Transcript Document

The Quasi-Simultaneous Approach for
Partitioned Systems in Hemodynamics
Gerk Rozema, Natasha Maurits, Arthur Veldman
Simple example
Introduction
When modeling complex systems often a modular
approach is chosen. A fully simultaneous treatment of
subsystems (strong coupling) requires an intensive
merging of the submodels at algorithmic and software
levels or the introduction of subiterations. Weak coupling
methods on the other hand are cheap but prone to
numerical stability problems. The quasi-simultaneous
method combines the best of both worlds. Here it is
presented in an unsteady setting.
Consider two masses m1 and m2 connected by a solid rod.
The weak approach
converges whenever
approach
with
Weak coupling method
Consider a partitioned system
where f typically denotes a force and  represents a
displacement (or its time derivative). In a strong coupling
method  and f are solved simultaneously. A weak coupling
method uses an f or  at the old time level and the system
is solved by substitution (direct method), e.g.
yields a hierarchical solution method
which is an iteration process with iteration matrix
The quasi-simultaneous
an ‘estimate’ of m1 converges when
Hence, in case
a certain amount of interaction will
make the process stable (i.e. convergent). Therefore it is
concluded that by using an interaction law stability can be
achieved for arbitrary mass ratios.
Applications
The quasi-simultaneous approach can be applied to any
partitioned system as long as suitable approximations are
available. Examples from fluid dynamics are boundary layer
interaction (steady), fluid-structure interaction and 0D-3D
flow coupling. Recent applications include a 3D compliant
carotid artery bifurcation (fluid-structure interaction) and its
coupling to a 0D circulation model. A 0D approximation of
the 3D flow model is used as interaction law.
It converges if and only if its spectral radius is smaller than
unity.
Quasi-simultaneous method
In a quasi-simultaneous approach, a simple approximation
I1 of M1 is utilized to obtain a better approximation of M1
(interaction law)
Carotid
bifurcation
The approximation of the outer equation is solved
simultaneously with the inner equation
3D compliant carotid
artery bifurcation
The iteration matrix reads
It converges whenever the spectral radius is smaller than
unity. Because of the simplicity of the interaction law, the
quasi-simultaneous approach adds only little complexity
and the computational effort per time step is hardly
effected. Moreover the stability problems are solved as is
demonstrated below with a simple unsteady mechanical
system.
Computational Mechanics & Numerical Mathematics
University of Groningen
P.O. Box 800, 9700 AV Groningen
University Medical Center Groningen
Department of Neurology
P.O. Box 30.001, 9700 RB Groningen
0D circulation
model
R uG