A new unconditionally stable fractional step method for non‐linear coupled thermomechanical problems

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Abstract
A new class of time‐stepping algorithms for strongly coupled thermomechanical problems is presented which retains the computational convenience of traditional staggered algorithms without upsetting the unconditional stability property characteristic of fully implicit (monolithic) schemes. This class of methods was first introduced by the authors (in the general context of thermoplasticity) in the USCM held at Chicago, July 1991, and in the IUTAM symposium held at Hannover, Germany, September 1991 The proposed schemes are fractional step methods associated with a two phase operator split of the full non‐linear system of thermoelasticity into an adiabatic elastodynamics phase, followed by a heat conduction phase at fixed configuration. This operator split is shown to inherit the contractivity property of the full problem of evolution, thus leading to unconditionally B‐stable product formula algorithms. In sharp contrast with this stability result, traditional staggered algorithms based on an isothermal mechanical phase followed by a heat conduction phase with an effective heat source are shown to lead, at best, to conditionally stable schemes. It is further shown that the actual implementation of these two classes of schemes is essentially identical. The numerical simulations presented include both dynamic and quasi‐static problems and are shown to closely replicate the stability estimates and non‐linear stability results derived for these two classes of staggered schemes.

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