Tetrahedral Partitions Research Articles

Reconstruction of volume data with quadratic super splines

We propose a new approach to reconstruct nondiscrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition. We discuss the smoothness and approximation properties of our model and compare to alternative piecewise polynomial constructions. We observe, as a nonstandard phenomenon, that the derivatives of our splines yield optimal approximation order for smooth data, while the theoretical error of the values is nearly optimal due to the averaging rules. Our approach enables efficient reconstruction and visualization of the data. As the piecewise polynomials are of the lowest possible total degree two, we can efficiently determine exact ray intersections with an isosurface for ray-casting. Moreover, the optimal approximation properties of the derivatives allow us to simply sample the necessary gradients directly from the polynomial pieces of the splines. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.

Acute Type Refinements of Tetrahedral Partitions of Polyhedral Domains

We present a new technique to perform refinements on acute type tetrahedral partitions of a polyhedral domain, provided that the center of the circumscribed sphere around each tetrahedron belongs to the tetrahedron. The resulting family of partitions is of acute type; thus, all the tetrahedra satisfy the maximum angle condition. Both these properties are highly desirable in finite element analysis.

Nonobtuse tetrahedral partitions

We show how to generate refinements of tetrahedral partitions, where no obtuse angles appear. Such partitions play an important role in deriving discrete maximum principles and maximum norm error estimates for the finite element method. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 327–334, 2000