The 2nd Joint International Conference on Multibody System Dynamics
May 29-June 1, 2012, Stuttgart, Germany
Multibody domain decomposition for parallel processing: a wave-based
approach to handling interface dynamics
C. Smoothey, W. O'Connor
UCD School of Mechanical & Materials Engineering
University College Dublin, Belfield, Dublin 4, Ireland
craig@smoothey.org william.oconnor@ucd.ie
Abstract
For many good reasons there is growing interest in ways to allow parallel processing of multibody
dynamics problems. Some recent approaches include “Domain Decomposition” [1] and “Divide and
Conquer” [4]. This paper explores a new approach, reported as work in progress, with initial,
promising results. The strategy is an extension of work done on wave analysis of lumped systems in
another context [3]. In the approach, a larger system is subdivided into smaller subsystems, which are
solved in parallel. Interconnection points are boundaries for each. Dynamic coupling across boundaries
is handled in terms of transmitted and reflected motion components (or "waves"), in both directions,
across the boundaries.
The first test case was to model a 1-D system of an undamped, uniform mass-spring string as if
divided into two distinct strings (Fig.1). The uniform dynamics across the imagined boundary imply
minimal impedance discontinuity, maximizing the challenge of getting the transfer right. The central
mass of the original system was taken as the boundary element, common to the two subsystems. Each
subsystem saw this boundary as “active”, like an actuator (see Fig. 1), partly reflecting and partly
absorbing incident motion. Wave analyzers [4] were used to determine the right travelling wave in the
left system and the left travelling wave in the right system. The sum of these two waves was used to
drive the actuators that represent the central mass in the two subsystems. These actuators were then, in
effect, partly absorbing the wave in one subsystem and simultaneously launching it into the other
subsystem, and partly reflecting.
Figure 2: The left travelling wave
analyzer.
Figure 1: The original and split systems.
When implemented, the motion of the split uniform system was found to represent the motion of the
original uniform system as accurately as desired. The long-term accuracy was limited only by the
order of the wave transfer function, G, utilized in the wave analyzers. The structure of the left
travelling wave analyzer is shown in Fig.2. The only data required by the control system is the position
of the actuator and of its nearest mass. The wave transfer function, G, estimates the motion transfer
between adjacent masses in a system imagined to extend forward to infinity. It can be approximated
quite well by second order systems, where the effects of the extension to infinity are modeled simply
by a viscous damper. This gives sufficiently good results for many purposes. For perfect results, for
example to reproduce very accurately the response of the original system to an impulse continuing for
a long time, higher order models of G [3] were used.
Usually a system of interest will not be uniform. The division of the larger system into subsystems can
then be chosen so that boundaries occur where there is an impedance discontinuity. The uniform
system was modified to be “bi-uniform”, with different parameters on each side of the boundary. The
left half of the system was configured with k=1, m=1. The right half of the system was configured
with k=4, m=1. The wave transfer function GL2R moving across the discontinuity from left hand to
right was
L
X i ( s) 
GL2R 
2
2
X i 1 ( s) s   L  2R   2R G R ( s)
2
(1)
where GR was the wave transfer function of a hypothetical infinite uniform system configured as in the
right-half of the discontinuous system and ωL, ωR are the √(k/m) values in the left, right systems. A
symmetrical GR2L wave transfer function was also constructed for the opposite direction. The control
system was the same as in Fig.1 and Fig.2 except that two of the four transfer functions changed. A
simulation was conducted with a mass-spring string containing 41 masses. A unit impulse was
launched into the left side of the system. Figures 3, 4 show the mass displacements of the original and
split systems after 180 seconds. The centre black line is the interface point. The maximum absolute
error of the bi-uniform, split system was 351 nanometres where the maximum absolute displacement is
195 millimetres. The results were obtained using 11th order G estimates. In practice such high, longterm accuracy would not be required so lower order G estimates would more than suffice for most
engineering purposes, especially when damping is present.
Figure 3: Original bi-uniform system after 180
seconds simulation time.
Figure 4: Split bi-uniform system after 180 seconds
simulation time.
A similar approach was then tested for a completely non-uniform cascaded mass-spring system, again
split in the middle. The results were remarkably similar to the bi-uniform case, with arbitrarily high
accuracy achievable. Work is now in progress on a flexing system of n-links with torsional springs at
the joints. Detailed results will be available in time for the full paper. The background ideas for waveresolving in such flexural systems have already been successfully implemented in other applications.
The proposed method promises various benefits for parallel processing. Obviously each subsystem
would be solved simultaneously by different processor cores, utilizing wave-based coupling between
the subsystems. These subsystems would clearly be significantly smaller and more manageable than
the whole. The computational overhead for wave resolution is small. Furthermore, because the
interface “wave” variables are components of the motion of the local systems, it is not necessary to
solve explicitly for the action-reaction forces and torques at the subsystem interfaces. Thus, as in
analytical mechanics in general (vs. vector mechanics), the internal, constraint forces are handled
automatically and need not be determined explicitly in the solution scheme. This is a significant extra
benefit of the method.
References
[1] Imanishi, E.; Nanjo, T.: “Fast Simulation Of Flexible Multibody Dynamics Using Improved
Domain Decomposition Technique”, Proceedings of the ECCOMAS Thematic Conference on
Multibody Dynamics 2011, July 4-7, 2011, Brussels.
[2] McKeown, D.J.: “Wave Based Control of Flexible Mechanical Systems” (Doctoral Dissertation,
University College Dublin). Dublin, 2009.
[3] O'Connor, W.J.: “Wave-Echo Control of Lumped Flexible Systems”, Journal of Sound and
Vibration, Vol. 298, pp. 1001-1018, 2006.
[4] Poursina, M.; Anderson K.S.: “Constant Temperature Simulation Of Articulated Polymers Using
Divide-And-Conquer Algorithm”, Proceedings of the ECCOMAS Thematic Conference on
Multibody Dynamics, 2011, July 4-7, 2011, Brussels.