Simultaneous Analysis of Continuously Embedded Reissner-Mindlin Shells in 3D Bulk Domains

TL;DR

This work formulates a linear Reissner–Mindlin shell model for all level-set–defined shells embedded in a bulk domain Ω and solves them simultaneously using Bulk Trace FEM, a method that couples a conforming 3D bulk mesh with non-conforming shell geometries. The approach relies on Tangential Differential Calculus to define geometry and operators on implicit level-set surfaces, and employs the co-area formula to derive a robust weak form weighted by ||∇φ|| for integration over all shells. A mixed ansatz for the displacement u and the in-plane difference vector w is used, with a stabilization term ensuring a well-conditioned system for the tangential constraint of w. Numerical results on classical benchmarks and generalized test cases demonstrate higher-order convergence for residuals and stored-energy errors, illustrating the viability of simultaneous, continuous embedding of multiple shells for design optimization and reinforcement in bulk materials.

Abstract

A mechanical model and numerical method for the simultaneous analysis of Reissner-Mindlin shells with geometries implied by a continuous set of level sets (isosurfaces) over some three-dimensional bulk domain is presented. A three-dimensional mesh in the bulk domain is used in a tailored FEM formulation where the elements are by no means conforming to the level sets representing the shape of the individual shells. However, the shell geometries are bounded by the intersection curves of the level sets with the boundary of the bulk domain so that the boundaries are meshed conformingly. This results in a method which was coined Bulk Trace FEM before. The simultaneously considered, continuously embedded shells may be useful in the structural design process or for the continuous reinforcement of bulk domains. Numerical results confirm higher-order convergence rates.

Figures (22)

• Figure 1: Some bulk domain $\Omega$, level-set function $\phi\left(\boldsymbol x\right)$, and implied level sets $\Gamma^{c}$.
• Figure 2: The bulk domain $\Omega$ resulting from some larger $\Omega^{\star}$ (light gray) and a prescribed level-set interval $\left[\phi^{\min},\phi^{\max}\right]$. Some selected level sets $\Gamma^{c}$ in this interval are also shown.
• Figure 3: One has to ensure that the topology of the level sets varies smoothly within the bulk domain. In this example where $\phi$ features some local extremum within the bulk domain $\Omega$, some level sets are closed in $\Omega$ whereas others feature a boundary. Hence, the topology does not vary smoothly and this combination of $\phi$ and $\Omega$ is not valid.
• Figure 4: Vector fields in the domain $\Omega$ and on the boundary $\partial\Omega$ shown on an arbitrarily chosen level set $\Gamma^c$ with constant $c \in \left(\phi_{\min}, \phi_{\max}\right)$. The right figure shows a zoom of the left one. Normal vectors $\boldsymbol n$ with respect to the level sets $\Gamma^{c}$ in $\Omega$ are shown in blue. Normal vectors $\boldsymbol m$ with respect to $\partial\Omega$ are red, tangential vectors $\boldsymbol t$ are gray and co-normal vectors $\boldsymbol q$ are green.
• Figure 5: Sketch of the kinematic relations of some Reissner--Mindlin shells embedded in the bulk domain $\Omega$ where one middle surface $\Gamma^c$ of a shell is highlighted. The relations introduced in this section are depicted. Blue colour refers to the undeformed configuration and red colour refers to the deformed configuration.