Domain-Decomposition-Based Fluid Structure Interaction Algorithms for Highly Nonlinear and Anisotropic Elastic Arterial Wall Models in 3 D
Applied Mechanics, Statics and Dynamics
Final Report Abstract
This project considered the simulation of fluid-structure interaction (FSI) of large human blood vessels such as coronary arteries. Our focus has been, from the beginning of the project, on the application of sophisticated nonlinear structural models for the arterial wall since we are interested in quantities inside the arterial wall, such as the transmural stresses, which are associated with atherogenesis, i.e., the narrowing of arteries due to plaque formation. In this context, monolithic FSI coupling algorithms are most suitable. To the best of our knowledge, this project was the first to consider fluid-structure interaction using nonlinear hyperelastic, anisotropic, and incompressible wall models. Our results have shown that a suitable discretization of the structure is very important in our FSI context: We have found that at least P2 or F¯ finite elements are necessary for reasonable stress approximations in the vessel wall, since the accuracy of P1 discretization is poor, in comparison. Mesh convergence was not achieved for P1 elements even for relatively large structural meshes. This is of practical relevance since FSI simulations in biomechanics using linear finite elements for the structure are not uncommon. To discretize the fluid problem, we use inf-sup stable Taylor-Hood (P2 − P1 ) elements and piecewise quadratic (P2 ) elements to discretize the geometry problems. Note that our most recent simulations using different hyperelastic, anisotropic material models indicate that using F¯ elements for the structure is numerically more stable than using P2 elements, i.e., in certain cases Newton convergence is achieved using F¯ elements but not using P2 finite elements. By comparing the response of commonly used, simpler structural models, such as Neo–Hooke, with the results obtained with more sophisticated, anisotropic models, we found significant differences; this indicates that simple wall models should not be used in our context. For the realistic simulation of arteries, various approaches have been developed to capture all relevant aspects influencing the transmural stress distribution in the wall. Aside from anisotropic material models, these include a new, anisotropic growth-induced method for the calculation of residual stresses combined with an extended approach for realistic fiber orientations based on the adaptation to the principle stress state. Furthermore, a new and stretch-dependent chemo-mechanical model for the incorporation of the smooth muscle activation was developed. All these aspects have been analyzed in combined simulations and the results show that indeed all aspects need to be considered to allow for realistic stress distributions. New solver algorithms have been developed in this project. We solve the Newton system using the GMRES method, preconditioned by a monolithic Dirichlet-Neumann preconditioner where, for the blocks, efficient preconditioners for the fluid, for the structure, and for the geometry problem are needed. We have found that adding a coarse space helps to reduce the time to solution for our FSI problems, except for very small time steps. Therefore, for the blocks, new parallel preconditioners have been developed, i.e., monolithic two-level overlapping Schwarz methods for fluid problems and three-level overlapping Schwarz methods for the structure and geometry problems. These methods and their implementations are new and use exotic coarse spaces for scalability, which can be constructed without an explicit mesh hierarchy. This project has initially been built on the LifeV software library, which makes heavy use of the Trilinos software infrastructure. Additional (stand-alone) software has been developed during the project, most notably, FROSch (see https://shylu-frosch.github.io) and FEDDLib. The FROSch software collects the new parallel implementations of our two- and three-level domain decomposition solver framework and has been extended significantly over time. The FROSch framework is now part of the Trilinos ShyLU package. The FEDDLib collects a new simulation framework for fluids, solids, and fluid-structure interaction. An open source release of the FEDDLib is planned.
Publications
- An algorithmic scheme for the automated calculation of fiber orientations in arterial walls. Comput. Mech., 58(5):861–878, 2016
S. Fausten, D. Balzani, and J. Schröder
(See online at https://doi.org/10.1007/s00466-016-1321-z) - A parallel implementation of a two-level overlapping Schwarz method with energy-minimizing coarse space based on Trilinos. SIAM J. Sci. Comput., 38(6):C713–C747, 2016
A. Heinlein, A. Klawonn, and O. Rheinbach
(See online at https://doi.org/10.1137/16M1062843) - Numerical modeling of fluid-structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains. Int. J. Numer. Meth. Bio., 32(10):e02756, 2016
D. Balzani, S. Deparis, S. Fausten, D. Forti, A. Heinlein, A. Klawonn, A. Quarteroni, O. Rheinbach, and J. Schröder
(See online at https://doi.org/10.1002/cnm.2756) - A combined growth and remodeling framework for the approximation of residual stresses in arterial walls. Zeitschrift für Angewandte Mathematik und Mechanik, 98:2072– 2100, 2018
A. Zahn and D. Balzani
(See online at https://doi.org/10.1002/zamm.201700273) - Monolithic overlapping Schwarz domain decomposition methods with GDSW coarse spaces for incompressible fluid flow problems. SIAM J Sci. Comput., 41(4):C291–C316, 2019
A. Heinlein, C. Hochmuth, and A. Klawonn
(See online at https://doi.org/10.1137/18M1184047) - A three-level extension of the GDSW overlapping Schwarz preconditioner in three dimensions. Lecture Notes in Computational Science and Engineering, 138:185–192, 2020
Alexander Heinlein, Axel Klawonn, Oliver Rheinbach, and Friederike Röver
(See online at https://doi.org/10.1007/978-3-030-56750-7_20) - FROSch: A fast and robust overlapping Schwarz domain decomposition preconditioner based on Xpetra in Trilinos. Lecture Notes in Computational Science and Engineering, 138:176–184, 2020
A. Heinlein, A. Klawonn, S. Rajamanickam, and O. Rheinbach
(See online at https://doi.org/10.1007/978-3-030-56750-7_19) - Reduced dimension GDSW coarse spaces for monolithic Schwarz domain decomposition methods for incompressible fluid flow problems. Int. J. Numer. Meth. Eng., 121(6):1101–1119, 2020
A. Heinlein, C. Hochmuth, and A. Klawonn
(See online at https://doi.org/10.1002/nme.6258)
