Generating high-quality high-order parameterization for isogeometric analysis on triangulations

This paper presents an approach for automatically generating high-quality high-order parameterizations for isogeometric analysis on triangulations. A B-spline represented boundary geometry is parameterized into a collection of high-order Bézier triangles or tetrahedra in 2D and 3D spaces, respectively. Triangular Bézier splines are used to represent both the geometry and physical fields over the triangulation. By imposing continuity constraints on the Bézier ordinates of the elements, a set of global Cr smooth basis is constructed and used as the basis for analysis. To ensure high quality of the parameterization, both the parametric and physical meshes are optimized to reduce the shape distortion of the high-order elements relative to well-defined reference elements. The shape distortion is defined based on the Jacobian of the triangular Bézier splines, and its sensitivity with respect to the location of control points is derived analytically and evaluated efficiently. Moreover, a sufficient condition is derived to guarantee the generated mesh is free of local self-intersection, thanks to the convex hull property of triangular Bézier splines. By using a Heaviside projection function, the non-negative Jacobian determinant constraints are formulated efficiently as a single optimization constraint. Several 2D and 3D numerical examples are presented to demonstrate that high-quality high-order elements are generated using our approach.