A pure-Lagrangian finite element approach for solving thermo-electrical-mechanical models. Application to electric upsetting

Abstract

In this paper, we introduce a novel numerical procedure for solving fully coupled thermo-electrical-mechanical problems using implicit Runge–Kutta time integration within a purely Lagrangian finite element framework. Our formulation, grounded in continuum mechanics, accurately captures the interdependence of mechanical, thermal, and electrical effects under large deformations. It features a fully coupled thermo-electrical-mechanical Lagrangian model with an elasto-viscoplastic constitutive law, considers six primary variables –velocity, temperature, electric potential, plastic deformation gradient, an internal strain hardening variable, and a Lagrange multiplier for enforcing contact conditions– and employs a pure-Lagrangian description. This ensures the computational domain remains fixed and known a priori, simplifies the tracking of free surfaces, and eliminates convective terms. To validate our approach, we solve several axisymmetric benchmark problems and analyze convergence rates in both time and space. Moreover, our numerical results show excellent agreement with the solution obtained using commercial packages for an in-die electric upsetting process.