Adaptive space time finite element methods for dynamic nonlinear thermomechanical coupled problems

In this contribution, we consider nonlinear thermomechanical coupled problems of the following type: Find a displacement field and a temperature distribution, which fulfil the equation of motion with a temperature dependentstress field and the heat equation including a heat source depending on the displacement. The continuous model is discretized with a space time Petrov Galerkin method using continuous and piecewise d-linear basis functionsin space and time, which can be reduced to a time stepping scheme due to the discontinuous piecewise constant temporal test functions, see for instance [1].The coupled system is solved by a staggered scheme. The aim is to control the discretization error as well as the error in the solution scheme in a nonlinear goal functional. To this end,we use the dual weighted residual (DWR) method. In detail, we combine the ideas from [1] for goal oriented error estimation in nonlinear time dependentproblems and from [2] for balancing discretization and numerical error in static problems.The discretization error is measured by the usual DWR techniques discussedfor time dependent problems,e. g., in [1], while the numerical error is estimated by testing the residuum with the discrete dual solution, cf. [2]. However, the error estimator can only be evaluated after solving the complete primal and dual problem.Thus, we have to design a special solution algorithm. The main idea is to solve the primal problem doingonly one step of the staggered solution scheme in each time step. Then the discretization and the numerical error is estimated. If the numerical error is dominant, an additional loop over all time steps including one step of the staggered scheme is executed. Otherwise, the mesh is adaptively refined. By this approach, we can reduce the numerical effort for the accurate solving of thermomechanical problems by adopting the solution scheme to the special needs of the underlying problem and the chosen goal functional