On the linear independence of truncated hierarchical generating systems

Zore, Urška; Jüttler, Bert; Kosinka, Jiří

doi:10.1016/j.cam.2016.04.014

Cited by 16 publications

(19 citation statements)

References 30 publications

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“…Among the most popular subdivision schemes are the Catmull-Clark [20] and Doo-Sabin [21] schemes on quadrilateral meshes, and Loop's scheme on triangular meshes [22]. The Loop subdivision basis functions have been extensively studied, most interestingly regarding their smoothness [23,24], (local) linear independence [25,26], approximation power [27], and the robust evaluation around so called extraordinary vertices [28].…”

Section: Introductionmentioning

confidence: 99%

On numerical integration in isogeometric subdivision methods for PDEs on surfaces

Jüttler

Mantzaflaris

Perl

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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Subdivision surfaces offer great flexibility in capturing irregular topologies combined with higher order smoothness. For instance, Loop and Catmull-Clark subdivision schemes provide C 2 smoothness everywhere except at extraordinary vertices, where the generated surfaces are C 1 smooth. The combination of flexibility and smoothness leads to their frequent use in geometric modeling and makes them an ideal candidate for performing isogeometric analysis of higher order problems with irregular topologies, e.g. thin shell problems. In the neighborhood of extraordinary vertices, however, the resulting surfaces (and the functions defined on them) are non-polynomial. Thus numerical integration based on quadrature has to be performed carefully to preserve the expected higher order consistency. This paper presents a detailed case study of different quadrature schemes for isogeometric discretizations of partial differential equations on closed surfaces with Loop's subdivision scheme. As model problems we consider elliptic equations with the Laplace-Beltrami and the surface bi-Laplacian operator as well as the associated eigenvalue problem for the Laplace-Beltrami operator. A particular emphasis is on the robustness of the approach in the vicinity of extraordinary vertices. Based on a series of numerical experiments, different quadrature schemes are compared. Mid-edge quadrature, which can easily be implemented via a lookup table, turns out to be a preferable choice due to its robustness and efficiency.

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Section: Introductionmentioning

confidence: 99%

On numerical integration in isogeometric subdivision methods for PDEs on surfaces

Jüttler

Mantzaflaris

Perl

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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“…One of the natural candidates is subdivision surfaces with the Catmull-Clark subdivision scheme [2]. Recently, (truncated) hierarchical refinement has been presented for this basis in [22,20]. One of the freely available CAGD packages in which subdivision surface geometries can be designed is Blender TM , which has been used in this context.…”

Section: Subdivision Surface Approximation Spacesmentioning

confidence: 99%

Finite-Element/Boundary-Element Coupling for Inflatables: Effective Contact Resolution

Opstal

2016

Advances in Computational Fluid-Structure Interaction and Flow Simulation

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A fluid-structure interaction technique for the simulation of inflatable structures is introduced. The presented finite-element/boundary-element (FEBE) technique couples an isogeometric finite element discretization of a flexible shell structure with an isogeometric boundary element discretization of a Stokes fluid. This technique was introduced in [10] for planar problems and extended to 3D in [11]. One of the marked advantages of this approach is the contact mechanism, lubrication, which is an inherent attribute of the flow model. Its role in preventing contact was theoretically substantiated, but only demonstrated in the planar setting. In the present work, we demonstrate the effectiveness of the contact mechanism in the isogeometric and three-dimensional setting, discussing various aspects pertaining to the accurate resolution of the traction responsible for contact prevention.An extensive body of literature already covers the interface of the fields of fluidstructure interaction and contact mechanics. If the problem under consideration requires that the gap between two contact surfaces may vanish, an interface tracking technique may be applied, sometimes locally [5, 19], e.g. [3, 7, 21]. If however, a problem allows for a finite gap to be maintained between the contacting surfaces, Arbitrary Lagrangian-Eulerian [4] and Space-time [18] techniques can been used to compute problems as challenging as the disreefing of parachute clusters [15,16,17] and 1000 spheres falling through a tube [6]. In this work a finite (but arbitrarily small) gap will likewise be maintained in contact regions.The target application for the current approach is inflatable structures. These typically undergo large deformations with ubiquitous self-contact during the inflation process, which often starts from a complex, folded initial configuration. Correct simulation is contingent to the resolution of every single contact mode throughout the

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“…Therefore, partition of unity needs to be satisfied when developing different types of basis functions for isogeometric analysis. It can be satisfied by introducing a truncation mechanism to hierarchical B-splines [7] or subdivision surfaces [8,9]. The truncated B-spline basis functions decrease the overlapping of basis functions from different hierarchical levels.…”

Section: Introductionmentioning

confidence: 99%