SUMMARYThe efficient modelling of three-dimensional contact problems is still a challenge in non-linear implicit structural analysis. We use a primal-dual active set strategy (SIAM J. Optim. 2003; 13:865-888), based on dual Lagrange multipliers (SIAM J. Numer. Anal. 2000; 38:989-1012) to handle the non-linearity of the contact conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to achieve a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way.For the application to thin-walled structures we adapt a three-dimensional non-linear shell formulation including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface-oriented shell element, which allows the application of contact conditions directly to nodes lying on the contact surface. Shell typical locking phenomena are treated with the enhanced-assumed-strain-method and the assumed-natural-strain-method.The discretization in time is done with the implicit Generalized-method (J. Appl. Mech. 1993; 60:371-375) and the Generalized Energy-Momentum Method (Comp. Methods Appl. Mech. Eng. 1999; 178:343-366) to compare the development of energies within a frictionless contact description.In order to conserve the total energy within the discretized frictionless contact framework, we follow an approach from Laursen and Love (Int. J. Numer. Methods Eng. 2002; 53:245-274), who introduced a discrete contact velocity to update the velocity field in a post-processing step.Various examples show the good performance of the primal-dual active set strategy applied to the implicit dynamic analysis of thin-walled structures.
SUMMARYFor extending the usability of implicit FE codes for large-scale forming simulations, the computation time has to be decreased dramatically. In principle this can be achieved by using iterative solvers. In order to facilitate the use of this kind of solvers, one needs a contact algorithm which does not deteriorate the condition number of the system matrix and therefore does not slow down the convergence of iterative solvers like penalty formulations do. Additionally, an algorithm is desirable which does not blow up the size of the system matrix like methods using standard Lagrange multipliers. The work detailed in this paper shows that a contact algorithm based on a primal-dual active set strategy provides these advantages and therefore is highly efficient with respect to computation time in combination with fast iterative solvers, especially algebraic multigrid methods.
SUMMARYThe implicit finite element (FE) simulation of incremental metal cold forming processes is still a challenging task. We introduce a dynamic, overlapping domain decomposition method to reduce the computational cost and to circumvent the need for sophisticated remeshing procedures. The two FE domains interchange information using the elastoplastic operator split and the mortar method.
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