Fluid–structure interaction in blood flows on geometries based on medical imaging
Introduction
The main objective of this article is to present a feasible strategy to simulate blood flows in compliant large vessels. Such simulations raise many difficulties. We specifically address two of them: first, the mesh generation for the fluid and the wall and second, the efficient and robust resolution of the nonlinear fluid–structure problem.
A structural model for large arteries has to be three-dimensional and has to handle large displacements. Given that the wall of the blood vessels is thin, it is convenient to use shell elements. For “realistic” simulations, a procedure that starts with medical images and automatically generates a finite element mesh has to be designed. The most efficient and reliable procedures able to achieve this goal produce tetrahedral meshes and thus triangular meshes of the arteries surfaces. In this context, it could be appropriate to use triangular shell elements. Nevertheless, it is usually admitted that quadrilateral shell elements are more reliable in general situations. We therefore have to couple quadrilateral meshes for the structure and tetrahedral meshes for the fluid. A possible strategy could be to use a method—for example interpolation or mortar element [5]—to manage non-matching grids. But the practical implementation of such techniques on general 3D geometries is rather involved. In the present work, we propose an alternative which consists in converting pairs of triangles of the tetrahedral mesh skin in order to produce a quadrilateral mesh. Of course, this procedure has to be carried out very carefully to ensure the quality of the resulting mesh. By the way, the method proposed in this paper is a general way to generate a non-structured quadrilateral mesh of a surface. As a result of this approach, the vertices of the quadrilateral and the tetrahedral meshes can be matched, which makes the message passing between the fluid and structure solvers very simple.
Once suitable meshes have been generated, the fluid–structure problem has to be tackled. To this purpose, partitioned schemes are very convenient: they allow to use the available software with minor changes, the most adapted schemes for both fluid and structure solvers, and they ensure that the fluid–structure solver will automatically inherit future improvements in fluid or structure algorithms. For a presentation of partionned and simultaneous procedures for fluid–structure problems we refer to [36], [40], [2], [3]. In the family of partitioned schemes, loosely-coupled algorithms (see [15] and the references therein) are very attractive since they typically require one (or a few) resolution(s) of fluid and structure at each time step. Such algorithms are very efficient in aeroelasticity [15], [26], [34], but it has been observed in many works [12], [23], [33] that strongly-coupled algorithms seem to be mandatory in blood flows. Partitioned strongly-coupled methods yield the resolution of a nonlinear problem on the fluid–structure interface [25], which may be very time-consuming. In particular, classical fixed-point methods are too expensive, and sometimes not robust enough (even if acceleration techniques improve their efficiency [29], [30], [31]). This fact, which is now well-understood [10], [25], is mainly due to the added-mass effect (see for example [32] for a presentation of this effect in various applications). We present in this article a Jacobian-free Newton–Krylov method based on the following basic idea: a simplified problem which takes into account the added-mass effect can be used to efficiently evaluate an approximate product “Jacobian times vector” in a Krylov method. The resulting algorithm appears to be efficient and robust in the context of blood flows.
The article is organized as follows. We present in Section 2 the basic equations of the problem. Section 3 is devoted to the description of the mesh generation algorithms. In Section 4, the shell model, the load computation and the time marching algorithms are detailed. The inexact Newton method is described in Section 5 and some numerical results are presented in Section 6.
Section snippets
General formulation of the fluid–structure problem
Let Ω(t) be a time-dependent domain of occupied by a continuum medium. It is assumed that, for all time t, and ΩF(t) ∩ ΩS(t) = ∅, where ΩF(t) is occupied by a fluid and ΩS(t) is occupied by an elastic solid. We denote by the fluid–structure interface. The domain Ω(t) is the current configuration of the system. Let be a reference configuration. We define the deformation of the continuum medium:the deformation gradient:
Unstructured mesh generation
It will be explained in the next section that it is convenient to use quadrilateral elements for the structure, whereas the fluid domain is generally made of tetrahedra. We address in this section the construction of both meshes.
In the context of numerical simulations, mesh adaptation is often used in order to improve the accuracy of the numerical solution as well as to reduce the number of degrees of freedom (i.e., the mesh size) required to capture the behavior of the solution. Schematically,
Discretization
In our computations, the fluid equations are discretized in space with P1/P1 stabilized finite elements on tetrahedra (or Q1/Q1 on hexahedra). Below, we give some details regarding the structure discretization. We next present how the load induced by the fluid on the structure is computed and we close this section with the time-advancing schemes.
A Jacobian-free Newton–Krylov algorithm
Classical fixed-point method are very expensive to solve the above problem. A theoretical explanation of this fact is given in [10] in a simplified framework. Thus, we advocate to look for a zero of using an inexact Newton algorithm [22], [23]:
- (1)
Initialization: .
- (ii)
Approximate Jacobian resolution: .
- (iii)
Update the solution: .
In step (ii), denotes a suitable approximation of and in step (iii) λk is determined with a linesearch
Numerical results
The purpose of this section is to illustrate the capabilities of the methods introduced in this article. We basically solve the same physical problem on three different geometries: the fluid is initially at rest and a pressure of 103 N/m2 has been imposed at the inlet for 0.005 seconds. The physical parameters are: μ = 0.006 Ns/m2, ρF = 103 kg/m3, E = 3 × 105 N/m2, ν = 0.3, ρS = 1.2103 kg/m3 and the thickness of the the shell is 10−4 m. The solution is a pressure wave that propagates along the vessel.
The first
Acknowledgments
The authors wish to thank M. Thiriet and S. Salmon for the mesh of the aneurism, K. Perkold, D. Liepsch and A. Leuprecht for the hexahedral mesh of the carotid, L. Formaggia, A. Veneziani and the biomedical group of the CRS4 for the geometry of the second carotid.This work has been partially supported by the Research Training Network “Mathematical Modelling of the Cardiovascular System” (HaeMOdel), contract HPRN-CT-2002-00270 of the European Community.
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