Intrinsic mixed-dimensional beam-shell-solid couplings in linear Cosserat continua via tangential differential calculus

TL;DR

This work addresses mixed-dimensional coupling in linear elasticity by deriving a unified Cosserat micropolar continuum in 3D and reducing it to shell, plate, and beam models using tangential differential calculus. Coupling across volume, shell, plate, and beam domains is formulated as a single mixed-dimensional variational problem using consistent Sobolev trace operators, eliminating the need for mortar elements. The authors provide explicit reduced-energy functionals for each dimensional model, demonstrate intrinsic agreement of displacement and rotation DOF at interfaces, and validate the approach with three numerical examples in NGSolve. In the limit $L_c\to 0$, the Cosserat model recovers Navier–Cauchy behavior, indicating the method's compatibility with classical elasticity while enabling design for sandwich, composite, and fibre-reinforced parts. The framework offers a simple, dimension-agnostic coupling mechanism with potential for seamless intersections, at the cost of additional rotational DOFs and computational expense, and opens avenues for optimized mixed-dimensional structure design.

Abstract

We present an approach to the coupling of mixed-dimensional continua by employing the mathematically enriched linear Cosserat micropolar model. The kinematical reduction of the model to lower dimensional domains leaves its fundamental degrees of freedom intact. Consequently, the degrees of freedom intrinsically agree even at the interface with a domain of a different dimensionality. Thus, this approach circumvents the need for intermediate finite elements or mortar methods. We introduce the derivations of all models of various dimensions using tangential differential calculus. The coupling itself is then achieved by defining a mixed-dimensional action functional with consistent Sobolev trace operators. Finally, we present numerical examples involving a three-dimensional silicone-rubber block reinforced with a curved graphite shell on its lower surface, a three-dimensional silver block reinforced with a graphite plate and beams, and lastly, intersecting silver shells reinforced with graphite beams.

Figures (12)

• Figure 2.1: The domain $V \subset \mathbb{R}^3$ of the linear Cosserat micropolar model with Neumann $A_N$ and Dirichlet $A_D$ boundaries under internal forces $\mathbf{f}$ and couple-forces $\bm{M}$. The model can be intuitively understood as the superposition of a linear elastic Navier--Cauchy model with an infinitesimally thin fibre-matrix that reinforces the material with respect to bending and torsion.
• Figure 3.1: Mapping of the flat reference domain to a shell in three-dimensional space $\mathbf{x}:\Omega \subset \mathbb{R}^3 \to V \subset \mathbb{R}^3$. The middle surface of the shell is mapped via $\mathbf{r}:\omega \subset \mathbb{R}^2 \to A \subset \mathbb{R}^3$, and the thickness via $\zeta \mathbf{n}$ with $\zeta \in [-h/2, h/2]$.
• Figure 4.1: A three-dimensional beam mapped from the reference domain $\Omega \subset \mathbb{R}^3$ to the physical domain $V\subset \mathbb{R}^3$ via $\mathbf{x}: \Omega \subset \mathbb{R}^3 \to V \subset \mathbb{R}^3$. The cross-section $\omega \subset \mathbb{R}^2$ of the beam is left unchanged by the mapping. The centroid line of the beam is mapped by $\mathbf{r}:[0,l] \to \mathbb{R}^3$.
• Figure 5.1: Trace operators from a meshed three-dimensional domain $V \subset \mathbb{R}^3$ of three tetrahedral elements to domains of codimensions one $\Xi_1 \subset \mathbb{R}^{3-1}$ of two triangles and two $\Xi_2 \subset \mathbb{R}^{3-2}$ of a single line. The operators yield valid finite element spaces on the lower dimensional entities. Namely, restricting the continuous three-dimensional Lagrange space $\mathcal{CG}^p(V)$ onto $\Xi_1$ or $\Xi_2$ via $\mathop{\mathrm{\mathrm{tr}}}\nolimits_A$ or $\mathop{\mathrm{\mathrm{tr}}}\nolimits_s$ yields the equivalent Lagrange space on these lower dimensional domains, being $\mathcal{CG}^p(\Xi_1)$ and $\mathcal{CG}^p(\Xi_2)$. Analogously, the tangential trace operator $\mathop{\mathrm{\mathrm{tr}}}\nolimits_A^t$ yields a valid surface Nédélec finite element space $\mathcal{N}_{II}^{p-1}(\Xi_1) \supset \nabla_t\mathcal{CG}^p(\Xi_1)$ from the volumetric Nédélec space $\mathcal{N}_{II}^{p-1}(V) \supset \nabla \mathcal{CG}^p{V}$, and the application of the second tangential trace operator $\mathop{\mathrm{\mathrm{tr}}}\nolimits_s^t$ onto a curve is a valid discontinuous Lagrange space on it $[\mathcal{DG}^{p-1}(\Xi_2)]\mathbf{t} = \nabla_t \mathcal{CG}^p(\Xi_2)$.
• Figure 6.1: Illustration of the volumetric domain $V$ with Dirichlet boundary conditions on $A_D^1$ and $A_D^2$ and body force $\mathbf{f}$. The domain is subsequently reinforced with a stiff shell defined on its lower surface $\Xi$.