A Strict Energy-Decreasing Momentum-Conserving Time Finite Element Method for Dynamic Finite Thermoviscoelasticity

Abstract eng:
This paper presents a new strict energy-decreasing momentum-conserving monolithic time finite element method for dynamic finite thermoviscoelasticity. The algorithm exactly preserves the total linear and total angular momentum, and leads to the exact physical dissipation by conduction of heat (see [3]) as well as by the finite viscoelastic material law presented in [2]. In the thermodynamic equilibrium state, the method is exactly total energy conserving. Algorithmic extensions also allow a strict numerical damping of undesired high-frequent oscillations (compare [1]). The method is unconditionally nonlinear stable and extremely robust, especially compared to standard time integration methods for thermo-mechanical coupling as the trapezoidal rule. The algorithm relies on a Lagrangian formulation of the hyperbolic equations of motion, the parabolic entropy evolution equation and the strain-based viscous evolution equations. The stresses and the entropy emanate from an isotropic thermoviscoelastic free energy, and the heat flux arises from Fourier's law of heat conduction. These fully coupled nonlinear PDE's and ODE’s are discretised by a new time finite element method. The hyperbolic equations of motion and the viscous evolution equations are discretised by a continuous finite element method in time, and the parabolic entropy evolution equation is solved by a new discontinuous finite element method. The strict energy-decreasing mainly relies on a specific extension of the stress and the discontinuous Galerkin method. In order to preserve the accuracy order of the time finite element method, the corresponding coupled system of nonlinear algebraic equations is solved by a consistent linearized monolithic solution strategy. New convergence criteria take the energy consistency into account, and are free of the scaling in the primary variables. Representative numerical simulations show a superior performance of the new time integration algorithm with mechanical and thermal Boundary-Conditions as isothermal environments and supports, convective heat transfer, external mechanical pressure, heat transfer by radiation and transient traction. References [1] Armero F. Energy-dissipative momentum-conserving time-stepping algorithms for finite strain multiplicative plasticity, Computer Methods in Applied Mechanics and Engineering, 195:48624889, 2006. [2] Gross M. and Betsch P. Energy-momentum consistent finite element discritization of dynamic finite viscoelasticity. Int. J. Numer. Methods Engrg., 81:1341-1386, 2010. [3] Romero I. Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics. Part I: Monolithic integrators and their application to finite strain thermoelasticity. Computer Methods in Applied Mechanics and Engineering, 199:1841-1858, 2010.

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