Mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities
The Method of Auxiliary Mapping (MAM), introduced by Babuka and Oh [2], is an effective method for dealing with singularities in elasticity [16]. MAM was extended to boundary element method (BEM) in the framework of mesh free particle methods [17], reproducing polynomial particle methods [20], and also to infinite domain problems [18]. Similarly, we consider NURBS geometrical mappings that are able to generate crack singularities for isogeometric analysis of elliptic boundary value problems. However, the mapping techniques proposed in this paper are different from MAM. In order to generate singular shape functions, MAM uses conformal mappings that locally change the physical domain, whereas the NURBS mappings used for design of engineering system are not allowed to alter the physical domain for isogeometric analysis. Moreover, unlike MAM, the proposed method makes it possible to independently control the radial and angular direction of the function to be approximated as far as the point singularities are concerned. We prove error estimates in Sobolev norms and demonstrate that the proposed mapping technique is highly effective for isogeometric analysis of elliptic boundary value problems with singularities. Mesh refinements to deal with singularities are compared with the mapping technique in the isogeometic analysis framework.
mapping techniques isogeometric analysis nurbs geometrical mapping b-spline basis functions knot vectors control points