Boolean operations on sets using surface data

Satoh, Toshiyuki; Chiyokura, Hiroaki

doi:10.1145/112515.112536

Cited by 20 publications

(6 citation statements)

References 9 publications

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“…The trimming surfaces, S, are then separated by the iso-surface value t, in direction dir, S iso , into two groups. The separation is done by applying B-rep intersection and subtraction Boolean operations Satoh and Chiyokura (1991); Thomas (1986) between S (that forms a closed B-rep), and S iso . See lines 3 and 4 in Algorithm 1, where S L and S R are the result of the intersection and subtraction Boolean operations between S and S iso respectively.…”

Section: Subdivision Into Bézier Trimmed Trivariatesmentioning

confidence: 99%

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Massarwi

Antolín

Elber

2019

Computer Aided Geometric Design

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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...

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Section: Subdivision Into Bézier Trimmed Trivariatesmentioning

confidence: 99%

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Massarwi

Antolín

Elber

2019

Computer Aided Geometric Design

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“…Earlier techniques for offsetting 3D models (Rossignac and Requicha, 1986) (Satoh and Chiyokura, 1991) (Forsyth , 1995) are computationally expensive and unstable. To overcome the difficulties, new offsetting methods based on the discrete representation of the 3D model have become popular.…”

Section: Prior Workmentioning

confidence: 99%

Simple offset algorithm for generating workpiece solid model for milling simulation

Inui

Umezu

Tsukahara³

2017

JAMDSM

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In the milling simulation of a large stamping die for an automobile part, a workpiece solid model is necessary as an input data. The initial shape of the workpiece is generally equivalent to an offset shape of the stamping die CAD model. Accurate offset shape is not a requirement on the workpiece solid model for the milling simulation. In this paper, we propose a novel method named "Simple Offset" for fast and stable generation of a simplified offset shape of a polyhedral solid model. In this method, surface points of the offset shape are sampled using 3 mutually perpendicular grids in the model space. The final offset shape is obtained by properly connecting the sampled points. Simplification level of the offset shape can be controlled by changing the resolution of the grids for sampling the surface points. An experimental system is implemented and some computation results are demonstrated.

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“…Rossignac and Requicha (1986) studied filleting and rounding based on regularized sets and presented their relations to surface offsetting. However, most earlier work on surface offsetting such as (Rossignac and Requicha 1986) (Farouki 1985) (Satoh and Chiyokura 1991) (Frosyth 1995) all offset surfaces of models first, then trim or extend these offset surfaces to reconstruct a closed 3D model. Due to the complexity of trimming and extending operations, the approach is difficult to implement robustly.…”

Section: Review Of Related Workmentioning

confidence: 99%

Filleting and Rounding Using a Point-Based Method

Chen¹,

Wang

Rosen

et al. 2005

Volume 2: 31st Design Automation Conference, Parts a and B

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Rounds and fillets are important design features. We introduce a new point-based method for constant radius rounding and filleting. Based on the mathematical definitions of offsetting operations, discrete offsetting operations are introduced. Steps of our approach are discussed and analyzed. The methodology has been implemented and tested. We present the experimental results on accuracy, memory and running time for various input geometries and radius. Based on the test results, the method is very robust for all kinds of geometries.

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Boolean operations on sets using surface data

Cited by 20 publications

References 9 publications

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Volumetric untrimming: Precise decomposition of trimmed trivariates into tensor products

Simple offset algorithm for generating workpiece solid model for milling simulation

Filleting and Rounding Using a Point-Based Method

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