Optimizing domain parameterization in isogeometric analysis based on Powell–Sabin splines

Abstract

Please cite this article as: H. Speleers, C. Manni, Optimizing domain parameterization in isogeometric analysis based on Powell-Sabin splines, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractWe address the problem of constructing a high-quality parameterization of a given planar physical domain, defined by means of a finite set of boundary curves. We look for a geometry map represented in terms of Powell-Sabin B-splines. Powell-Sabin splines are C 1 quadratic splines defined on a triangulation, and thus the parameter domain can be any polygon.The geometry map is generated by the following three-step procedure. First, the shape of the parameter domain and a corresponding triangulation are determined, in such a way that its number of corners matches the number of corners of the physical domain. Second, the boundary control points related to the Powell-Sabin B-spline representation are chosen so that they parameterize the boundary curve of the physical domain. Third, the remaining inner control points are obtained by solving a nimble optimization problem based on the Winslow functional.The proposed domain parameterization procedure is illustrated numerically in the context of isogeometric Galerkin discretizations based on Powell-Sabin splines. It turns out that the flexibility rising from the generality of the parameter domain has a beneficial effect on the quality of the parameterization and also on the accuracy of the computed approximate solution.

Adaptive isogeometric analysis with hierarchical box splines

“…Box spline geometries may circumvent this problem but the development of optimal domain parameterizations in this context deserves a separate study and it is beyond the scope of this paper. For related work on domain parameterizations using splines on triangulations we refer to [10,31]. The aim of the current manuscript is solely to show the potential of the hierarchical box spline framework, suitably combined with simple geometric mappings, with respect to standard tensor-product counterparts.…”

On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

“…These macro-triangles are used in interpolation problems over triangulations, see e.g. [3,11], and recently also in Isogeometric analysis [35]. Typically, Gaussian quadrature for quadratic polynomials (10) is used over every micro-triangle.…”

Structural Design Optimization Using Isogeometric Analysis: A Comprehensive Review

“…Representative parameterization methods include analysis suitable T-splines in Scott et al [Scott, Li, Sederberg et al (2012); da Veiga, Buffa, Cho et al 2012], an analysis-suitable parameterization framework using harmonic method in Xu et al [Xu, Mourrain, Duvigneau et al (2013)], a method of using mapped B-spline basis functions in [Yuan and Ma (2014)] and an optimized trivariate B-spline solids parameterization approach in Wang et al [Wang and Qian (2014)]. Representative analysis techniques for shape design optimization problems in-clude the T-Spline based IGA in Ha et al [Ha, Choi and Cho (2010) [Speleers and Manni (2015)], isogeometric B-Rep analysis for trimmed surfaces in Philipp et al [Philipp, Breitenberger, D'Auria et al (2016)], an immersed method termed immersogeometric methods in Wu et al [Wu, Kamensky, Wang et al (2017)], an combination of immersed method and bound-ary method in Marco et al [Marco, Ródenas, Fuenmayor et al (2018)], and triangulations based IGA in Wang et al [Wang, Xia, Wang et al (2018)]. It should be noted that among these methods, the boundary element method is popular for its certain advantages over the domain-based methods, e.g.…”