A higher‐order Trace finite element method for shells

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 which extends the classical models based on curvilinear coordinates to implicit geometries. The shell model is described by partial differential equations 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.

Implicit Analysis of Reissner–Mindlin shells with the Trace FEM

“…(4) [7]. For the enforcement of essential boundary conditions the non-symmetric version of Nitsche's method is employed [7].…”

“…As well-known in the context of FDMs, the three major issues have to be addressed properly: (1) the accurate integration of the implicitly defined geometry, (2) the stabilization of the stiffness matrix and (3) the enforcement of boundary conditions. Special attention is required when higher-order convergence rates, see, e.g., [5,7], are sought. The numerical results confirmed the successful implementation of a higher-order accurate Trace FEM for linear Reissner-Mindlin shells, provided that the fields are sufficiently smooth.…”

“…The Trace FEM is a fictitious domain method (FDM) for PDEs on implicitly defined manifolds [6,7]. The degrees of freedom (DoFs) are implied by a 3D background mesh and the discrete weak form is then integrated on the trace, i.e., the implicitly defined midsurface of the shell Γ.…”

A Consistent Higher-Order Isogeometric Shell Formulation

“…In this paper, we employ the rotation-free Kirchhoff-Love shell formulated in the frame of the tangential differential calculus (TDC) [97,98]. The resulting shell formulation is independent of the concrete geometry definition and enables shell analysis on implicitly defined surfaces where a parametrization is not available nor needed, see [99]. However, herein, the midsurface is explicitly specified by a single NURBS patch.…”

“…In fact, there is no higher-order accurate shell formulation in the context of trimming. It is, however, noted that the already mentioned works from Fries et al [45] on membranes and Schöllhammer et al [99] on linear RM shells are presented in the framework of higher-order Trace FEM.…”