Meshing, Geometric Modeling and Numerical Simulation 3

Chapter 9Some Algorithms and Formulas

As in the first two volumes, a few formulas and algorithms are given here related to the topics presented in this volume. Some of these formulas are already available, scattered throughout the different volumes, but it seemed interesting to gather them here to avoid having to look for them elsewhere. This is the case for the Bernstein polynomials and the Bézier forms given here at the beginning of the first section. This section ends with the expression of surfaces (volumes) of curved elements, given in Chapter 6 for some elements only.

We then revisit localization problems. Addressed in Volume 1 for triangulations (therefore of degree 1), we look at what happens for curved meshes. This will be an opportunity to see how to find the coordinates in the parameter space of a current point of an element, which is again a trivial problem for simplexes (degree 1) and already more sensitive, even for a quadrilateral element (of degree 1 × 1) before even considering other degrees.

Space-filling curves, seen in Volume 1 for insertion algorithms, reviewed in Volume 2 under the topic of parallelism and also found in this volume regarding renumbering (and partitioning) methods, have not been detailed in terms of their actual construction; this is therefore an opportunity to come back to that point.

9.1. Bernstein polynomials and Bézier forms

9.1.1. Bernstein polynomials

Their expressions depend on the chosen system of parameters, namely barycentric ...