Isogeometric unstructured tetrahedral and mixed-element Bernstein-Bézier discretizations

A well-known problem in the field of isogeometric analysis is that of surface-to-volume parameterization. In CAD packages, solid objects are represented by a collection of NURBS or T-spline surfaces, but to perform engineering analysis for many real world problems, we must find a way to parameterize the volumes of these objects as well. This has proven to be difficult using traditional isogeometric methods, as the tensor-product nature of trivariate NURBS and T-splines limit their ability to provide analysis suitable parameterizations of arbitrarily complex volumes. To overcome the limitations of trivariate NURBS and T-splines, we propose the use of unstructured Bernstein-Bézier discretizations. In earlier work, we demonstrated the feasibility of this approach in two dimensions through the construction of an automatic mesh generation environment capable of generating geometrically exact unstructured meshes comprised of rational Bézier triangles. This paper extends the concepts presented in our previous work to unstructured tetrahedral and mixed-element meshes in three dimensions. The main contributions of this paper are three-fold. First, we present a framework for creating geometrically exact meshes comprised of rational Bézier hexahedra, tetrahedra, wedges and pyramids through degree elevation of suitable linear meshes. We additionally discuss how our approach may be applied to higher-order mesh generation for more traditional finite element approaches. Next, we propose two quality metrics for three-dimensional curvilinear meshes, and discuss some important considerations for mesh quality of higher-order meshes. Finally, we demonstrate the analysis suitability of the meshes constructed using our framework through numerical examples.