Abstract
The Kirchhoff–Love shell theory is recasted in the frame of the tangential differential calculus (TDC) where differential operators on surfaces are formulated based on global, three-dimensional coordinates. As a consequence, there is no need for a parametrization of the shell geometry implying curvilinear surface coordinates as used in the classical shell theory. Therefore, the proposed TDC-based formulation also applies to shell geometries which are zero-isosurfaces as in the level-set method where no parametrization is available in general. For the discretization, the TDC-based formulation may be used based on surface meshes implying element-wise parametrizations. Then, the results are equivalent to those obtained based on the classical theory. However, it may also be used in recent finite element approaches as the TraceFEM and CutFEM where shape functions are generated on a background mesh without any need for a parametrization. Numerical results presented herein are achieved with isogeometric analysis for classical and new benchmark tests. Higher-order convergence rates in the residual errors are achieved when the physical fields are sufficiently smooth.
Highlights
The mechanical modeling of shells leads to partial differential equations (PDEs) on manifolds where the manifolds are curved surfaces in the three-dimensional space
We summarize the advantages of the tangential differential calculus (TDC)-based formulation of Kirchhoff–Love shells: (1) the definition of the boundary value problem (BVP) does not need a parametrization of the surface, (2) the TDC-based formulation is suitable for very recent finite element technologies such as CutFEM and TraceFEM, (3) the implementation is advantagous in finite element (FE) codes where other PDEs on manifolds are considered as well due to the split of FE technology and application
The linear Kirchhoff–Love shell theory is reformulated in terms of the TDC using a global Cartesian coordinate system and tensor notation
Summary
The mechanical modeling of shells leads to partial differential equations (PDEs) on manifolds where the manifolds are curved surfaces in the three-dimensional space. We summarize the advantages of the TDC-based formulation of Kirchhoff–Love shells: (1) the definition of the BVP does not need a parametrization of the surface (though it can handle the classical situation where a parametrization is given), (2) the TDC-based formulation is suitable for very recent finite element technologies such as CutFEM and TraceFEM (though the typical approach based on the Surface FEM or IGA is possible and demonstrated ), (3) the implementation is advantagous in finite element (FE) codes where other PDEs on manifolds are considered as well due to the split of FE technology and application. For the proof of equivalence of both cases we refer to, e.g., [18]
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