Algorithmic formulation and numerical implementation of coupled electromagnetic‐inelastic continuum models for electromagnetic metal forming

Abstract

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.

Identification of plastic material parameters with error estimation

“…[5]) and ρ = 2.7 · 10 3 kg/m 3 . All identifi cations and the corresponding error analysis are performed using pseudo experimental data which was generated from simulation data and normally distributed measurement deviations; this also justifi es the very simple approach for the isotropic hardening function chosen in (2). To illustrate the importance of the inertia effects in this process, the mass density ρ, which is usually obtained from literature, is analyzed here also.…”

“…Given the material model [1] and its numerical implementation [2], an identifi cation of the corresponding material parameters is possible. In particular, the viscoplastic behavior is determined by the flow rulė…”

Modelling and Simulation of 3D electromagnetic metal forming processes

“…The algorithmic realization of the model is discussed in [6], for further details we refer to these publications. In particular, this special case is based on the quasi-static approximation to Maxwell's equations, in which the wave character of the electromagnetic (EM) fields is neglected.…”

“…This approach is known from fluid structure interaction as the fictive boundary method. A more recent approach applies an incrementally progressing meshupdating algorithm [6] and is based on the arbitrary Lagrangian Eulerian solution scheme (ALE). Here, the position of the nodal coordinates of the EM-Mesh is adopted by means of solving the Laplace problem.…”

Error estimates for discretizations of coupled electromagnetic‐mechanical problems

“…Accounting for the resulting high strain rates, Svendsen and Chanda formulated a material model [1,2] for a wide class of engineering materials under the influence of strong electromagnetic fields based on the Perzyna model of elasto-viscoplasticity. The evolution of the electromagnetic field is determined by Maxwell's equations under the quasistatic hypothesis, since the occurring wave lengths are much longer than the distances relevant for the forming process (see, e.g., [3]). The two systems are coupled via the Lorentz force representing the source term in the impulse balance for the mechanical structure and via the distribution of conductivity, which depends on the position of the moving structure.…”