Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Abstract

3D objects, modeled using Computer Aided Geometric Design (CAGD) tools, are traditionally represented using a boundary representation (B-rep), and typically use spline functions to parameterize these boundary surfaces. However, recent development in physical analysis, in isogeometric analysis (IGA) in specific, necessitates a volumetric parametrization of the interior of the object. IGA is performed directly by integrating over the spline spaces of the volumetric spline representation of the object. Typically, tensor-product B-spline trivariates are used to parameterize the volumetric domain.A general 3D object, that can be modeled in contemporary B-rep CAD tools, is typically represented using trimmed B-spline surfaces. In order to capture the generality of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi and Elber (2016) proposed the use of trimmed trivariates volumetric elements. However, the use of trimmed geometry makes the integration process more difficult since integration over trimmed B-spline basis functions is a highly challenging task Xu et al. (2017). In this work, we propose an algorithm that precisely decomposes a trimmed B-spline trivariate into a set of (singular only on the boundary) tensor-product B-spline trivariates, that can be utilized to simplify the integration process, in IGA. The trimmed B-spline trivariate is first subdivided into a set of trimmed Bézier trivariates, at all its internal knots. Then, each trimmed Bézier trivariate, is decomposed into a set of mutually exclusive tensor-product B-spline trivariates, that precisely cover the entire trimmed domain. This process, denoted untrimming, can be performed in either the Euclidean space or the parametric space of the trivariate. We present examples of the algorithm on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results.are the control points of T , and B i,d is the i'th univariate B-spline basis functions of degree d.3D geometric objects, generated with contemporary computer aided geometric design (CAGD) systems, are almost solely exploiting a boundary representation (B-rep), where these objects' boundaries are represented as trimmed parametric surfaces. In recent years, with recent developments of additive manufacturing and 3D printing as well as analysis, the demand for a full volumetric representation (V-rep) is increasing. Volumetric modeling of the inside of the object, as oppose to only its boundary (B-rep), can be used to describe different volumetric properties such as materials or stresses, in scalar, vector or tensor fields. Developments in physical analysis, isogeometric analysis (IGA) in specific Cottrell et al. (2009), employs a parametrization of the object's volume. In IGA, the analysis is performed by integrating in the same spline spaces that describe the geometry.Tensor product trivariates are limited to a cuboid topology, making them difficult to use in creating general 3D objects, having an arbitrar...

Isogeometric Analysis on V-reps: First results

“…An alternative approach for a high-order re-parameterization of the trimmed Bézier elements has been recently proposed in [43] for the case of 3D V-reps in which F is a non-trivial map.…”

Sum factorization techniques in Isogeometric Analysis

“…Since (16) and (20) are linear in #X d the bound extends to 1 < s d < p. If s δ < p for some δ = d, then the same bound is achieved by reordering the directions. Lemma 9 yields R ≤ 2 d−1 d p. Thus, we obtain C k p d+1 N, and C app k d p N.…”

“…A more detailed analysis can be performed starting directly from (16) and (20). In both formulas there is a product for δ = 1, .…”

Interpolatory Catmull-Clark volumetric subdivision over unstructured hexahedral meshes for modeling and simulation applications

“…On the other hand, in three-dimensional isogeometric analysis [Lai et al (2017)], volumetric modeling plays a key role as a geometric foundation for numerical simulation [Zhang et al (2007), Xu et al (2017)]. Trivariate NURBS solids [Xu et al (2013a), Xu et al (2013b), Xu et al (2018), Xu et al (2015), Xu et al (2014), Pan et al (2020), Pan et al (2018), Massarwi et al (2019)] and trivariate T-spline solids [Wang et al (2012), Zhang et al (2012), Wang et al (2013), Zhang et al (2013), Liu et al (2014), Wei et al (2017a)] have been used as modeling and numerical tools in isogeometric modeling and analysis. However, because of their tensorproduct structure, trivariate NURBS and T-spline solids have some limitations on the construction of analysis-suitable volumetric parameterizations from arbitrary complex geometries [Wei et al (2017b), Wei et al (2018), Chen et al (2019)].…”