Mapped B-spline basis functions for shape design and isogeometric analysis over an arbitrary parameterization

• Presents a novel method for both shape design and isogeometric analysis using mapped basis functions.
• The space spanned by mapped B-spline basis functions is an extension of uniform B-splines over an arbitrary parameterization.
• The continuity of the resulting surfaces can be arbitrary higher order, including at extraordinary points.
• The proposed method can be further extended to other basis functions as well as to non-quadrilateral meshes.

It is well-known that B-spline surfaces are defined by a regular array of control vertices. In case of models with arbitrary topology, it is extremely difficult to maintain continuity conditions among neighboring surfaces. The scenario is also true for applications in isogeometric analysis (IGA). This paper presents a novel method for shape design and isogeometric analysis from a quadrilateral control mesh of arbitrary topology using mapped B-spline basis functions. Based on an arbitrary input quadrilateral control mesh, a global parameterization of the final surface is first defined through a Gravity Center Method (GCM). A re-parameterization method is then applied to map a B-spline basis function to others that are explicitly defined and are 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. For practical implementation, the surface can be evaluated patch by patch. Depending on the order of the B-spline basis function used for mapping to others, the global continuity of the resulting surface, including at extraordinary points, can be arbitrary higher order. In the present paper, a uniform cubic B-spline basis function is used and the resulting surface is globally C2C2 continuous. Several numerical examples are provided to demonstrate the proposed method for both shape design and isogeometric analysis.