An Objective FE-Formulation for Cosserat Shells Based on Spherical Bézier Interpolation
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An objective Bézier Finite Element (FE) formulation for geometrically exact Cosserat shell structures undergoing large deformations is presented. The approach extends classical bilinear interpolation [1] in ℝ³ to a spherical bilinear interpolation on the rotation manifold SO(3), further generalized to high order than 1 by means of the De Casteljau’s algorithm, as in done [2,3]. This enables the use of high order bi-variate Bézier surface representations for both the mid-surface (in ℝ³) and the frames (or unit matrices in SO(3)), ensuring an objective and path-independent multiplicative up-dating iterative numerical scheme. Recursive expressions are derived for the interpolation of finite rotations, as well as for the spin and curvature vectors, resulting to a generalization of Bézier basis functions in SO(3). To mitigate locking effects, a mixed formulation is employed. Several numerical examples are investigated in order to highlight the convergence, objectivity and path-independence properties of the proposed formulation. REFERENCES [1] C. Sansour and H. Bednarczyk. The Cosserat surface as a shell model, theory and finite-element formulation, Computer Methods in Applied Mechanics and Engineering, Vol. 120(1–2), pp. 1–32, 1995. [2] L. Greco, A. Cammarata, D. Castello, M. Cuomo. An objective FE-formulation for Cosserat rods based on the spherical Bézier interpolation, Computer Methods in Applied Mechanics and Engineering, Vol. 425, pp. 116947, 2024. [3] L. Greco, A. Cammarata, D. Castello, M. Cuomo. Spherical B-spline interpolation with application to a mixed isogeometric cosserat rod formulation, Computer Methods in Applied Mechanics and Engineering, Vol. 446, part A, pp. 118239, 2025.
