Conclusion

The resulting interpolants are smooth and bounded within their respective domains and the sum of any set of interpolants is one. The functions are linearly independent. Examples of the smooth and bounded behavior of the functions are shown in the figures. The convex polygon shape functions automatically satisfy constant and linear fields. This is not the case for the concave or multiply connected domain representation. Nevertheless, the linearity constraint can be imposed on the representation.

This type of interpolant is distinctly different from the available interpolant and shape function formulations, since each interpolant is constructed to satisfy the required conditions exactly: smoothness, boundedness, linearity on sides, and the linear field conditions. Using this algebraic construction, closed-form interpolants satisfying different field properties can also be derived.