Development and Implementation of Nagata Patches Interpolation Algorithms

Abstract

The main objective of this work is the development and implementation of Nagata patches interpolation algorithms to be used in the description of tools for the numerical simulation of sheet metal forming. Surface description accuracy is of paramount importance when modelling contact problems. However, most FEM researchers still resort to polyhedral models to describe contact surfaces, which can oversimplify the original system by neglecting the curvature. A simple algorithm for interpolating discretized surfaces and recover the original geometry was recently proposed by Nagata (2005). The main idea behind this parametric surface description is the quadratic interpolation of a curved segment, from the position and normal vectors at the end points. In this work, Nagata patches algorithms are first applied to interpolate polyhedral meshes of simple geometries (cylinder, sphere and torus) where the normal vectors in each node are provided by analytical functions. The use of triangular or quadrilateral Nagata patches is compared, both in terms of efficiency and robustness of the local interpolation algorithm. Afterwards, the interpolation algorithms are applied using different normal vectors approximations, to analyse the influence of the normal vector accuracy in the Nagata interpolation accuracy. Several methods for estimating the normal vector from polyhedral models are analyzed and their efficiency is studied, using the same simple geometries. Finally, the Nagata patch algorithms are applied to interpolate polyhedral meshes, using the interpolation available in the original CAD geometry to estimate the normal vectors. This algorithm allows bridging the gap between CAD and CAE models, since it allows the interpolation of discretized surfaces recovering the original CAD geometry. Tools for Nagata patch visualization and qualitative and quantitative analysis were also developed and presented. Finally, some guidelines for polyhedral mesh generation, in order to guarantee accurate Nagata patch interpolation, are proposed.