Shape design and isogeometric analysis over an arbitrary parameterization using mapped basis functions

This talk presents a novel method for shape design and isogeometric analysis from a quadrilateral control mesh of arbitrary topology using mapped basis functions. Based on an arbitrary input quadrilateral control mesh, a global parametrization of the final surface is first defined through a Gravity Center Method (GCM). A re-parametrization method is then applied to map a given basis function to others tailored to each of the control vertices which can be either regular or extraordinary ones. The final surface is defined by all the input control vertices with their corresponding mapped basis functions patch by patch. Depending on the continuity of the original basis function used for mapping to others, the global continuity of the resulting surface, including at extraordinary positions, can be of an arbitrarily higher order. In this talk, a uniform cubic B-spline basis function is used to illustrate the method and the resulting surface is globally continuous. Several examples are provided to demonstrate the proposed method for both shape design and isogeometric analysis. Other basis functions can also be used for mapping purpose and the method can also be extended to non-quadrilateral control meshes.