Advertisement

An Accelerated Newton Method for Nonlinear Materials in Structure Mechanics and Fluid Mechanics

  • Thomas RichterEmail author
  • Carolin Mehlmann
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

We analyze a modified Newton method that was first introduced by Turek and coworkers. The basic idea of the acceleration technique is to split the Jacobian A(x) into a “good part” \(A^{\prime }_1(x)\) and into a troublesome part \(A^{\prime }_2(x)\). This second part is adaptively damped if the convergence rate is bad and fully taken into account close to the solution, such that the solver is a blend between a Picard iteration and the full Newton scheme. We will provide first steps in the analysis of this technique and discuss the effects that accelerate the convergence.

Notes

Acknowledgements

The authors acknowledge the financial support by the Deutsche Forschungsgemeinschaft (314838170), GRK 2297 MathCoRe, the Federal Ministry of Education and Research of Germany (05M16NMA) and the German Federal Environmental Foundation.

References

  1. 1.
    P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Computational Mathematics. vol. 35 (Springer, Berlin, 2011)Google Scholar
  2. 2.
    W.D. Hibler, A dynamic thermodynamic sea ice model. J. Phys. Oceanogr. 9, 815–846 (1979)CrossRefGoogle Scholar
  3. 3.
    M. Losch, A. Fuchs, J.F. Lemieux, A. Vanselow, A parallel Jacobian-free Newton-Krylov solver for a coupled sea ice-ocean model. J. Comput. Phys. 257, 901–911 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Mandal, A. Ouazzi, S. Turek, Modified Newton Solver for Yield Stress Fluids. Proceedings of the ENUMATH 2015 (Springer, Berlin, 2016)Google Scholar
  5. 5.
    S. Mandal, Efficient FEM solver for quasi-Newtonian problems with application to granular materials. Dissertation, Technical University Dortmund (2016)Google Scholar
  6. 6.
    C. Mehlmann, T. Richter, A modified global Newton solver for viscous-plastic sea ice models. Ocean Model. 116, 96–107 (2017)CrossRefGoogle Scholar
  7. 7.
    T. Wick, Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation. Comput. Methods. Appl. Mech. Eng. 325, 577–611 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Magdeburg, Institute for Analysis and NumericsMagdeburgGermany

Personalised recommendations