Subdivision shells with exact boundary control and non‐manifold geometry

Abstract

SUMMARYWe introduce several extensions to subdivision shells that provide an improved level of shape control over shell boundaries and facilitate the analysis of shells with non-smooth and non-manifold joints. To this end, extended subdivision schemes are used that enable to relax the continuity of the limit surface along prescribed crease edges and to create surfaces with prescribed limit positions and normals. Furthermore, shells with boundaries in the form of conic sections, such as circles or parabolas, are represented with rational subdivision schemes, which are defined in analogy to rational b-splines. In terms of implementation, the difference between the introduced and conventional subdivision schemes is restricted to the use of modified subdivision stencils close to the mentioned geometric features. Hence, the resulting subdivision surface is in most parts of the domain identical to standard smooth subdivision surfaces. The particular subdivision scheme used in this paper constitutes an extended version of the original Loop's scheme and is as such based on triangular meshes. As in the original subdivision shells, surfaces created with the extended subdivision schemes are used for interpolating the reference and deformed shell configurations. At the integration points, the subdivision surface is evaluated using a newly developed discrete parameterization approach. In the resulting finite elements, the only degrees of freedom are the mid-surface displacements of the nodes and additional Lagrange parameters for enforcing normal constraints. The versatility of the newly developed elements is demonstrated with a number of geometrically nonlinear shell examples.

Isogeometric shape optimisation of shell structures using multiresolution subdivision surfaces

“…The equilibrium configurations of the shell with prescribed loading are obtained from minimising (4). Note that for a well posed problem also the displacements on some parts of the boundary have to be prescribed in addition to the loading.…”

“…To this end, we use Gauss integration with 3 points for Loop and 4 points for Catmull-Clark subdivision. The basis function values at the Gauss points are evaluated with the simplified version of the algorithm proposed by Stam [29,30], see [3,4]. Specifically, since the Gauss points are relatively far from extraordinary vertices there are no efficiency gains from the eigendecomposition considered in [29,30].…”

“…As subdivision schemes triangular Loop and quadrilateral Catmull-Clark scheme are used. In order to preserve corners and edges of the original geometry, at vertices and edges on the boundary modified subdivision stencils are applied [4,25].…”

“…Specifically, Loop subdivision surfaces were used for discretising the Kirchhoff-Love shells and representing their geometry. As an extension of this approach, the treatment of industrially prevalent non-manifold shell * Corresponding author geometries and the inclusion of out-of-plane shear deformations relevant for thicker shells were proposed in [4] and [5], respectively. More recently the isogeometric analysis of shells and beams using NURBS basis functions were introduced in [6,7].…”

Phase-field modeling of fracture in linear thin shells

“…Instead, here we focus on methods relying on smooth basis functions. Finite element methods with high order continuity have been proposed, either based on subdivision surfaces [13,34] or on isogeometric analysis [35,36,37]. The higher order continuity of the meshfree basis functions has also been exploited for this purpose [14,15], but since meshfree basis functions are defined in physical space, these methods were applied to simple geometries with a single parametric patch.…”