Keywords: scalability, dynamics, Newmark method, impact problem, geometric non-linearity, material non-linearity.

This paper is concerned with application of our TFETI based algorithm suggested in [1] to the impact problems accompanied by geometric and material non-linear effects. The TFETI method belongs to the class of non-overlapping spatial decompositions. Its key concept is based on the idea that the compatibility between sub-domains is ensured by Lagrange multipliers. The primal variables, which are displacements, are eliminated so that we solve the problem for the dual variables or Lagrange multipliers. The difference between the well-known FETI and TFETI is that the latter method, unlike the former one, also enforces the Dirichlet boundary conditions in terms of the Lagrange multipliers, which has a significant bearing on some segments of computations because of lesser sensitivity to round-off errors. The TFETI technique converts the original problem to the quadratic programming one and if applied to the contact problems, this approach transforms the general non-penetration constraints to the simple bound constraints. After eliminating the primal variables, the original problem is reduced to a small and relatively well conditioned one. Our approach stems from an optimal algorithm to the solution of strictly convex bound constrained quadratic programming problems with preconditioning by the conjugate projector to the subspace defined by the trace of the rigid body modes on the fictitious interfaces between sub-domains. Application of this algorithm to transient problems shows that if the time steps and the ratio of the decomposition and discretisation parameters are kept uniformly bounded, then the cost of time step computations is proved to be proportional to the number of nodal variables. The time integration requires a special treatment to guarantee stability and removal of the undesired non-physical oscillations of solution along the contact interfaces. We applied the contact stabilised Newmark scheme introduced in [2] that reduces the solution to a sequence of quadratic programming problems. We show that our transient algorithm can be applied to the solution of problems exhibiting geometric and material non-linearities. Results of numerical experiments document that the proposed algorithm is highly accurate and exhibits both parallel and numerical scalabilities which is quite essential for its application to high performance computers.

1

Z. Dostál, T. Kozubek, T. Brzobohatý, A. Markopoulos, O. Vlach, "Scalable TFETI with preconditioning by conjugate projector for transient contact problems of elasticity", Computer Methods in Applied Mechanics and Engineering, submitted.

2

R. Krause, M. Walloth, "A time discretization scheme based on Rothe's method for dynamical contact problems with friction", Computer Methods in Applied Mechanics and Engineering, 199, 1-19, 2009. doi:10.1016/j.cma.2009.08.022

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