Classical and all-floating FETI methods for the simulation of arterial tissues

Abstract

High-resolution and anatomically realistic computer models of biological soft tissues play a significant role in the understanding of the function of cardiovascular components in health and disease. However, the computational effort to handle fine grids to resolve the geometries as well as sophisticated tissue models is very challenging. One possibility to derive a strongly scalable parallel solution algorithm is to consider finite element tearing and interconnecting (FETI) methods. In this study we propose and investigate the application of FETI methods to simulate the elastic behavior of biological soft tissues. As one particular example we choose the artery which is - as most other biological tissues - characterized by anisotropic and nonlinear material properties. We compare two specific approaches of FETI methods, classical and all-floating, and investigate the numerical behavior of different preconditioning techniques. In comparison to classical FETI, the all-floating approach has not only advantages concerning the implementation but in many cases also concerning the convergence of the global iterative solution method. This behavior is illustrated with numerical examples. We present results of linear elastic simulations to show convergence rates, as expected from the theory, and results from the more sophisticated nonlinear case where we apply a well-known anisotropic model to the realistic geometry of an artery. Although the FETI methods have a great applicability on artery simulations we will also discuss some limitations concerning the dependence on material parameters.

Keywords: all-floating FETI; artery; biological soft tissues; parallel computing.

Figure 1

Diagrammatic model of the major components of a healthy elastic artery, from [5]. The intima, the innermost layer is negligible for the modeling of healthy arteries, it plays a very important role in the modeling of diseased arteries, though. The two predominant directions of the collagen fibers in the media and the adventitia are indicated with black curves.

Figure 2

Decomposition of a domain Ω₀ into four subdomains Ω_0,i, i = 1, … , 4.

Figure 3

Fully redundant classical FETI (a) and all-floating FETI (b) formulation: Ω_0,i, i = 1, … , 5, denote the local subdomains, the black dots correspond to the subdomain vertices and the dashed lines correspond to the constraints (34). The gray strip indicates Dirichlet boundary conditions. Note that the number of constraints for the all-floating approach rises with the number of vertices on the Dirichlet boundary.

Figure 4

Mesh of an aorta seen from above showing the brachiocephalic artery, and the left common carotid and subclavian arteries. The fine mesh consists of 5 418 594 tetrahedrons and 1 055 901 vertices, while colors indicate the displacement field with an internal pressure of 1 mmHg. Additionally, the splits show the decomposition of the mesh into 480 subdomains (left). Coarser mesh consisting of 720 060 tetrahedrons and 150 725 vertices used in Section 5.3 with 5 selected vertices A–E (right); colors show the distribution of the stress magnitude σ_mag according to (56) with an internal pressure of 300 mmHg. For both images red indicates high and blue low values.

Figure 5

Mesh of a segment of a common carotid artery from two different points of view. The mesh consists of 9 195 336 tetrahedrons and 1 621 365 vertices. Color indicates the distribution of the stress magnitude σ_mag according to (56) due to an internal pressure of 1 mmHg, red indicates high and blue low values. Additionally, the splits show the decomposition of the mesh into 512 subdomains.

Figure 6

Distribution of the stress magnitude σ_mag inside the aorta (left); values of high stress in red and of low stress in blue. To the right the fiber directions (black curves) and the two layers (adventitia in red and media in orange) of the carotid artery are shown.

Figure 7

Stress magnitude σ_mag versus relative displacement u_rel (left) and evolution of the displacement norm u_norm over the load steps up to an internal pressure p of 300 mmHg (right). The plots were generated using data at the specific points A–E, as shown in Fig. 4 (right).

Figure 8

Comparison of all-floating FETI (gray) and classical FETI (black) for a time stepping scheme. Average iteration numbers of one time step (left) and solving times in seconds for one time step (right) over 572 load steps.

Figure 9

Computation times (in s) for a simulation of the anisotropic arterial model with the aorta mesh (left) and the carotid artery mesh (right) using a varying number of cores.