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Cross fields are auxiliary in the generation of quadrangular meshes. A method to generate cross fields on surface manifolds is presented in this paper. Algebraic topology constraints on quadrangular meshes are first discussed. The duality between quadrangular meshes and cross fields is then outlined, and a generalization to cross fields of the Poincaré-Hopf theorem is proposed, which highlights some fundamental and important topological constraints on cross fields. A finite element formulation for the computation of cross fields is then presented, which is based on Ginzburg-Landau equations and makes use of edge-based Crouzeix-Raviart interpolation functions. It is first presented in the planar case, and then extended to a general surface manifold. Finally, application examples are solved and discussed.

Purpose -Proposes a new quasi-static vector hysteresis model based on an energy approach, where dissipation is represented by a friction-like force. Design/methodology/approach -The start point is the local energy balance of the ferromagnetic material. Dissipation is represented by a friction-like force, which derives from a non-differentiable convex functional. Several elementary hysteresis cells can be combined, in order to increase the number of free parameters in the model, and therefore improve the accuracy. Findings -A friction-like force is a good way to represent magnetic dissipation at the macroscopic level. The proposed method is easy to implement and non-differentiability amounts in this case to a simple "if" statement.Research limitations/implications -The next steps are the extension to dynamic hysteresis and the in-depth analysis of the identification process, which is only sketched in this paper. Practical implications -This vector model, which is based on a reasonable phenomenological description of local magnetic dissipation, enables the numerical analysis of rotational hysteresis losses on a sound theoretical basis. Originality/value -It proposes a simple, general purpose macroscopic model of hysteresis that is intrinsically a vector one, and not the vectorization of a scalar model.

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