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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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, 2016
J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1976)
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)
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)
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)
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)
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)
P.G. Ciarlet, Mathematical Elasticity, Vol. I : Three-Dimensional Elasticity. Studies in Mathematics and Its Applications (North-Holland, Amsterdam, 1988)
R.S. Dembo, S.C. Eisenstat, T. Steihaug, Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)
S.C. Eisenstat, H.F. Walker, Globally convergent inexact Newton methods. SIAM J. Optim. 4, 393–422 (1994)
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)
P.J. Lanzkron, D.J. Rose, J.T. Wilkes, An analysis of approximate nonlinear elimination. SIAM J. Sci. Comput. 17, 538–559 (1996)
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)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Gong, S., Cai, XC. (2018). A Nonlinear Elimination Preconditioned Newton Method with Applications in Arterial Wall Simulation. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_33
Download citation
DOI: https://doi.org/10.1007/978-3-319-93873-8_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93872-1
Online ISBN: 978-3-319-93873-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)