchmarkMethod 1169218862334318758814486214.250?
proposedMethod 2170118862332318658834486412.35913.27
combinedMethod 3162117982227303856104278811.60918.53
Tab.1  Number of iteration and analysis time for arch truss
Fig.2  Load-displacement curve for arch truss
Fig.3  33-member truss
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
246810
benchmarkMethod 1705842109920204035245275.797?
proposedMethod 2717854114019533901158684.48422.65
combinedMethod 37097979941856443187874.02330.60
Tab.2  Number of iteration and analysis time for 33-member truss
Fig.4  Load-displacement curve for 33-member truss
Fig.5  Truss tower. (a) 3D view; (b) X-Z view; (c) X-Y view
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
13579
benchmarkMethod 16877548791027117093085.281?
proposedMethod 26877548791026116993044.61912.54
combinedMethod 3658719838979111588764.21020.28
Tab.3  Number of iteration and analysis time for truss tower
Fig.6  Load-displacement curve for truss tower
Fig.7  Shallow truss. (a) X-Z view; (b) X-Y view
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
246810
benchmarkMethod 139039039444620137365.016?
proposedMethod 238838839144420137134.6886.54
combinedMethod 336936937342319435424.5639.03
Tab.4  Number of iteration and analysis time for shallow truss
Fig.8  Load-displacement curve for shallow truss
Fig.9  Circular truss dome. (a) 3D view; (b) X-Y view
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
13579
benchmarkMethod 11354131813841472159614405213.735?
proposedMethod 21354131813841472159614403195.2508.65
combinedMethod 31291125713191404152213734182.42314.65
Tab.5  Number of iteration and analysis time for circular truss dome
Fig.10  Load-displacement curve for circular truss dome
Fig.11  Portal frame
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
246810
benchmarkMethod 1247282418125041265472907526370772.938?
proposedMethod 2247262418125040265472907526367263.14013.43
combinedMethod 3235752305523875253132772525140559.37518.60
Tab.6  Number of iteration and analysis time for circular portal frame
Fig.12  Load-displacement curve for circular portal frame
Fig.13  Five-story frame
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
13579
benchmarkMethod 110979949869869871000612.141?
proposedMethod 2109599298498498699929.17124.46
combinedMethod 3104394693893893995187.90734.87
Tab.7  Number of iteration and analysis time for circular five-story frame
Fig.14  Load-displacement curve for circular five-story frame
Fig.15  Toggle frame
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
246810
benchmarkMethod 182785088191795988404.843?
proposedMethod 282585088291895988404.17113.88
combinedMethod 378981384387891684504.20313.21
Tab.8  Number of iteration and analysis time for toggle frame
Fig.16  Load-displacement curve for toggle frame
Fig.17  Cylindrical roof
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
13579
benchmarkMethod 129303082402177164064467532889.760?
proposedMethod 229303082402177164064467532549.17611.79
combinedMethod 327942943392273593899448542209.65623.53
Tab.9  Number of iteration and analysis time for cylindrical roof
Fig.18  Load-displacement curve for cylindrical roof
Fig.19  Load-displacement curve of central node of cylindrical roof including the load limit points
Fig.20  Deformed shape of cylindrical roof
Fig.21  Cantilever plate
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
246810
benchmarkMethod 110645125011150710164102121083091735.86?
proposedMethod 210645125011150710164102121083081537.1611.45
combinedMethod 3101501191910976981997441035771456.4716.10
Tab.10  Number of iteration and analysis time for cantilever plate
Fig.22  Load-displacement curve for cantilever plate
Fig.23  Cook membrane (dimensionless)
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
13579
benchmarkMethod 13762972812682582869360.250?
proposedMethod 23752972812682582867323.35910.24
combinedMethod 33582842682562462737304.48415.48
Tab.11  Number of iteration and analysis time for cook membrane
Fig.24  Load-displacement curve for cook membrane (dimensionless)
Fig.25  Shallow spherical cap
procedure methodsignnumber of iteration for each steptotal iterationstime (s)ET
246810
benchmarkMethod 117772726348719321764240481063.062?
proposedMethod 21777272634871932176424047941.25011.46
combinedMethod 31694259936811843168023288903.79714.98
Tab.12  Number of iteration and analysis time for shallow spherical cap
Fig.26  Load-displacement curve for shallow spherical cap
Sstiffness matrix
X, xdisplacement
Xjexact displacement
Frinternal forces in real condition
Prexternal force in real condition
Fffictitious internal forces
X˙velocity
X¨acceleration
R, rresidual forces
Mmass matrix
Cdamping matrix
hfictitious time
niteration number
i, jnumerator of degrees of freedom
ndofnumber of degrees of freedom
Eanalysis error
rreduction rate in analysis error
Iidentity matrix
λeigenvalues of matrix
ETanalysis time criterion
  
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