Transfinite Patches for Isogeometric Analysis

This paper extends the well-known transfinite interpolation formula, which was developed in late sixties by the applied mathematician William Gordon at the premises of General Motors as an extension of the pre-existed Coons interpolation formula. Here, a conjecture is formulated, which claims that the meaning of the involved blending functions can be enlarged, so that to include any linear independent and complete set of functions, including piecewise-linear, trigonometric functions, Bernstein polynomials, B-splines, NURBS and so on. In this sense, NURBS-based isogeometric analysis and aspects of T-splines may be considered as special cases. Applications are provided for the accuracy in the interpolation through the L2-error norm, of closed-formed functions prescribed at the nodal points of the transfinite patch, which represent the solution of partial differential equations under boundary conditions of Dirichlet type.