Bicubic B-spline blending patches with optimized shape

Two constructions of bicubic B-spline patches with fixed boundary conditions are described. Their goal is to minimize functionals taken for measures of patch badness. The first construction is numerically solving the triharmonic equation −Δ3p=0. The functional minimized in the second construction is the sum of a term determined by the surface shape (the distribution of mean curvature) and a term introduced to overcome the problem of ambiguity of minimum of the first term. In addition to boundary conditions one can impose constraints, e.g. fix constant parameter curves of the patch.

Research highlights
► Constructions of smooth blending patches by shape optimization are proposed.
► The boundary conditions for the surface make it possible to obtain the curvature continuity.
► It is possible to impose constraints, e.g. interpolation conditions, in order to improve the construction results.
► The hernia effect, related to the optimization criteria, and methods for its compensation, are discussed.
► The numerical optimization procedure has a high convergence rate and it is suitable for a parallel implementation.