1. Computational Mechanics & Numerical Mathematics University of Groningen Linking an artery to the circulation: Introducing a Quasi-Simultaneous Coupling Approach for Partitioned systems in Hemodynamics G. Rozema, A.E.P. Veldman, N.M. Maurits University of Groningen, University Medical Center Groningen The Netherlands

2. Computational Mechanics & Numerical Mathematics University of Groningen Consider fluid and structure subsystems F and S, relating the interface variables x and f: In weak coupling methods the subsystems are solved successively: Drawback: stability only if  (S -1 F) is smaller than unity. In strong coupling methods subiterations are performed between subsystems. Drawback: expensive The weak coupling method, FSI x: motion f: force S: structure model F: fluid model

3. Computational Mechanics & Numerical Mathematics University of Groningen The stability condition for fluid-structure interaction in an elastic artery Stability depends on the system parameters, in particular the mass ratio. For example the stability condition for FSI in an elastic artery is: For  t  0 the limit on the mass ratio remains! Structure mass fluid mass wall stiffness r 0 : artery radius  f : fluid density  s : wall density h: wall thickness E: Young’s modulus l: artery length

4. Computational Mechanics & Numerical Mathematics University of Groningen Theoretical and experimental stability bound for FSI in an elastic artery Theoretical stability bound is in good agreement with experimental results. Physiologically realistic simulations are in the unstable region, making the weak coupling method unsuitable for this application. Solid line: theoretical stability bound Markers: experimental results

5. Computational Mechanics & Numerical Mathematics University of Groningen Consider subsystems M 1 and M 2 : In the quasi-simultaneous method a simple approximation I 1 of M 1 is utilized as follows: The approximate model I 1 is solved simultaneously with M 2, without subiterations: stability is achieved at low computational cost. The quasi-simultaneous method

6. Computational Mechanics & Numerical Mathematics University of Groningen The iteration operator of the quasi-simultaneous method: The method is stable if the spectral radius of the iteration operator is smaller than unity. For example assuming scalar subsystems yields:  the approximation I 1 of M 1 needs to be ‘half as good’ as M 1. In practice, simple models often satisfy this criterion! Summary: Stability of the strong method for the cost of the weak method. The stability of the quasi-simultaneous method

7. Computational Mechanics & Numerical Mathematics University of Groningen Example: 3D elastic artery coupled to electric network circulation model Weak coupling method is unstable. Solution: the quasi-simultaneous method, using an RLC approximation of the elastic artery to stabilize the system.