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2018 | OriginalPaper | Chapter

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

Authors : Shihua Gong, Xiao-Chuan Cai

Published in: Domain Decomposition Methods in Science and Engineering XXIV

Publisher: Springer International Publishing

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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.

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Literature
1.
go back to reference 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 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
2.
go back to reference J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1976)MathSciNetCrossRef J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1976)MathSciNetCrossRef
3.
go back to reference 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)CrossRef 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)CrossRef
4.
go back to reference 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. 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)
5.
go back to reference 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)CrossRef 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)CrossRef
6.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
7.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
8.
go back to reference P.G. Ciarlet, Mathematical Elasticity, Vol. I : Three-Dimensional Elasticity. Studies in Mathematics and Its Applications (North-Holland, Amsterdam, 1988) P.G. Ciarlet, Mathematical Elasticity, Vol. I : Three-Dimensional Elasticity. Studies in Mathematics and Its Applications (North-Holland, Amsterdam, 1988)
9.
10.
11.
go back to reference 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)MathSciNetCrossRef 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)MathSciNetCrossRef
12.
go back to reference P.J. Lanzkron, D.J. Rose, J.T. Wilkes, An analysis of approximate nonlinear elimination. SIAM J. Sci. Comput. 17, 538–559 (1996)MathSciNetCrossRef P.J. Lanzkron, D.J. Rose, J.T. Wilkes, An analysis of approximate nonlinear elimination. SIAM J. Sci. Comput. 17, 538–559 (1996)MathSciNetCrossRef
13.
go back to reference 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)MATH 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)MATH
Metadata
Title
A Nonlinear Elimination Preconditioned Newton Method with Applications in Arterial Wall Simulation
Authors
Shihua Gong
Xiao-Chuan Cai
Copyright Year
2018
DOI
https://doi.org/10.1007/978-3-319-93873-8_33

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