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.

The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications

“…Numerical results were obtained using LifeV (carotid artery) and the finite element library redbKIT v2.1 (github.com/redbKIT/redbKIT/releases) (AAA), P2 finite elements, a Newmark unconditionally stable scheme for the time discretization and an exponential vessel wall law. Holzapfel and Steinbach (2014). As in classical FETI methods, Lagrange multipliers are introduced to glue the solution together at the subdomain interfaces.…”

5. Recent development of robust monolithic fluid-structure interaction solvers

“…In [54], a similar technique has been employed in developing algebraic multigrid methods for solving the linearized FSI system using three hyperelastic structure models, namely Neo-Hookean, Mooney-Rivlin and anisotropic two-layer thick-walled artery (see [8,46,1]). In particular, the W-cycles of the special matrix-graph based algebraic multigrid methods for individual fields have been applied in each smoothing step [50,86,87,40,94].…”

Numerical simulation of fluid–structure interaction problems with hyperelastic models: A monolithic approach

“…As we observe, the simulation results obtained from the model of Neo-Hookean and Mooney-Rivlin material are quite similar to each other (the speed and magnitude of the pressure waves). This is due to the fact that these two models have only one term difference in the energy functional; see (2) and (4). The pressure waves obtained from the model of the anisotropic two-layer thick walled artery travels with slower speed and smaller magnitude than the other two models.…”

“…In the monolithic approach, the linearization of the nonlinear coupled system turns out to be a nontrivial task and requires tedious work on both the analytical derivation and computer implementation. One difficulty considered in this work results from the hyperelastic nonlinear material law as for the thick-walled artery with the media and adventitia layer (see [42,33]), for which the second and fourth order tensors of the energy functional with respect to the right Cauchy-Green tensor demand heavy amount of computational effort in each Newton iteration; see, e.g., [41,16] for an introduction on the basic tools used to derive these quantities under the Lagrangian framework and e.g., [4] for the simulation of such arterial tissues. Thanks to our previous work in [52], the linearization for the hyperelastic models tackled in a partitioned FSI solver is reused in this work.…”