A structure-preserving discretisation of SO(3)-rotation fields for finite Cosserat micropolar elasticity

TL;DR

This paper presents a novel geometric structure-preserving interpolation method for discretizing SO(3)-rotation fields in finite Cosserat micropolar elasticity, ensuring physical constraints and stability in the asymptotic limit.

Contribution

The paper introduces $ ext{Γ}$-SPIN, a new interpolation technique that maintains objectivity and curvature representation while reducing locking effects in Cosserat models.

Findings

Ensures stable behavior as Cosserat couple modulus tends to infinity.

Matches the polar part of the deformation tensor in discretization.

Demonstrates improved accuracy on complex curved domains.

Abstract

We introduce a new method, dubbed Geometric Structure-Preserving Interpolation ($\Gamma$-SPIN) to preserve physics-constraints inherent in the material parameter limits of the finite-strain Cosserat micropolar model. 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. The construction allows the discrete Cosserat rotation tensor to match the polar part of the discrete deformation tensor. 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.