Domain-Decomposition-Based FSI Algorithms

DFG D-A-CH Projekt Domain-Decomposition-Based FSI Algorithms

Fluid-Structure Interaction with a Polyconvex Hyperelastic Structure

Daniel Balzani (TUD), Simone Deparis (EPFL), Axel Klawonn (UzK), Oliver Rheinbach (TUBAF), Jörg Schröder (UDE), Alfio Quarteroni (EPFL)

Transmural stress distributions are a major driving factor for atherogenesis. We therefore consider Fluid-Structure Interaction (FSI) in arteries using sophisticated, highly nonlinear material models for the vessel wall. Our goal is to enable the accurate prediction of the distribution of stresses in arterial walls under physiological loading conditions. The use of anisotropic hyperelastic and almost incompressible material models, together with a suitable discretization from a three-field formulation, is a main contribution of this paper. Additionally, the influence of viscoelasticity is investigated.

A simple but nonsymmetric curved geometry is proposed which is suitable as a benchmark tested for fluid-structure interaction simulations in biomechanics with nonlinear structural models. In our computations, a ramp phase is used to bring the setting to a physiological blood pressure, followed by one or several heartbeats. The FSI problem is solved using a monolithic approach, and the linearized block systems are solved iteratively using parallel domain decomposition preconditioners. Based on the curved benchmark geometry, and corresponding physiologically-based boundary conditions, the influence of different material models, spatial discretizations, and meshes of varying refinement is investigated. Our results indicate that linear shape functions in the structure are not sufficient to provide good approximations of the wall stresses.

Recent Publications on Domain-Decomposition-Based FSI Algorithms

A. Heinlein, A. Klawonn, O. Rheinbach, and F. Röver: A Three-Level Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions, pp. 187-204. Springer International Publishing, Cham, 2019. https://doi.org/10.1007/978-3-030-14244-5_10

A. Heinlein, A. Klawonn, O. Rheinbach, and F. Röver: A three-level extension of the GDSW overlapping Schwarz preconditioner in three dimensions. Accepted 2019 for publication in Springer LNCSE. Preprint

A. Heinlein, A. Klawonn, S. Rajamanickam, and O. Rheinbach: FROSch: A fast and robust overlapping Schwarz domain decomposition preconditioner based on Xpetra in Trilinos. Accepted 2019 for publication in Springer LNCSE. Preprint

Daniel Balzani, Simone Deparis, Simon Fausten, Davide Forti, Alexander Heinlein, Axel Klawonn, Alfio Quarteroni, Oliver Rheinbach, and Jörg Schröder: "Numerical Modeling of Fluid-Structure Interaction in Arteries with Anisotropic Polyconvex Hyperelastic and Anisotropic Viscoelastic Material Models at Finite Strains". Int. J. Numer. Methods Biomed. Eng. (IJNMBE) http://dx.doi.org/10.1002/cnm.2756 TUBAF Preprint 2015-03

Daniel Balzani, Simone Deparis, Simon Fausten, Davide Forti, Alexander Heinlein, Axel Klawonn, Alfio Quarteroni, Oliver Rheinbach, and Jörg Schröder, "Aspects of Arterial Wall Simulations: Nonlinear Anisotropic Material Models and Fluid Structure Interaction", Proceedings of the WCCM XI, 2014. Download at CIMNE . Also Preprintreihe der Fakultät für Mathematik und Informatik der TU Bergakademie Freiberg, ISSN 1433-9407, 2014, Nr. 2014-03. TUBAF Preprint .

Simone Deparis, Davide Forti, Alexander Heinlein, Axel Klawonn, Alfio Quarteroni, and Oliver Rheinbach, "A Comparison of Preconditioners for the Steklov-Poincare Formulation of the Fluid-Structure Coupling in Hemodynamics". PAMM, Volume 15, Issue 1, pp. 93-94, October 2015. http://dx.doi.org/10.1002/pamm.201510037



FETI-DP Methods for Optimal Control Problems in Linear Elasticity with Control Constraints


Cores Problem Size #GMRES(30)Solution Time
4 321604 13 18.7s
16 1283205 30 25.9s
64 5126404 42 31.1s
256 20492804 52 34.9s
576 46099204 55 36.7s
1024 81945604 56 37.9s

Recycling of Lagrange Multipliers in Active Set iterations reduce computational cost:

Active Set/GMRES Active Set/GMRES+Recycling
31+31+31+31+31 31+28+24+17+5

Roland Herzog and Oliver Rheinbach, "FETI-DP methods for optimal control problems", Domain Decomposition Methods in Science and Engineering XXI, Jocelyne Erhel, Martin J. Gander, Laurence Halpern, Geraldine Pichot, Taoufik Sassi, and Olof B. Widlund (Eds.), Springer-Verlag, Lecture Notes in Computational Science and Engineering, Vol. 98, pp. 387-395, 2014. http://dx.doi.org/10.1007/978-3-319-05789-7_36 Preprint at INRIA

Roland Herzog and Oliver Rheinbach, "Scalability of a FETI-DP Method for Optimal Control Problems", Volume 14, Issue 1, pages 837–838, December 2014, http://dx.doi.org/10.1002/pamm.201410399