Geração de malhas de alta ordem para análise isogeométrica utilizando elementos de Bézier racionais

Abstract

Isogeometric Analysis (IGA) is a numerical method that has been receiving increasing attention in the last decade. The main goal of IGA is to closely couple geometric modeling with numerical analysis in Computer-Aided Engineering (CAE) systems. To that end, in IGA both components use the same geometric representation, e.g., Bézier curves and surfaces, NURBS, and T-Splines. However, in many cases, especially in CAE systems based on the Finite Element Method, the geometric representation in a CAD system cannot be directly employed in numerical simulation, as only the boundary of the geometry is parametrized. In these cases, an interior parametrization must be constructed before performing isogeometric analysis. The domain parametrization techniques for NURBS and T-Splines are not enough robust to handle complex geometries, e.g. geometries with multiple holes and narrow regions. In addition, the use of trimmed surfaces in IGA has considerable complexity, either in the construction of their parameterization or when these models are employed directly in numerical simulations. On the other hand, the use of high-order unstructured mesh generator algorithms to construct domain parametrization is more robust in these cases. This work presents an algorithm for generation of unstructured geometrically exact meshes composed of rational Bézier triangles of arbitrary degree and applies it to plane models described by NURBS curves in a Boundary-Representation (B-Rep) scheme. The proposed algorithm is capable of generating high quality coarse meshes even when high curvature segments are considered. The proposed algorithm attains superior performance when compared to a well-known algorithm in the literature and produces meshes with similar quality in comparison to meshes obtained through quality optimization. The algorithm is generalized to surface models in 3D, where geometry is given by trimmed NURBS. Structured meshing algorithms for generation of rational Bézier triangles and quadrilaterals are also presented. Moreover, a shell formulation based on degenerated solid for rational Bézier elements is presented. The algorithms developed are used in elasticity and heat transfer problems, and static and free vibration analysis of shells, using the elements developed in this work. The performance of rational Bézier elements is assessed in several numerical examples, demonstrating its convergence under mesh refinement.