Computational Mechanics

, Volume 64, Issue 3, pp 717–739 | Cite as

Performance of preconditioned iterative linear solvers for cardiovascular simulations in rigid and deformable vessels

  • Jongmin SeoEmail author
  • Daniele E. Schiavazzi
  • Alison L. Marsden
Original Paper


Computing the solution of linear systems of equations is invariably the most time consuming task in the numerical solutions of PDEs in many fields of computational science. In this study, we focus on the numerical simulation of cardiovascular hemodynamics with rigid and deformable walls, discretized in space and time through the variational multiscale finite element method. We focus on three approaches: the problem agnostic generalized minimum residual and stabilized bi-conjugate gradient (BICGS) methods, and a recently proposed, problem specific, bi-partitioned (BIPN) method. We also perform a comparative analysis of several preconditioners, including diagonal, block-diagonal, incomplete factorization, multigrid, and resistance based methods. Solver performance and matrix characteristics (diagonal dominance, symmetry, sparsity, bandwidth and spectral properties) are first examined for an idealized cylindrical geometry with physiologic boundary conditions and then successively tested on several patient-specific anatomies representative of realistic cardiovascular simulation problems. Incomplete factorization preconditioners provide the best performance and results in terms of both strong and weak scalability. The BIPN method was found to outperform other methods in patient-specific models with rigid walls. In models with deformable walls, BIPN was outperformed by BICG with diagonal and incomplete LU preconditioners.


Cardiovascular simulation Iterative linear solvers Preconditioning Fluid-structure interaction 



This work was supported by NIH grant (NIH R01-EB018302), NSF SSI grants 1663671 and 1339824, and NSF CDSE CBET 1508794. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) [35], which is supported by National Science Foundation grant number ACI-1548562. We thank Mahidhar Tatineni for assisting on building Trilinos on Comet cluster, which was made possible through the XSEDE Extended Collaborative Support Service (ECSS) program [1]. The authors also thank Michael Saunders, Michael Heroux, Mahdi Esmaily, Ju Liu, and Vijay Vedula, for fruitful discussions that helped in the preparation of this paper. The authors would like to thank the two anonymous reviewers whose comments greatly contributed to improve the completeness of the present study. We also acknowledge support from the open source SimVascular project at

Supplementary material


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Pediatrics and Institute for Computational and Mathematical Engineering(ICME)Stanford UniversityStanfordUSA
  2. 2.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameUSA
  3. 3.Department of Pediatrics, Bioengineering and ICMEStanford UniversityStanfordUSA

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