This paper sets out to demonstrate three things: (i) implicit integration with absolute nodal coordinate formulation (ANCF) is effective in handling very stiff systems when an accurate computation of the sensitivity matrix is part of the solution sequence, (ii) parallel computing can provide a vehicle for ANCF to tackle very large kinematically constrained problems with millions of degrees of freedom and produce results in a matter of seconds, and (iii) large systems of equations associated with implicit integration can be solved in parallel by relying on an iterative approach that avoids costly matrix factorizations, which would be prohibitively expensive and memory intensive. For (iii), the approach adopted relies on a Krylov–subspace method that is invoked in the Newton stage at each time step of the numerical solution process. The proposed approach is validated against a commercial package and several simple systems for which analytical solutions are available. A set of numerical experiments demonstrates the scaling of the parallel solution method and provides insights in relation to the size of ANCF problems that are tractable using graphics processing unit (GPU) parallel computing and implicit numerical integration.

References

References
1.
Von Dombrowski
,
S.
,
2002
, “
Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates
,”
Multibody Syst. Dyn.
,
8
, pp.
409
432
.10.1023/A:1021158911536
2.
Dufva
,
K.
, and
Shabana
,
A. A.
,
2005
, “
Analysis of Thin Plate Structures Using the Absolute Nodal Coordinate Formulation
,”
Proc. IMechE Part K: J Multibody Dyn.
,
219
, pp.
345
355
.
3.
Gerstmayr
,
J.
, and
Shabana
,
A.
,
2006
, “
Analysis of Thin Beams and Cables Using the Absolute Nodal Co-Ordinate Formulation
,”
Nonlinear Dyn.
,
45
, pp.
109
130
.10.1007/s11071-006-1856-1
4.
Shabana
,
A. A.
,
2008
,
Computational Continuum Mechanics
,
Cambridge University Press
,
New York
.
5.
Melanz
,
D.
,
2012
, “
On the Validation and Applications of a Parallel Flexible Multi-body Dynamics Implementation
,” M.S. thesis, University of Wisconsin-Madison, Madison, WI.
6.
Shabana
,
A. A.
,
2005
,
Dynamics of Multibody Systems
,
3rd ed.
,
Cambridge University Press
,
New York
.
7.
Haug
,
E. J.
,
1989
,
Computer-Aided Kinematics and Dynamics of Mechanical Systems. Volume I: Basic Methods
,
Allyn and Bacon
,
Boston, MA
.
8.
Haug
,
E. J.
,
Wu
,
S.
, and
Yang
,
S.
,
1986
, “
Dynamics of Mechanical Systems With Coulomb Friction, Stiction, Impact, and Constraint Addition-Deletion-I
,”
Mech. Mach. Theory
,
21
, pp.
401
406
.10.1016/0094-114X(86)90088-1
9.
Hairer
,
E.
, and
Wanner
,
G.
,
1996
,
Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems
, Vol.
14
,
2nd ed.
,
Springer
,
Berlin
.
10.
Khude
,
N.
,
Heyn
,
T.
,
Jay
,
L.O.
, and
Negrut
,
D.
,
2007
, “
A Comparison of Low Order Numerical Integration Formulas for Time Domain Analysis of Mechanical Systems
,”
Proceedings of the ECCOMAS Thematic Conference
,
Milan, Italy
.
11.
Negrut.
,
D.
,
Rampalli
,
R.
,
Ottarsson
,
G.
, and
Sajdak
,
A.
,
2007
, “
On an Implementation of the HHT Method in the Context of Index 3 Differential Algebraic Equations of Multibody Dynamics
,”
ASME J. Comput. Nonlinear Dyn.
,
2
(
1
), pp.
73
85
.10.1115/1.2389231
12.
Hussein
,
B.
,
Negrut
,
D.
, and
Shabana
,
A.
,
2008
, “
Implicit and Explicit Integration in the Solution of the Absolute Nodal Coordinate Differential/Algebraic Equations
,”
Nonlinear Dyn.
,
54
, pp.
283
296
.10.1007/s11071-007-9328-9
13.
Newmark
,
N. M.
,
1959
, “
A Method of Computation for Structural Dynamics
,”
ASCE J. Eng. Mech.
, pp.
67
94
.
14.
Arnold
,
M.
, and
Bruls
,
O.
,
2007
, “
Convergence of the Generalized-Alpha Scheme for Constrained Mechanical Systems
,”
Multibody Syst. Dyn.
,
18
, pp.
185
202
.10.1007/s11044-007-9084-0
15.
Khude
,
N.
,
Jay
,
L.
,
Schaffer
,
A.
, and
Negrut
,
D.
,
2007
, “
A Discussion of Low Order Numerical integration Formulas for Rigid and Flexible Multibody Dynamics
,”
Proceeding of the 6th ASME International Conference on Multibody Systems, Nonlinear Dynamics and Control
,
Las Vegas, NV
, Paper No. DETC2007-35666.
16.
Bottasso
,
C. L.
,
Bauchau
,
O.
, and
Cardona
,
A.
,
2007
, “
Time-Step-Size-Independent Conditioning and Sensitivity to Perturbations in the Numerical Solution of Index Three Differential Algebraic Equations
,”
Siam J. Sci. Comput.
,
29
, pp.
397
414
.10.1137/050638503
17.
van der
Vorst
,
H. A.
,
1992
, “
BI-CGSTAB: a Fast and Smoothly Converging Variant of BI-CG for the Solution of Nonsymmetric Linear Systems
,”
SIAM J. Sci. Stat. Comput.
,
13
, pp.
631
644
.10.1137/0913035
18.
Atkinson
,
K. E.
,
1989
,
An Introduction to Numerical Analysis
,
2nd ed.
,
Wiley
,
New York
.
19.
Saad
,
Y.
,
2003
,
Iterative Methods for Sparse Linear Systems
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.
20.
NVIDIA
,
2011
, “
Compute Unified Device Architecture Programming Guide 4.0
,” Available at http://developer.download.nvidia.com/compute/cuda/4_0/toolkit/docs/NVIDIA_CUDA_ProgrammingGuide_4.0.pdf
21.
Flynn
,
M. J.
,
1972
, “
Some Computer Organizations and Their Effectiveness
,”
IEEE Trans. Comput.
,
100
, pp.
948
960
.10.1109/TC.1972.5009071
22.
Tasora
,
A.
, and
Anitescu
,
M.
,
2011
, “
A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics
,”
Comput. Methods Appl. Mech. Eng.
,
200
, pp.
439
453
.10.1016/j.cma.2010.06.030
23.
ABAQUS
,
2012
, “
Simulia
,” Available at http://www.3ds.com/products/simulia/portfolio/abaqus/overview/
24.
Taylor
,
R. L.
,
1999
,
FEAP, a Finite Element Analysis Program: Version 7.1 j User Manual
,
Department of Civil and Environmental Engineering, University of California, Berkeley
.
25.
Hoberock
,
J.
, and
Bell
,
N.
,
2010
, “
Thrust: A Parallel Template Library
,” Available at http://thrust.googlecode.com
26.
Gerstmayr
,
J.
, and
Irschik
,
H.
,
2008
, “
On the Correct Representation of Bending and Axial Deformation in the Absolute Nodal Coordinate Formulation With an Elastic Line Approach
,”
J. Sound Vib.
,
318
, pp.
461
487
.10.1016/j.jsv.2008.04.019
You do not currently have access to this content.