Kirchhoff–Love shell theory based on tangential differential calculus

We propose a reformulation of the linear Reissner–Mindlin shell theory in terms of tangential differential calculus. An advantage of our approach is that shell analysis on implicitly defined surfaces is enabled and a parametrization of the surface is not required. In addition, the implementation is more compact and intuitive compared to the classical approach. The numerical results confirm, that this approach is equivalent to the classical theory based on local coordinates.

show abstract

“…For a more detailed introduction to the TDC, we refer to, e.g., [3,4]. For a more detailed introduction to the TDC, we refer to, e.g., [3,4].…”

Section: Tangential Differential Calculus (Tdc)mentioning

confidence: 99%

Reissner–Mindlin shell theory based on tangential differential calculus

Schöllhammer

Fries

2019

Computer Methods in Applied Mechanics and Engineering

Self Cite

View full text Add to dashboard Cite

show abstract

“…Most importantly, the use of the tangential differential calculus (TDC) was found highly useful and generalizes the mechanical models in the sense that they become valid for explicit and implicit geometry definitions. 20,31,[45][46][47][48] By contrast, in flow and transport applications on curved surfaces, the general coordinate-free definition of the boundary value problems is a standard for a long time, [27][28][29]39,49 thus enabling the application of the Trace FEM earlier than in structural mechanics as proposed herein. Herein, to the best knowledge of the authors, this is the first time, that a Trace FEM approach is applied to curved Reissner-Mindlin shells.…”

Section: F I G U R Ementioning

confidence: 99%

A higher‐order Trace finite element method for shells

Schöllhammer

Fries

2020

Int J Numer Methods Eng

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In contrast to the Kirchhoff-Love shell theory [3], transverse shear deformations are considered and the displacement field u Ω is defined as u Ω = u + ζw, where u is the deflection of the middle surface and w is the difference vector, modelling the rotation of the shell director. As usual in the Reissner-Mindlin theory, the out of plane drilling moment is neglected and therefore, the difference vector is tangential.…”

Section: Reissner-mindlin Shell Equationsmentioning

confidence: 99%