A two-level time step technique for the partitioned solution of one-dimensional arterial networks

Abstract

a b s t r a c tIn this work a numerical strategy to address the solution of the blood flow in one-dimensional arterial networks through a topology-based decomposition is presented. Such decomposition results in the local analysis of the blood flow in simple arterial segments. Hence, iterative methods are used to perform the strong coupling among the segments, which communicate through non-overlapping interfaces. Specifically, two approaches are considered to solve the associated nonlinear interface problem: (i) the Newton method and (ii) the Broyden method. Moreover, since the modeling of blood flow in compliant vessels is tackled using explicit finite element methods, we formulate the coupling problem using a two-level time stepping technique. A local (inner) time step is used to solve the local problems in single arteries, meeting thus local stability conditions, while a global (outer) time step is employed to enforce the continuity of physical quantities of interest among the one-dimensional segments. Several examples of application are presented. Firstly a study about spurious reflections produced at interfaces as a consequence of the twolevel time stepping technique is carried out. Secondly, the application of the methodologies to physiological scenarios is presented, specifically addressing the solution of the blood flow in a model of the entire arterial network. The effects of non-uniformities of the material properties, of the variation of the radius, and of viscoelasticity are taken into account in the model and in the (local) numerical scheme; they are quantified and commented in the arterial network simulation.

Geometric multiscale modeling of the cardiovascular system, between theory and practice

“…This method may be used in association with an operator splitting technique [60,116], where the flow rate is split into two components, one satisfying the pure elastic problem and the second one the visco-elastic correction.…”

“…General statements on the most restrictive conditions are difficult to draw, as they depend on the specific problem (the vascular district for the 3D model as well as the portion of the network for the 1D one). For this reason, in partitioned schemes where the different solvers are called in an either sequential or parallel fashion, it is worth resorting to multi-level time stepping techniques, where the different geometric components are solved with individual convenient time steps and matching interface conditions are fulfilled thanks to suitable synchronization procedures [28,116] -see also Sect. 6.4.2 In the remainder of this Section we keep considering the time discretization of the coupled problems already introduced in Sect.…”

Numerical simulation of left ventricular assist device implantations: Comparing the ascending and the descending aorta cannulations

“…The spatial discretization is accomplished using P1 finite elements. For more details see Malossi et al 29 and references therein.…”

On the continuity of mean total normal stress in geometrical multiscale cardiovascular problems

“…The fluid problem (3) is coupled with the 1-D structural model through the pressure-area constitutive relation. As in [17,18] we take into account the elastic and viscoelastic responses of the vessel wall, such that…”

“…In addition, a F and j F are the Coriolis and friction coefficients, respectively, whose definitions are given, e.g., in [17]. The fluid problem (3) is coupled with the 1-D structural model through the pressure-area constitutive relation.…”