Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products

3D objects, modeled using Computer Aided Geometric Design 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. 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 B\'ezier trivariates, at all its internal knots. Then, each trimmed B\'ezier 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 on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results.Comment: 18 pages, 32 figures. Contribution accepted in International Conference on Geometric Modeling and Processing (GMP 2019

Special Session on Industry 4.0

No abstract available

Special Session on Industry 4.0

No abstract available

WISDOM Project - X. The morphology of the molecular ISM in galaxy centres and its dependence on galaxy structure

We use high-resolution maps of the molecular interstellar medium (ISM) in the centres of 86 nearby galaxies from the millimetre-Wave Interferometric Survey of Dark Object Masses (WISDOM) and Physics at High Angular Resolution in Nearby GalaxieS (PHANGS) surveys to investigate the physical mechanisms setting the morphology of the ISM at molecular cloud scales. We show that early-type galaxies tend to have smooth, regular molecular gas morphologies, while the ISM in spiral galaxy bulges is much more asymmetric and clumpy when observed at the same spatial scales. We quantify these differences using non-parametric morphology measures (Asymmetry, Smoothness, and Gini), and compare these measurements with those extracted from idealized galaxy simulations. We show that the morphology of the molecular ISM changes systematically as a function of various large-scale galaxy parameters, including galaxy morphological type, stellar mass, stellar velocity dispersion, effective stellar mass surface density, molecular gas surface density, star formation efficiency, and the presence of a bar. We perform a statistical analysis to determine which of these correlated parameters best predicts the morphology of the ISM. We find the effective stellar mass surface (or volume) density to be the strongest predictor of the morphology of the molecular gas, while star formation and bars maybe be important secondary drivers. We find that gas self-gravity is not the dominant process shaping the morphology of the molecular gas in galaxy centres. Instead effects caused by the depth of the potential well, such as shear, suppression of stellar spiral density waves, and/or inflow, affect the ability of the gas to fragment

MATHICSE Technical Report: Volumetric Untrimming: Precise decomposition of trimmed trivariates into tensor products

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 specic, 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 B--spline trivariates, at all its internal knots. Then, each trimmed Bezier 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

Reduced order modeling for physically-based augmented reality

In this work we explore the possibilities of reduced order modeling for augmented reality applications. We consider parametric reduced order models based upon separate (affine) parametric dependence so as to speedup the associated data assimilation problems, which involve in a natural manner the minimization of a distance functional. The employ of reduced order methods allows for an important reduction in computational cost, thus allowing to comply with the stringent real time constraints of video streams, i.e., around 30 Hz. Examples are included that show the potential of the proposed technique in different situations