Bézier projection: A unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis

Thomas, Derek C.; Scott, Michael A.; Evans, John A.; Tew, Kevin; Evans, Emily J.

doi:10.1016/j.cma.2014.07.014

Cited by 88 publications

(71 citation statements)

References 88 publications

(137 reference statements)

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“…In addition, this method provides a equivalent treatment for T-spline and NURBS. A detailed explanation of the Bézier extraction can be found in [57][58][59].…”

Section: A Finite Element Structurementioning

confidence: 99%

“…Adopting the idea of the Bézier extraction operation, presented by Borden et al [57] and Scott et al [58], the T-spline and NURBS can be incorporated efficiently via modifying the shape functions in the classical BEM and FEM. Thomas et al [59] developed a generalization of such technique. In recent years, the application of the T-spline in the IGA framework to different areas has been developed [60][61][62][63][64].…”

Section: Introductionmentioning

confidence: 99%

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A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

Maestre¹,

Pallarès²,

Cuesta³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. In this paper we describe a three-dimensional Isogeometric BEM in the time domain based on the T-spline and Non Uniform Rational B-Splines (NURBS) basis to study the free surface behavior. Traditionally, the Lagrange polynomials have been used to discretize the geometry and the BEMs variables. The Lagrange basis are discontinues across the elements although the physical variables are continuous. In dynamics problems with high mesh distortions this approach can produce numerical instabilities that can be quickly propagated. This problem could be overcome using T-spline or NURB basis. The main advantages of this approach are: (1) the control of the continuity and smoothness of the T-spline and NURBS basis, which makes the model numerically stable without the need of artificial smooth techniques;(2) the high order geometrical approximation by non-rational splines;(3) the refinement capabilities without affecting the geometry and BEM's variables; and (4) the direct integration with computer aid geometrical design tools. We use the concept of the Bézier extraction operation which provides an element point of view of the T-Spline and NURBS similar to the traditional finite element. The boundary integral Equation is solved at each time step by the GMRES algorithm and the time marching scheme is performed with a fourth-order Runge-Kutta method to update the model. Additionally, the hydrodynamic force is calculated by an auxiliary boundary equation. Some numerical benchmark examples are analysed to show the accuracy and the stability of the method 7967 J. Maestre, J. Pallarés and I. Cuesta

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“…In addition, this method provides a equivalent treatment for T-spline and NURBS. A detailed explanation of the Bézier extraction can be found in [57][58][59].…”

Section: A Finite Element Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

Maestre¹,

Pallarès²,

Cuesta³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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show abstract

“…As a result, exact geometry is introduced into the analysis. The smooth basis can then be leveraged in analysis [5,6,7] and lead to innovative approaches to model design [8,9,10], analysis [11,12,13,14], optimization [15], and adaptivity [16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

Robust Poisson Surface Reconstruction

Estellers

Scott

Tew

et al. 2015

Lecture Notes in Computer Science

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Abstract. We propose a method to reconstruct surfaces from oriented point clouds with non-uniform sampling and noise by formulating the problem as a convex minimization that reconstructs the indicator function of the surface's interior. Compared to previous models, our reconstruction is robust to noise and outliers because it substitutes the leastsquares fidelity term by a robust Huber penalty; this allows to recover sharp corners and avoids the shrinking bias of least squares. We choose an implicit parametrization to reconstruct surfaces of unknown topology and close large gaps in the point cloud. For an efficient representation, we approximate the implicit function by a hierarchy of locally supported basis elements adapted to the geometry of the surface. Unlike ad-hoc bases over an octree, our hierarchical B-splines from isogeometric analysis locally adapt the mesh and degree of the splines during reconstruction. The hierarchical structure of the basis speeds-up the minimization and efficiently represents clustered data. We also advocate for convex optimization, instead isogeometric finite-element techniques, to efficiently solve the minimization and allow for non-differentiable functionals. Experiments show state-of-the-art performance within a more flexible framework.

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“…Next, we will describe how, using a p-refinement algorithm for NURBS [36], a consistent mixed formulation can be derived using order elevation, Bézier extraction and Bézier projection. Subsequently, weak forms are derived in a standard manner,including the mass coupling between the local scale and the global scale.…”

Section: Introductionmentioning

confidence: 99%

Isogeometric analysis of fluid‐saturated porous media including flow in the cracks

Vignollet

May

Borst

2016

Numerical Meth Engineering

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SUMMARYAn isogeometric model is developed for the analysis of fluid transport in pre-existing faults or cracks that are embedded in a fluid-saturated deformable porous medium. Flow of the interstitial fluid in the porous medium and fluid transport in the discontinuities are accounted for and are coupled. The modelling of a fluid-saturated porous medium in general requires the interpolation of the displacements of the solid to be one order higher than that of the pressure of the interstitial fluid. Using order elevation and Bézier projection, a consistent procedure has been developed to accomplish this in an isogeometric framework. Particular attention has also been given to the spatial integration along the isogeometric interface element in order to suppress traction oscillations that can arise for certain integration rules when a relatively high dummy stiffness is used in a poromechanical model.

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Bézier projection: A unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis

Cited by 88 publications

References 88 publications

A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

A 3d Isogeometrical Boundary Element Analysis for Non-Linear Gravity Wave Propagation

Robust Poisson Surface Reconstruction

Isogeometric analysis of fluid‐saturated porous media including flow in the cracks

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