A Polyconvexity-Inspired Mixed Formulation and Structure-Preserving Discretization for Coupled Nonlinear Electro-Thermo-Elastodynamics

  • Hille, Moritz (Karlsruhe Institute of Technology)
  • Franke, Marlon (Karlsruhe Institute of Technology)
  • Zähringer, Felix (Karlsruhe Institute of Technology)
  • Ortigosa, Rogelio (Technical University of Cartagena)
  • Betsch, Peter (Karlsruhe Institute of Technology)
  • Gil, Antonio (Swansea University)

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We present a novel mixed framework in combination with a structure-preserving space-time discretization in order to simulate coupled nonlinear electro-thermo-elastodynamical problems in a consistent manner [1]. The mixed environment is facilitated by a polyconvex framework for elastodynamics proposed in [2]. To obtain the mixed framework, the properties of the tensor cross product [3] are utilized to derive a Hu-Washizu type extension of the strain energy function. The strong form of the formulation obtained in this way is subsequently extended by the energy balance as well as Gauss’s and Faraday’s law to cover the thermodynamic and electrostatic contribution, respectively. An appropriate polyconvexity-inspired internal energy function is chosen to obtain a nonlinear, fully coupled electro-thermo-elastodynamical formulation utilizing the properties of the underlying mixed framework. On top of this, a structure-preserving, second order accurate time integration scheme based on discrete derivatives in the sense of [4] is presented yielding a stable and robust formulation even for large time steps. Eventually, the numerical performance of the newly devised method is investigated in representative numerical examples. REFERENCES [1] Franke, M., Zähringer, F., Hille, M., Ortigosa, R., Betsch, P. and Gil, A. J. A novel mixed and energy-momentum consistent framework for coupled nonlinear thermo-electroelastodynamics. Int. J. Numer. Methods Eng. (2023) doi: 10.1002/nme.7209 [2] Betsch, P., Janz, A. and Hesch, C. A mixed variational framework for the design of energymomentum schemes inspired by the structure of polyconvex stored energy functions. Comput. Methods Appl. Mech. Engrg. (2018) 335:660–696. [3] Bonet, J., Gil, A. J. and Ortigosa, R. On a tensor cross product based formulation of large strain solid mechanics. Int. J. Solids Structures (2016) 84:49–63. [4] Gonzales, O. Time integration and discrete Hamiltonian systems. Journal of Nonlinear Science (1996) 6(5):449–467.