Isogeometric triangular Bernstein-Bezier discretizations: Automatic mesh generation and geometrically exact finite element analysis

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-Bezier 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-Bezier triangles. In addition to presenting finite element analysis methodologies based on rational Bernstein-Bezier 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. (C) 2016 Elsevier B.V. All rights reserved.