Isogeometric triangular Bernstein–Bézier discretizations: Automatic mesh generation and geometrically exact finite element analysis

Abstract

Isogeometric analysis (IGA) was introduced as a way to bypass the design-to-analysis bottleneck inherent in the traditional Computer Aided Design (CAD) through Finite Element Analysis (FEA) paradigm. However, an outstanding problem in the field of IGA is that of surface-to-volume parameterization. In CAD packages, solid objects are represented by a collection of NURBS or T-spline bounding surfaces, but to perform engineering analysis on 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 IGA, as the tensor-product nature of trivariate NURBS and T-splines limit their ability to create analysis suitable parameterizations of arbitrarily complex volumes.To overcome the limitations of trivariate NURBS and T-splines, we propose the use of rational Bernstein–Bézier tetrahedra to create analysis suitable volumetric parameterizations for isogeometric analysis. In this paper, which is part one of a two part series, we present the methodology for discretizing two dimensional geometries using rational Bernstein–Bézier triangles. In addition to presenting finite element analysis methodologies based on rational Bernstein–Bézier triangles, we also introduce two new mesh generation strategies for automatically creating high quality, geometrically exact curvilinear meshes. We assess the quality of our mesh generation schemes using a suite of challenging two-dimensional geometries, and we verify the accuracy of our proposed numerical discretization approach using the method of manufactured solutions.