Computer Science Department
School of Computer Science, Carnegie Mellon University


A Bézier-Based Approach to Unstructured Moving Meshes

David Cardoze, Alexandre Cunha*, Gary L. Miller,
Todd Phillips**, Noel Walkington**

September 2003

Keywords: Mesh generation, computational geometry, Bézier curves, Bézier triangles, B-splines, finite element method, quadratic elements

We present in this report a new framework for maintaining good quality of two dimensional triangular moving meshes. The use of curved elements is the key idea that allows us to avoid excessive re nement and still obtain good quality meshes consisting of a low number of well shaped elements. We use B-splines curves to model object boundaries and objects are meshed with second order Bézier triangles. As the mesh evolves according to a non-uniform flow velocity field, we keep track of object boundaries and, if needed, carefully modify the mesh to keep it well shaped by applying a combination of vertex insertion and deletion, edgeflipping, and curve smoothing operations at each time step. Our algorithms for these tasks are extensions of known algorithms for meshes build of straight-sided elements and are designed for any fixed-order Bézier elements and B-splines. We discuss a calculus of geometric primitives for Bézier curves and triangles that we employ to implement such operations. Although in this work we have concentrated on quadratic elements, most of the operations are valid for elements of any order and they generalize well to higher dimensions. We present results of our scheme for a set of objects mimicking red blood cells subject to a a priori computed flow velocity field. As a pure geometric exploration, our method does not account for neither refinement nor coarsening dictated by the simulation results.

21 pages

*Laboratory for Mechanics, Algorithms and Computing, Carnegie Mellon University
**Department of Mathematics, Carnegie Mellon University

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