Implementation of a direct procedure for critical point computations using preconditioned iterative solvers

  • Authors:

  • Reijo Kouhia;Miroslav Tma;Jari MäKinen;Alexis Fedoroff;Heikki MarjamäKi

  • Affiliations:

  • Tampere University of Technology, Department of Mechanics and Design, P.O. Box 589, FI-33101 Tampere, Finland;Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou ví 2, 18207 Prague 8, Czech Republic;Tampere University of Technology, Department of Mechanics and Design, P.O. Box 589, FI-33101 Tampere, Finland;Aalto University, Department of Civil and Structural Engineering, P.O. Box 12100, FI-00760 Aalto, Finland;Tampere University of Technology, Department of Mechanics and Design, P.O. Box 589, FI-33101 Tampere, Finland

  • Venue:
  • Computers and Structures
  • Year:
  • 2012

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Abstract

Computation of critical points on an equilibrium path requires the solution of a non-linear eigenvalue problem. These critical points could be either bifurcation or limit points. When the external load is parametrized by a single parameter, the non-linear stability eigenvalue problem consists of solving the equilibrium equations along the criticality condition. Several techniques exist for solution of such a system. Their algorithmic treatment is usually focused for direct linear solvers and thus use the block elimination strategy. In this paper special emphasis is given for a strategy which can be used also with iterative linear solvers. Comparison to the block elimination strategy with direct linear solvers is given. Due to the non-uniqueness of the critical eigenmode a normalizing condition is required. In addition, for bifurcation points, the Jacobian matrix of the augmented system is singular at the critical point and additional stabilization is required in order to maintain the quadratic convergence of the Newton's method. Depending on the normalizing condition, convergence to a critical point with negative load parameter value can happen. The form of the normalizing equation is critically discussed. Due to the slenderness of the buckling sensitive structures the resulting matrices are ill-conditioned and a good preconditioner is mandatory for efficient solution.