A Higher-order Trace Finite Element Method for Shells

TL;DR

This paper introduces a higher-order Trace Finite Element Method for Reissner-Mindlin shells that uses a background mesh and level-set geometry, enabling accurate solutions for PDEs on implicit surfaces.

Contribution

It develops a novel higher-order Trace FEM for shells based on level-set geometry and Tangential Differential Calculus, addressing key numerical challenges.

Findings

Method achieves higher-order accuracy with smooth solutions.

Numerical integration and stabilization techniques are effective.

Boundary conditions are successfully enforced using Nitsche's method.

Abstract

A higher-order fictitious domain method (FDM) for Reissner-Mindlin shells is proposed which uses a three-dimensional background mesh for the discretization. The midsurface of the shell is immersed into the higher-order background mesh and the geometry is implied by level-set functions. The mechanical model is based on the Tangential Differential Calculus (TDC) which extends the classical models based on curvilinear coordinates to implicit geometries. The shell model is described by PDEs on manifolds and the resulting FDM may typically be called Trace FEM. The three standard key aspects of FDMs have to be addressed in the Trace FEM as well to allow for a higher-order accurate method: (i) numerical integration in the cut background elements, (ii) stabilization of awkward cut situations and elimination of linear dependencies, and (iii) enforcement of boundary conditions using Nitsche's method. The numerical results confirm that higher-order accurate results are enabled by the proposed method provided that the solutions are sufficiently smooth.