Isogeometric analysis using manifold-based smooth basis functions

Abstract

We present an isogeometric analysis technique that builds on manifold-based smooth basis functions for geometric modelling and analysis. Manifold-based surface construction techniques are well known in geometric modelling and a number of variants exist. Common to all is the concept of constructing a smooth surface by blending together overlapping patches (or, charts), as in differential geometry description of manifolds. Each patch on the surface has a corresponding planar patch with a smooth one-to-one mapping onto the surface. In our implementation, manifold techniques are combined with conformal parameterisations and the partition-of-unity method for deriving smooth basis functions on unstructured quadrilateral meshes. Each vertex and its adjacent elements on the surface control mesh have a corresponding planar patch of elements. The star-shaped planar patch with congruent wedge-shaped elements is smoothly parameterised with copies of a conformally mapped unit square. The conformal maps can be easily inverted in order to compute the transition functions between the different planar patches that have an overlap on the surface. On the collection of star-shaped planar patches the partition of unity method is used for approximation.The smooth partition of unity, or blending functions, are assembled from tensor-product b-spline segments defined on a unit square. On each patch a polynomial with a prescribed degree is used as a local approximant. In order to obtain a mesh-based approximation scheme the coefficients of the local approximants are expressed in dependence of vertex coefficients. This yields a basis function for each vertex of the mesh which is smooth and non-zero over a vertex and its adjacent elements. Our numerical simulations indicate the optimal convergence of the resulting approximation scheme for Poisson problems and near optimal convergence for thin-plate and thin-shell problems discretised with structured and unstructured quadrilateral meshes.

Manifold-based isogeometric analysis basis functions with prescribed sharp features

“…The discussion is focused on their application in finite element analysis, so that the underlying manifold concepts from differential geometry and the partition of unity interpolation are only mentioned in passing. For a more comprehensive discussion on manifold-based basis functions and surface construction in geometry we refer to [18,21,25].…”

“…The parameter β can be chosen relatively freely [21]. For instance, when β = 4/v j the scaling term drops and the map Λ I qr is composed of a conformal map and a rotation as in [25]. The second expression in (21) shows that the radius |z| will remain constant when β = 1.…”

“…In the chart domainΩ 2 the arrangement of the crease edges is rotationally symmetric so that it is Type 1 and in the chart domainΩ 1 it is asymmetric so that it is Type 2. For the chartΩ 1 the mapping (25) and for the chartΩ 2 the mapping (19) is to be used. Hence, in comparison to the smooth case illustrated in Figure 6 only the mapping for the chartΩ 1 is different while the other mappings are the same.…”

“…The new chart domains are parameterised with a quasi-conformal map so that each element on it can have a different shape depending on the arrangement of the crease edges. In contrast to the conformal map used in [25] the proposed quasi-conformal map is not anglepreserving but still provides easily computable smooth transition functions. The proposed new construction leads to three different types of chart domains.…”

“…One of them is specifically designed to deal with crease arrangements which lead to creased sectors with concave corners. As in [25] we assemble the smooth partition of unity blending functions from tensor-product B-spline segments defined over a unit square. However, similar to [27], the knot-interval and the B-spline coefficients are chosen such that the assembled blending functions do not require normalisation.…”