3,576 research outputs found
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
Lagrangian submanifolds from tropical hypersurfaces
We prove that a smooth tropical hypersurface in can be lifted
to a smooth embedded Lagrangian submanifold in . This
completes the proof of the result announced in the article "Lagrangian pairs
pants" arXiv:1802.02993. The idea of the proof is to use Lagrangian pairs of
pants as the main building blocks.Comment: 59 pages, 14 Figures. This article completes the proof of the result
announced in "Lagrangian pairs pants" arXiv:1802.0299
A simple approach to the numerical simulation with trimmed CAD surfaces
In this work a novel method for the analysis with trimmed CAD surfaces is
presented. The method involves an additional mapping step and the attraction
stems from its sim- plicity and ease of implementation into existing Finite
Element (FEM) or Boundary Element (BEM) software. The method is first verified
with classical test examples in structural mechanics. Then two practical
applications are presented one using the FEM, the other the BEM, that show the
applicability of the method.Comment: 20 pages and 16 figure
Conversion of trimmed NURBS surfaces to Catmull-Clark subdivision surfaces
This paper introduces a novel method to convert trimmed NURBS surfaces to untrimmed subdivision surfaces with Bézier edge conditions. We take a NURBS surface and its trimming curves as input, from this we automatically compute a base mesh, the limit surface of which fits the trimmed NURBS surface to a specified tolerance. We first construct the topology of the base mesh by performing a cross-field based decomposition in parameter space. The number and positions of extraordinary vertices required to represent the trimmed shape can be automatically identified by smoothing a cross field bounded by the parametric trimming curves. After the topology construction, the control point positions in the base mesh are calculated based on the limit stencils of the subdivision scheme and constraints to achieve tangential continuity across the boundary. Our method provides the user with either an editable base mesh or a fine mesh whose limit surface approximates the input within a certain tolerance. By integrating the trimming curve as part of the desired limit surface boundary, our conversion can produce gap-free models. Moreover, since we use tangential continuity across the boundary between adjacent surfaces as constraints, the converted surfaces join with G1 continuity. © 2014 The Authors.EPSRC, Chinese Government (PhD studentship) and Cambridge Trust
Isogeometric Analysis on V-reps: first results
Inspired by the introduction of Volumetric Modeling via volumetric
representations (V-reps) by Massarwi and Elber in 2016, in this paper we
present a novel approach for the construction of isogeometric numerical methods
for elliptic PDEs on trimmed geometries, seen as a special class of more
general V-reps. We develop tools for approximation and local re-parametrization
of trimmed elements for three dimensional problems, and we provide a
theoretical framework that fully justify our algorithmic choices. We validate
our approach both on two and three dimensional problems, for diffusion and
linear elasticity.Comment: 36 pages, 44 figures. Reviewed versio
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