The Helicoidal Modeling in Computational Finite Elasticity. Part II: Multiplicative Interpolation

The approximate, substitute model of the position and orientation fields over a finite, multi-coordinate domain is built in a consistent way with the helicoidal modeling of the continuum, as proposed in Part I for a discrete application of variational principles in computational elasticity. The set of the position and orientation, referred to as oriento-position, is made dependent multiplicatively on the nodal values through relative rototranslations, and an implicit interpolation formula is written by weighting the relative helices. The proposed interpolation scheme is frame-invariant and path-independent, and the resulting weighted average oriento-position is obtained numerically. A consistent linearization of the model field is carried out by developing explicit formulae for the derivatives, up to third-order, of orthogonal tensors. The parent interpolation of the orientation field, which can be useful by itself in the context of classical modeling, is also discussed.