Computational Mechanics & Numerical Mathematics University of Groningen

1. R u G The Quasi-Simultaneous Approach for Partitioned Systems in Hemodynamics Gerk Rozema, Natasha Maurits, Arthur Veldman Simple example Introduction Consider two masses m1 and m2 connected by a solid rod.The weak approach converges whenever The quasi-simultaneous approach with 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. 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. 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 It converges if and only if its spectral radius is smaller than unity. In a quasi-simultaneous approach, a simple approximation I1 of M1 is utilized to obtain a better approximation of M1 (interaction law) The approximation of the outer equation is solved simultaneously with the inner equation 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. 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. Quasi-simultaneous method Carotid bifurcation 3D compliant carotid artery bifurcation 0D circulation model Computational Mechanics & Numerical Mathematics University of Groningen P.O. Box 800, 9700 AV Groningen University Medical Center GroningenDepartment of NeurologyP.O. Box 30.001, 9700 RB Groningen