A fast and accurate dynamic relaxation scheme

doi:10.1007/s11709-018-0486-2

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 Author(s): 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:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-018-0486-2
https://academic.hep.com.cn/fsce/EN/Y2019/V13/I1/176

Fig.1 Arch truss

procedure method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	1	2	3	4	5	total iterations	time (s)	$E T$
benchmark	Method 1	1692	1886	2334	3187	5881	44862	14.250	?
proposed	Method 2	1701	1886	2332	3186	5883	44864	12.359	13.27
combined	Method 3	1621	1798	2227	3038	5610	42788	11.609	18.53

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	2	4	6	8	10	total iterations	time (s)	$E T$
benchmark	Method 1	705	842	1099	2020	4035	24527	5.797	?
proposed	Method 2	717	854	1140	1953	3901	15868	4.484	22.65
combined	Method 3	709	797	994	1856	4431	8787	4.023	30.60

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	1	3	5	7	9	total iterations	time (s)	$E T$
benchmark	Method 1	687	754	879	1027	1170	9308	5.281	?
proposed	Method 2	687	754	879	1026	1169	9304	4.619	12.54
combined	Method 3	658	719	838	979	1115	8876	4.210	20.28

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	2	4	6	8	10	total iterations	time (s)	$E T$
benchmark	Method 1	390	390	394	446	201	3736	5.016	?
proposed	Method 2	388	388	391	444	201	3713	4.688	6.54
combined	Method 3	369	369	373	423	194	3542	4.563	9.03

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	1	3	5	7	9	total iterations	time (s)	$E T$
benchmark	Method 1	1354	1318	1384	1472	1596	14405	213.735	?
proposed	Method 2	1354	1318	1384	1472	1596	14403	195.250	8.65
combined	Method 3	1291	1257	1319	1404	1522	13734	182.423	14.65

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	2	4	6	8	10	total iterations	time (s)	$E T$
benchmark	Method 1	24728	24181	25041	26547	29075	263707	72.938	?
proposed	Method 2	24726	24181	25040	26547	29075	263672	63.140	13.43
combined	Method 3	23575	23055	23875	25313	27725	251405	59.375	18.60

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	1	3	5	7	9	total iterations	time (s)	$E T$
benchmark	Method 1	1097	994	986	986	987	10006	12.141	?
proposed	Method 2	1095	992	984	984	986	9992	9.171	24.46
combined	Method 3	1043	946	938	938	939	9518	7.907	34.87

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	2	4	6	8	10	total iterations	time (s)	$E T$
benchmark	Method 1	827	850	881	917	959	8840	4.843	?
proposed	Method 2	825	850	882	918	959	8840	4.171	13.88
combined	Method 3	789	813	843	878	916	8450	4.203	13.21

Tab.8 Number of iteration and analysis time for toggle frame

Fig.16 Load-displacement curve for toggle frame

Fig.17 Cylindrical roof

procedure method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	1	3	5	7	9	total iterations	time (s)	$E T$
benchmark	Method 1	2930	3082	4021	7716	4064	46753	2889.760	?
proposed	Method 2	2930	3082	4021	7716	4064	46753	2549.176	11.79
combined	Method 3	2794	2943	3922	7359	3899	44854	2209.656	23.53

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	2	4	6	8	10	total iterations	time (s)	$E T$
benchmark	Method 1	10645	12501	11507	10164	10212	108309	1735.86	?
proposed	Method 2	10645	12501	11507	10164	10212	108308	1537.16	11.45
combined	Method 3	10150	11919	10976	9819	9744	103577	1456.47	16.10

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	1	3	5	7	9	total iterations	time (s)	$E T$
benchmark	Method 1	376	297	281	268	258	2869	360.250	?
proposed	Method 2	375	297	281	268	258	2867	323.359	10.24
combined	Method 3	358	284	268	256	246	2737	304.484	15.48

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 method	sign	number of iteration for each step					total iterations	time (s)	$E T$
procedure method	sign	2	4	6	8	10	total iterations	time (s)	$E T$
benchmark	Method 1	1777	2726	3487	1932	1764	24048	1063.062	?
proposed	Method 2	1777	2726	3487	1932	1764	24047	941.250	11.46
combined	Method 3	1694	2599	3681	1843	1680	23288	903.797	14.98

Tab.12 Number of iteration and analysis time for shallow spherical cap

Fig.26 Load-displacement curve for shallow spherical cap

S	stiffness matrix
X, x	displacement
$X j ∗$	exact displacement
F_r	internal forces in real condition
P_r	external force in real condition
F_f	fictitious internal forces
$X ˙$	velocity
$X ¨$	acceleration
R, r	residual forces
M	mass matrix
C	damping matrix
h	fictitious time
n	iteration number
i, j	numerator of degrees of freedom
ndof	number of degrees of freedom
E	analysis error
r	reduction rate in analysis error
I	identity matrix
$λ$	eigenvalues of matrix
$E T$	analysis time criterion

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