# Reissner-Mindlin shell theory based on tangential differential calculus

## TL;DR

This paper reformulates the Reissner-Mindlin shell theory using tangential differential calculus in a Cartesian framework, enabling analysis on both parametrized and level-set defined surfaces with improved generality.

## Contribution

It introduces a TDC-based shell formulation that does not depend on surface parametrization, facilitating applications in modern finite element methods like TraceFEM and CutFEM.

## Key findings

- Achieves optimal higher-order convergence rates in numerical tests.
- Applicable to both classical parametrized and level-set surface representations.
- Demonstrates the method's effectiveness with isogeometric analysis using NURBS.

## Abstract

The linear Reissner-Mindlin shell theory is reformulated in the frame of the tangential differential calculus (TDC) using a global Cartesian coordinate system. The rotation of the normal vector is modelled with a difference vector approach. The resulting equations are applicable to both explicitly and implicitly defined shells, because the employed surface operators do not necessarily rely on a parametrization. Hence, shell analysis on surfaces implied by level-set functions is enabled, but also the classical case of parametrized surfaces is captured. As a consequence, the proposed TDC-based formulation is more general and may also be used in recent finite element approaches such as the TraceFEM and CutFEM where a parametrization of the middle surface is not required. Herein, the numerical results are obtained by isogeometric analysis using NURBS as trial and test functions for classical and new benchmark tests. In the residual errors, optimal higher-order convergence rates are confirmed when the involved physical fields are sufficiently smooth.