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A Nonlinear Elimination Preconditioned Newton Method with Applications in Arterial Wall Simulation

  • Shihua Gong
  • Xiao-Chuan Cai
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)

Abstract

Arterial wall can be modeled by a quasi-incompressible, anisotropic and hyperelastic equation that allows large deformation. Most existing nonlinear solvers for the steady hyperelastic problem are based on pseudo time stepping, which often requires a large number of time steps especially for the case of large deformation. It is also reported that the quasi-incompressibility and high anisotropy have negative effects on the convergence of both Newton’s iteration and the linear Jacobian solver. In this paper, we propose and study a nonlinearly preconditioned Newton method based on nonlinear elimination to calculate the steady solution directly without pseudo time integration. We show numerically that the nonlinear elimination preconditioner accelerates Newton’s convergence in cases with large deformation, quasi-incompressibility and high anisotropy.

References

  1. 1.
    S. Balay, S. Abhyankar, M.F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, K. Rupp, B.F. Smith, S. Zampini, H. Zhang, H. Zhang, PETSc users manual. Technical report ANL-95/11 - Revision 3.7, Argonne National Laboratory, 2016Google Scholar
  2. 2.
    J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Balzani, P. Neff, J. Schröder, G.A. Holzapfel, A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43, 6052–6070 (2006)CrossRefGoogle Scholar
  4. 4.
    D. Balzani, S. Deparis, S. Fausten, D. Forti, A. Heinlein, A. Klawonn, A. Quarteroni, O. Rheinbach, J. 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. 32(10), e02756 (2016)Google Scholar
  5. 5.
    D. Brands, A. Klawonn, O. Rheinbach, J. Schröder, Modelling and convergence in arterial wall simulations using a parallel FETI solution strategy. Comput. Methods Biomech. Biomed. Eng. 11, 569–583 (2008)CrossRefGoogle Scholar
  6. 6.
    S. Brinkhues, A. Klawonn, O. Rheinbach, J. Schröder, Augmented Lagrange methods for quasi-incompressible materials–applications to soft biological tissue. Int. J. Numer. Methods Biomed. Eng. 29, 332–350 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    X.-C. Cai, X. Li, Inexact Newton methods with restricted additive Schwarz based nonlinear elimination for problems with high local nonlinearity. SIAM J. Sci. Comput. 33, 746–762 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    P.G. Ciarlet, Mathematical Elasticity, Vol. I : Three-Dimensional Elasticity. Studies in Mathematics and Its Applications (North-Holland, Amsterdam, 1988)Google Scholar
  9. 9.
    R.S. Dembo, S.C. Eisenstat, T. Steihaug, Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    S.C. Eisenstat, H.F. Walker, Globally convergent inexact Newton methods. SIAM J. Optim. 4, 393–422 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Huang, C. Yang, X.-C. Cai, A nonlinearly preconditioned inexact Newton algorithm for steady state lattice Boltzmann equations. SIAM J. Sci. Comput. 38, A1701–A1724 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    P.J. Lanzkron, D.J. Rose, J.T. Wilkes, An analysis of approximate nonlinear elimination. SIAM J. Sci. Comput. 17, 538–559 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. Logg, K.-A. Mardal, G. Wells, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, vol. 84 (Springer, Berlin, 2012)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchPeking UniversityBeijingPeople’s Republic of China
  2. 2.Department of Computer ScienceUniversity of Colorado BoulderBoulderUSA

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