Monolithic fluid–structure interaction (FSI) of blood flow with arterial walls is considered, making use of sophisticated nonlinear wall models. These incorporate the effects of almost incompressibility as well as of the anisotropy caused by embedded collagen fibers. In the literature, relatively simple structural models such as Neo-Hooke are often considered for FSI with arterial walls. Such models lack, both, anisotropy and incompressibility. In this paper, numerical simulations of idealized heart beats in a curved benchmark geometry, using simple and sophisticated arterial wall models, are compared: we consider three different almost incompressible, anisotropic arterial wall models as a reference and, for comparison, a simple, isotropic Neo-Hooke model using four different parameter sets. The simulations show significant quantitative and qualitative differences in the stresses and displacements as well as the lumen cross sections. For the Neo-Hooke models, a significantly larger amplitude in the in- and outflow areas during the heart beat is observed, presumably due to the lack of fiber stiffening. For completeness, we also consider a linear elastic wall using 16 different parameter sets. However, using our benchmark setup, we were not successful in achieving good agreement with our nonlinear reference calculation.

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In this contribution, a small strain single crystal plasticity framework in the context of an infeasible primal–dual interior point method (IPDIPM) is discussed with a focus on the numerical treatment. Related to rate-independent algorithms in the field of single-crystal plasticity, the use of the IPDIPM to solve the constrained optimization problem offers the advantage that it handles the naturally arising redundancy in the slip system intrisically through a barrier term. This formulation penalizes the approach of the unfeasible domain, whereas the penalization term gradually approaches zero in the algorithm. This paper focusses on the numerical treatment and presents different tangent operator formulations and compares their convergency behavior in a numerical example.

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In this contribution, Fluid-Structure-Interaction (FSI) in blood vessels, in detail the simulation of realistic arterial geometries, where the interaction of the blood flow and the vessel wall is of special interest, is considered. Based on pervious research, cf. [1], our existing framework for FSI-simulations is extended towards realistic arterial geometries. The inflow and outflow boundary conditions for the fluid, as well as the boundary conditions for the structure are enhanced and adjusted to the chosen patient-specific geometry. In detail, an inflow profile for arbitrary shaped inflow cross-sections and a zero pressure boundary condition at the outflow are applied. Furthermore, the vessel wall is discretized using realistic material parameters of the media layer. The geometry and material parameters are adopted from [2]. In order to deal with the increasing complexity of the boundary value problem parallel computing and a two-level overlapping Schwarz method with energy-minimizing coarse space are applied; cf. [3]. The numerical simulations are performed using the Open-Source software LifeV, in particular a code which has been developed in cooperation with the group of Prof. Quarteroni from the EPF Lausanne.@en

In this contribution, results regarding fluid-structure interaction (FSI) simulations for three-dimensional arterial walls are presented. In detail, a benchmark problem for FSI simulations in arteries of sufficient complexity, which combines sophisticated nonlinear models for the fluid and the structure, cf. [1], as well as a short segment from a patient-specific arterial geometry are considered. For the patient-specific arterial geometry a specific inflow profile suited for realistic geometries and simplified boundary conditions for the outflow are taken into account. (© 2017 Wiley-VCH Verlag GmbH Co. KGaA, Weinheim)@en

The accurate prediction of transmural stresses in arterial walls requires on the one hand robust and efficient numerical schemes for the solution of boundary value problems including fluid–structure interactions and on the other hand the use of a material model for the vessel wall that is able to capture the relevant features of the material behavior. One of the main contributions of this paper is the application of a highly nonlinear, polyconvex anisotropic structural model for the solid in the context of fluid–structure interaction, together with a suitable discretization. Additionally, the influence of viscoelasticity is investigated. The fluid–structure interaction problem is solved using a monolithic approach; that is, the nonlinear system is solved (after time and space discretizations) as a whole without splitting among its components. The linearized block systems are solved iteratively using parallel domain decomposition preconditioners. A simple – but nonsymmetric – curved geometry is proposed that is demonstrated to be suitable as a benchmark testbed for fluid–structure interaction simulations in biomechanics where nonlinear structural models are used. Based on the curved benchmark geometry, the influence of different material models, spatial discretizations, and meshes of varying refinement is investigated. It turns out that often-used standard displacement elements with linear shape functions are not sufficient to provide good approximations of the arterial wall stresses, whereas for standard displacement elements or F-bar formulations with quadratic shape functions, suitable results are obtained. For the time discretization, a second-order backward differentiation formula scheme is used. It is shown that the curved geometry enables the analysis of non-rotationally symmetric distributions of the mechanical fields. For instance, the maximal shear stresses in the fluid–structure interface are found to be higher in the inner curve that corresponds to clinical observations indicating a high plaque nucleation probability at such locations.@en

The simulation of the physiological loading situation of arteries with moderate atherosclerotic plaque may provide additional indicators for medical doctors to estimate if the plaque is likely to rupture and if surgical intervention is required. In particular the transmural stresses are important in this context. They depend strongly on the mechanical response and thus, a predictive material model capturing all characteristics of the material behavior is required. Here, polyconvex strain energy functions are considered for the hyperelastic behavior and a simplified viscoelastic model is proposed which does not take into account an isochoric strain energy for the fiber response. Based thereon, a comparative study is presented, investigating the influence of viscoelasticity on the mechanical behavior of a simplified arterial wall and a rather small impact is found. Realistic predictions of transmural stress distributions require a simulation of the interaction between the blood flow and the vessel wall. We recall the equations that model fluid-structure interaction and the monolithic Convective Explicit algorithm for their numerical approximation, addressing both the cases when the fluid-structure meshes are conforming and noncon-forming at the interface. We also present numerical experiments, using the monolithic approach, for the fluid structure interaction problem in a curved tube using a hyperelastic material model for the structure and an absorbing boundary condition. The fluid structure interaction using a highly nonlinear anisotropic structural model for the solid in this context is one of the main contributions of this paper.@en