We introduce the (q, h)-blossom of bivariate polynomials, and we define the bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces on rectangular domains using the tensor product. Using the (q, h)-blossom, we construct recursive evaluation algorithms for (q, h)-Bézier surfaces and we derive a dual functional property, a Marsden identity, and a number of other properties for bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces. We develop a subdivision algorithm for (q, h)-Bézier surfaces with a geometric rate of convergence. Recursive evaluation algorithms for quantum (q, h)-partial derivatives of bivariate polynomials are also derived.
The convergence and the accuracy of numerical solutions to partial differential equations strongly depend on the quality of the grids on which these solutions are computed. First and second order quality metrics for hexahedral grids cells are formulated and applied to evaluate the quality of three-dimensional geologic grid structures. Examples measuring the Jacobian and the orthogonality of geologic grids are given.
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