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PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping
Singularities in Isogeometric Analysis
Institute of Applied Geometry, Johannes Kepler University, Linz, Austria
T. Takacs, "Singularities in Isogeometric Analysis", in B.H.V. Topping, (Editor), "Proceedings of the Eighth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 45, 2012. doi:10.4203/ccp.100.45
Keywords: isogeometric analysis, computer aided design, singularity, parametrisation, regularity, refinement.
Isogeometric analysis is a method which uses the NURBS based representation from computer aided design (CAD) for numerical simulation. CAD models are represented by non-uniform rational B-splines (NURBS). These NURBS are used to parametrise the physical domain. The test functions for the numerical simulation are defined as the composition of the NURBS on the parameter domain with the inverse of the domain parametrisation. In this paper the influence of singularities in the parametrisation is analysed. If singularities occur, the inverse of the parametrisation is not differentiable. Hence the test function spaces from isogeometric analysis do not necessarily fulfill the desired regularity properties.
The H1 and H2 regularity results are presented for the test functions from isogeometric analysis in the presence of singularities. Three different parametrisations of a triangular domain and four different fillet patches are considered. In the first example one edge of the parameter domain collapses to one corner of the triangle. In the other two examples two edges of the parameter domain map to one edge of the triangle. Hence the parameter lines are collinear. For these triangular patches uniform and adaptive refinement schemes are compared. it is shown that certain local refinement schemes lead to geometrically regular splittings of the singular patches.
It is shown that, depending on the singularity, certain test functions do not possess the desired regularity. Furtheron, modification schemes are devloped to reconstruct test functions that are sufficiently regular. The schemes reproduce test functions that are in H1 or H2, respectively, generating function spaces suitable for numerical analysis. It is concluded that local refinement strategies combined with the presented modification scheme are an important tool to achieve good numerical results for singularly parametrised domains.
It is shown that in the case of fillet patches the choice of the inner control points has an effect on the regularity properties of the test functions. Different parametrisations of the same fillet patch are compared. The parametrisations differ only in the choice of the inner control points and in the specific representation of the boundary. In one case all test functions are in H1, in another case all but one function are not in H1. The findings of this paper suggest that a careful treatment of singularities is necessary. But, if done properly, it leads to a very convenient method for numerical simulation on very general physical domains.
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