Isogeometric analysis on triangulations

Jaxon, Noah; Qian, Xiaoping

doi:10.1016/j.cad.2013.08.017

Cited by 78 publications

(61 citation statements)

References 33 publications

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“…However, when we are interested in the region of space bounded (or excluded) by a shape, for example when performing an engineering simulation (Cottrell et al, 2009;Jaxon and Qian, 2014) or computing a medial axis transform (Bucklow, 2014), our task is often made difficult by the deficiencies of such a B-rep. For example, the intersection curve of two ✩ This paper has been recommended for acceptance by Gerald Farin. NURBS surfaces is not, in general, a NURBS curve, leading to unavoidable gaps in trimmed NURBS models (Skytt and Vuong, 2013). Although the gaps can be made arbitrarily small, the result is still a discontinuous representation of shape.…”

Section: Introductionmentioning

confidence: 99%

Watertight conversion of trimmed CAD surfaces to Clough–Tocher splines

Kosinka

Cashman²

2015

Computer Aided Geometric Design

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The boundary representations (B-reps) that are used to represent shape in ComputerAided Design systems create unavoidable gaps at the face boundaries of a model. Although these inconsistencies can be kept below the scale that is important for visualisation and manufacture, they cause problems for many downstream tasks, making it difficult to use CAD models directly for simulation or advanced geometric analysis, for example. Motivated by this need for watertight models, we address the problem of converting B-rep models to a collection of cubic C 1 Clough-Tocher splines. These splines allow a watertight join between B-rep faces, provide a homogeneous representation of shape, and also support local adaptivity. We perform a comparative study of the most prominent Clough-Tocher constructions and include some novel variants. Our criteria include visual fairness, invariance to affine reparameterisations, polynomial precision and approximation error. The constructions are tested on both synthetic data and CAD models that have been triangulated. Our results show that no construction is optimal in every scenario, with surface quality depending heavily on the triangulation and parameterisation that are used.

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Section: Introductionmentioning

confidence: 99%

Watertight conversion of trimmed CAD surfaces to Clough–Tocher splines

Kosinka

Cashman²

2015

Computer Aided Geometric Design

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“…Therefore, they are a natural choice for the functions in (2.4), where the reference domain Ω 0 is the unit square. On the other hand, splines defined on triangulations are an interesting alternative to tensor-product B-splines/NURBS in IgA [17,34,35], because they inherently support local refinement and they offer more flexibility in choosing the shape of the parameter domain.…”

Section: Isogeometric Analysismentioning

confidence: 99%

“…Examples are T-splines [3,5], hierarchical splines [12,38], LR-splines [8,18], and splines on triangulations [17,34,35].…”

Section: Introductionmentioning

confidence: 99%

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

Speleers

Manni

2015

Journal of Computational and Applied Mathematics

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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.

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“…This work is focused on another approach, based on the definition of bivariate splines on irregular triangulations, that is, Powell-Sabin (PS) splines [5,7,8]. PS splines are piecewise quadratic polynomial with C 1 continuity, defined on a unstructured triangulation of the domain.…”

Section: Introductionmentioning

confidence: 99%

A Powell-Sabin finite element scheme for partial differential equations

Giorgiani

Guillard²,

Nkonga³

2016

ESAIM: Proc.

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Abstract. In this paper are analyzed finite element methods based on Powell-Sabin splines, for the solution of partial differential equations in two dimensions. PS splines are piecewise quadratic polynomials defined on a triangulation of the domain, and exhibit a global C 1 continuity. Critical issues when dealing with PS splines, and described in this work, are the construction of the shape functions and the imposition of the boundary conditions. The PS finite element method is used at first to solve an elliptic problem describing plasma equilibrium in a tokamak. Finally, a transient convective problem is also considered, and a stabilized formulation is presented.

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Isogeometric analysis on triangulations

Cited by 78 publications

References 33 publications

Watertight conversion of trimmed CAD surfaces to Clough–Tocher splines

Watertight conversion of trimmed CAD surfaces to Clough–Tocher splines

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

A Powell-Sabin finite element scheme for partial differential equations

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