A unified finite strain theory for membranes and ropes

Abstract

The finite strain theory is reformulated in the frame of the Tangential Differential Calculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, membranes and three-dimensional continua are treated with one set of governing equations. Secondly, the reformulated boundary value problem applies to parametrized and implicit geometries. Therefore, the formulation is more general than classical ones as it does not rely on parametrizations implying curvilinear coordinate systems and the concept of co-and contravariant base vectors. This leads to the third unification: TDC-based models are applicable to two fundamentally different numerical approaches. On the one hand, one may use the classical Surface FEM where the geometry is discretized by curved one-dimensional elements for ropes and two-dimensional surface elements for membranes. On the other hand, it also applies to recent Trace FEM approaches where the geometry is immersed in a higher-dimensional background mesh. Then, the shape functions of the background mesh are evaluated on the trace of the immersed geometry and used for the approximation. As such, the Trace FEM is a arXiv:1909.12640v1 [cs.CE] 27 Sep 2019 fictitious domain method for partial differential equations on manifolds. The numerical results show that the proposed finite strain theory yields higher-order convergence rates independent of the numerical methodology, the dimension of the manifold, and the geometric representation type.

A higher‐order Trace finite element method for shells

“…As outlined in Reference 31, the enforcement of essential of boundary conditions is a challenging task in FDMs due to the fact that it is not possible to directly prescribe nodal values of the active background elements. The situation in the case of shells may be quite delicate due to complex boundary conditions, for example, membrane support, symmetry support, clamped edges, and, therefore, the treatment of boundary conditions requires special attention.…”

“…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.…”

“…However, the extension to multiple level-set functions, where the zero-isosurface of the master level-set function is restricted with additional slave-level-set functions has not been addressed so far. Herein, we employ the integration strategy outlined by the authors in References 11,12,30,31. The advantages of this particular approach are an optimal higher-order accurate integration and a natural extension to multiple level-set functions.…”

“…Regarding the integration of the weak form, suitable integration points with higher‐order accuracy for multiple level‐set functions have to be provided. Herein, the approach, which naturally extends to multiple level‐set functions as outlined by the authors in References 11,12,30,31 is employed. Other higher‐order integration schemes for implicitly defined surfaces with one level‐set function are presented, for example, in References 32,33.…”

“…The essential boundary conditions need to be enforced in a weak manner due to the fact that a strong enforcement by prescribing nodal values does not apply (because the nodes of the background mesh are not on the shell boundary). Therefore, a similar approach as presented by the authors in Reference 31 is employed. In particular, the nonsymmetric version of Nitsche's method is used, see, for example, References 31,36‐38.…”