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

Abstract

For an accurate description of electromagneto-thermomechanical systems, electromagnetic fields need to be described 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. 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 more challenging because the field equations in electromagnetism must also be solved within the solid body while following their own different set of transformation rules. We additionally 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 opensource implementation using the FEniCS package and applying this implementation to several engineering problems in electromagnetic structure interaction undergoing large deformations.

Laplace domain BEM for anisotropic transient elastodynamics

“…Moreover, in a further study, the presented framework can be readily applied to more complex structures and materials with coupled fields (piezoelectric, magnetoelectroelastic, etc). A lot of effort has been put to investigate different phenomena in such materials [40,41].…”

Coupling and Computation of Electromagnetism and Mechanics

“…For the CAUCHY stress, σ in N/m 2 , we need to define a constitutive equation. An engineer models with that term the aforementioned effects, it is even possible to derive the stress by using principles of thermodynamics, we refer to [8] in small deformations and to [9] for large deformations. The interaction is given by the electromagnetic force density, F F F in N/m 3 , and indeed one of the main challenges relies on the correct definition of this force density.…”

“…An engineer models with that term the aforementioned effects, it is even possible to derive the stress by using principles of thermodynamics, we refer to [8] in small deformations and to [9] for large deformations. The known function, f , is a specific body force in N/kg because of the gravity.…”

Modeling and simulation of nonlinear electro-thermo-mechanical continua with application to shape memory polymeric medical devices

“…Theoretical models have also been developed for thermomechanical and electromagneto-thermomechanical problems in the case of large mechanical deforma-tions [20,21,22,23]. Popular numerical implementations combine the Lagrangian approach for the mechanical problem and the Eulerian or arbitrary Lagrangian Eulerian approach for the electromagnetic problem [24,25]. In this paper we consider low frequency electromagnetic problems and solve for a scalar potential formulation only defined in the mechanical domain.…”