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SUMMARYThe purpose of this work is the algorithmic formulation and implementation of a recent coupled electromagnetic-inelastic continuum field model (Continuum Mech. Thermodyn. 2005; 17:1-16) for a class of engineering materials, which can be dynamically formed using strong magnetic fields. Although in general relevant, temperature effects are for the applications of interest here minimal and are neglected for simplicity. In this case, the coupling is due, on the one hand, to the Lorentz force acting as an additional body force in the material. On the other hand, the spatio-temporal development of the magnetic field is very sensitive to changes in the shape of the workpiece, resulting in additional coupling. The algorithmic formulation and numerical implementation of this coupled model is based on mixed-element discretization of the deformation and electromagnetic fields combined with an implicit, staggered numerical solution scheme on two meshes. In particular, the mechanical degrees of freedom are solved on a Lagrangian mesh and the electromagnetic ones on an Eulerian one. The issues of the convergence behaviour of the staggered algorithm and the influence of data transfer between the meshes on the solution is discussed in detail. Finally, the numerical implementation of the model is applied to the modelling and simulation of electromagnetic sheet forming.

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The increasing need for lightweight building components has led to the development of new methods to manufacture such components. A promising concept is the systematic application of high-speed metal forming methods. Electromagnetic forming is one such method. Here, the deformation of the workpiece is driven by the Lorentz force which results from the interaction of a current generated in the workpiece with a magnetic field generated by a coil adjacent to the workpiece. This force represents an additional volume-or body-force density contribution in the balance of linear momentum. The numerical treatment of the coupled set of partial differential equations for the mechanical and electromagnetic fields can be made more efficient from the computational point of view by using the finite element technology suggested here, which is based on reduced integration and hourglass stabilisation. The main idea behind this new technology is to expand the constitutive quantities in a Taylor expansion with respect to a point on the local coordinate axis in the thickness direction. The result is a weak system of equations which decomposes into a part to be evaluated in two Gauss points and in addition the so-called hourglass stabilisation to be computed analytically.