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

Abstract

We introduce a new method, dubbed \textbf{\textit{Geometric Structure-Preserving Interpolation (Γ-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 Nédélec 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.

Figures (11)

• Figure 1: Current configuration of a micropolar body $V_\varphi \subset \mathbb{R}^3$ as given by the deformation map $\bm{\varphi} : V \to V_\varphi$ and the independent orientation of material points $\overline{\bm{R}}:V \to \mathrm{SO}(3)$. Source terms in the domain are the body forces $\mathbf{f}_\varphi$ and the couple-forces $\bm{M}_\varphi$. Their fluxes on the boundary are given by the traction $\mathbf{t}_\varphi$ and couple-traction $\bm{T}_\varphi$. Notably, the orientation of material points is independent of deformation curves.
• Figure 2: The deformation tensor is defined such that $\bm{F} = \bm{F}_1$ in $V_1$ and $\bm{F} = \bm{F}_2$ in $V_2$ with $V = V_1 \cup V_2$. Its product with the normal vector $\mathbf{n}$ at the arbitrary interface $\Xi = V_1 \cap V_2$ defines two different vectors, depending on whether $\bm{F}_1$ or $\bm{F}_2$ is evaluated. Nevertheless, the deformation tensor satisfies $\bm{F} \in \mathbb{R}^3 \otimes \mathit{H}(\mathrm{curl},V)$ as long its tangential components $\bm{F} \mathop{\mathrm{\mathrm{Anti}}}\nolimits \mathbf{n}$ are continuous at the interface, which is naturally given for any $\bm{F} = \mathrm{D} \bm{\varphi}$ with the deformation map $\bm{\varphi} \in \mathbb{R}^3 \otimes \mathit{H}^1(V)$.
• Figure 3: Finite element mesh in the reference configuration (a) and displacement magnitude (b) after rigid body rotation. Norm of the Cosserat strain $\| {\overline{\bm{R}}_h^T \bm{F}_h - \bm{\mathbbm{1}}} \|$ in the standard approach (c), with the interpolation $\overline{\bm{R}}_h \mapsto \Pi_c \overline{\bm{R}}_h$ (d), and with the interpolation and polar projection $\overline{\bm{R}}_h \mapsto \mathop{\mathrm{\mathrm{polar}}}\nolimits (\Pi_c^{p-1}\overline{\bm{R}}_h)$ (e).
• Figure 4: Wide cantilever beam domain with the corresponding Dirichlet and loaded Neumann surfaces.
• Figure 5: Relative error with mesh refinement (a) without interpolation, (b) with interpolation onto $\mathcal{N}_{II}^1(V)$ and (c) with interpolation onto $\mathcal{N}_{II}^1(V)$ and the subsequent polar extraction. Solid lines represent the formulation with $\overline{\bm{R}}_h\in\mathcal{GE}^2(V)$ and dashed lines the enrichment with $\overline{\bm{R}}_h\in\mathcal{GE}^{2+}(V)$.