A structure-preserving & objective discretisation of SO(3)-matrix rotation fields for finite Cosserat micropolar continua

arXiv:2602.15147v1 Announce Type: new
Abstract: We introduce a new method, dubbed textbf{textit{Geometric Structure-Preserving Interpolation ({Gamma}-SPIN)}}, to simultaneously preserve physics-constraints inherent in the material parameter limits of the finite-strain Cosserat micropolar model, and satisfy objectivity under superimposed rigid body motions. The method advocates to interpolate the Cosserat rotation tensor using geodesic elements, which maintain objectivity and correctly represent curvature measures. At the same time, it proposes relaxing the interaction between the rotation tensor and the deformation tensor to alleviate locking effects. This relaxation is achieved in two steps. First, the regularity of the Cosserat rotation tensor is reduced by interpolating it into the N’ed’elec space. Second, the resulting field is projected back onto the Lie-group of rotations. Together, these steps define a lower-regularity projection-based interpolation. This construction allows the discrete Cosserat rotation tensor to match the polar part of the discrete deformation tensor while remaining objective. This ensures stable behaviour in the asymptotic regime as the Cosserat couple modulus tends to infinity, which constrains the model towards its couple-stress limit. We establish the consistency, stability, and optimality of the proposed method through several benchmark problems. The study culminates in a demonstration of its efficacy on a more intricate curved domain, contrasted with outcomes obtained from conventional interpolation techniques.