The QuasiSimultaneous Approach for Partitioned Systems in Hemodynamics

The Quasi-Simultaneous Approach for Partitioned Systems in Hemodynamics Gerk Rozema, Natasha Maurits, Arthur Veldman Introduction Simple example 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 m 1 and m 2 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 m 1 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 0 D-3 D flow coupling. Recent applications include a 3 D compliant carotid artery bifurcation (fluid-structure interaction) and its coupling to a 0 D circulation model. A 0 D approximation of the 3 D 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 I 1 of M 1 is utilized to obtain a better approximation of M 1 (interaction law) Carotid bifurcation The approximation of the outer equation is solved simultaneously with the inner equation 3 D 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 0 D circulation model R u. G