Theory and computation of electromagnetic fields and thermomechanical structure interaction for systems undergoing large deformations

The governing equations for electromagneto-thermomechanical systems are well established and thoroughly derived in the literature, but have been limited to small deformations. This assumption provides an “ease” in the formulation: electromagnetic fields are governed in a Eulerian frame, whereby the thermomechanics is solved in a Lagrangean frame. It is possible to map the Eulerian frame to the current placement of the matter and the Lagrangean frame to a reference placement. The assumption of small deformations eliminates the distinction between current and initial placement such that electromagnetism and thermomechanics are formulated in the same frame. We present a rigorous and thermodynamically consistent derivation of governing equations for fully coupled electromagneto-thermomechanical systems properly handling finite deformations. A clear separation of the different frames is necessary. There are various attempts to formulate electromagnetism in the Lagrangean frame, or even to compute all fields in the current placement. Both formulations are challenging and heavily discussed in the literature. In this work, we propose another solution scheme that exploits the capabilities of advanced computational tools. Instead of amending the formulation, we can solve thermomechanics in the Lagrangean frame and electromagnetism in the Eulerian frame and manage the interaction between the fields. The approach is similar to its analog in fluid structure interaction, but additionally challenging because the electromagnetic governing equations must also be solved within the solid body while following their own different set of transformation rules. We further present a mesh-morphing algorithm necessary to accommodate finite deformations to solve the electromagnetic fields outside of the material body. We illustrate the use of the new formulation by developing an open-source implementation using the FEniCS package and applying this implementation to several engineering problems in electromagnetic structure interaction undergoing large deformations.