Isogeometric Divergence-conforming B-splines for the Steady Navier-Stokes Equations

Evans, John A.; Hughes, Thomas J.

doi:10.21236/ada560496

Cited by 58 publications

(90 citation statements)

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“…In summary, the only promising weakly divergence-free pair of standard finite element spaces is P k /P disc k−1 , k = d, on barycentric-refined grids. In the context of IGA there are also proposals for constructing discrete spaces that allow the computation of weakly divergence-free velocity fields [42,70,71]. However, for incompressible flow problems there seems to be a similar situation as described for scalar convection-diffusion equations: compared with standard finite element methods, only rather few contributions with IGA can be found in the literature in recent years, e.g., in [85,43].…”

Section: The Stokes Equationmentioning

confidence: 99%

“…Hence, the explicit dependency on the viscosity is reduced from ν −3 to ν −1 . If one considers weakly divergence-free and inf-sup stable pairs of finite element spaces, like the Scott-Vogelius pair P k /P disc k−1 , k ≥ d, on barycentric-refined grids or weakly divergence-free IGA methods from [42,70,71], one can use the standard form n conv (·, ·, ·) of the convective term in the error analysis. That means, one has to bound only one term and not two terms as in other skew-symmetric formulations of the convective term.…”

Section: Time-dependent Navier-stokes Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

John

Knobloch

Novo

2018

Comput. Visual Sci.

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Section: The Stokes Equationmentioning

confidence: 99%

Section: Time-dependent Navier-stokes Equationsmentioning

confidence: 99%

Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

John

Knobloch

Novo

2018

Comput. Visual Sci.

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“…The construction of multi-patch NURBS discretizations with C 0 continuity at patch boundaries is conceptually straightforward. Constructing multi-patch divergence-conforming B-spline discretizations requires the use of a discontinuous Galerkin framework at patch boundaries to enforce tangential continuity in this region while maintaining the stability and conservation properties of the single-patch divergence-conforming B-spline discretization as explained and tested in [48,49].…”

Section: Future Workmentioning

confidence: 99%

“…To the best of our knowledge, the only work that has tried pointwise divergence-free Eulerian discretization in the context of immersed FSI methods is the recent paper [46]. In [46], divergence-conforming B-splines [47,48,49,50] were applied to the nonboundary-fitted FSI method developed in [38,39]. The method used in [38,46] defines a fluid subproblem and a Kirchhoff-Love shell subproblem with no-tailored discretizations.…”

Section: Introductionmentioning

confidence: 99%

Non-body-fitted fluid–structure interaction: Divergence-conforming B-splines, fully-implicit dynamics, and variational formulation

Casquero

Zhang

Bona-Casas

et al. 2018

Journal of Computational Physics

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Immersed boundary (IB) methods deal with incompressible visco-elastic solids interacting with incompressible viscous fluids. A long-standing issue of IB methods is the challenge of accurately imposing the incompressibility constraint at the discrete level. We present the divergence-conforming immersed boundary (DCIB) method to tackle this issue. The DCIB method leads to completely negligible incompressibility errors at the Eulerian level and various orders of magnitude of increased accuracy at the Lagrangian level compared to other IB methods. Furthermore, second-order convergence of the incompressibility error at the Lagrangian level is obtained as the discretization is refined. In the DCIB method, the Eulerian velocitypressure pair is discretized using divergence-conforming B-splines, leading to inf-sup stable and pointwise divergence-free Eulerian solutions. The Lagrangian displacement is discretized using non-uniform rational B-splines, which enables to robustly handle large mesh distortions. The data transfer needed between the Eulerian and Lagrangian descriptions is performed at the quadrature level using the same spline basis functions that define the computational meshes. This conduces to a fully variational formulation, sharp treatment of the fluid-solid interface, and a 0.5 increase in the convergence rate of the Eulerian velocity and the Lagrangian displacement measured in L 2 norm in comparison with using discretized Dirac delta functions for the data transfer. By combining the generalized-α method and a block-iterative solution strategy, the DCIB method results in a fully-implicit discretization, which enables to take larger time steps. Various two-and three-dimensional problems are solved to show all the aforementioned properties of the DCIB method along with mesh-independence studies, verification of the numerical method by comparison with the literature, and measurement of convergence rates.

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“…This suffered from difficulties with mass loss in the discrete fluid solution. In [35], we circumvented this issue by applying a modification of the divergence conforming (or div-conforming) discretization described in Evans and Hughes [74][75][76] and based on work by Buffa et al [77,78]. We refer interested readers to [74] for further information on div-conforming B-splines and [35,Section 2.2] for details of the implementation we used for immersogeometric FSI analysis.…”

Section: Fluid Subproblemmentioning

confidence: 99%

Error estimates for projection-based dynamic augmented Lagrangian boundary condition enforcement, with application to fluid–structure interaction

Kamensky

Hsu

et al. 2018

Math. Models Methods Appl. Sci.

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In this work, we analyze the convergence of the recent numerical method for enforcing fluid–structure interaction (FSI) kinematic constraints in the immersogeometric framework for cardiovascular FSI. In the immersogeometric framework, the structure is modeled as a thin shell, and its influence on the fluid subproblem is imposed as a forcing term. This force has the interpretation of a Lagrange multiplier field supplemented by penalty forces, in an augmented Lagrangian formulation of the FSI kinematic constraints. Because of the non-matching fluid and structure discretizations used, no discrete inf-sup condition can be assumed. To avoid solving (potentially unstable) discrete saddle point problems, the penalty forces are treated implicitly and the multiplier field is updated explicitly. In the present contribution, we introduce the term dynamic augmented Lagrangian (DAL) to describe this time integration scheme. Moreover, to improve the stability and conservation of the DAL method, in a recently-proposed extension we projected the multiplier onto a coarser space and introduced the projection-based DAL method. In this paper, we formulate this projection-based DAL algorithm for a linearized parabolic model problem in a domain with an immersed Lipschitz surface, analyze the regularity of solutions to this problem, and provide error estimates for the projection-based DAL method in both the [Formula: see text] and [Formula: see text] norms. Numerical experiments indicate that the derived estimates are sharp and that the results of the model problem analysis can be extrapolated to the setting of nonlinear FSI, for which the numerical method was originally proposed.

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Isogeometric Divergence-conforming B-splines for the Steady Navier-Stokes Equations

Cited by 58 publications

References 0 publications

Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

Non-body-fitted fluid–structure interaction: Divergence-conforming B-splines, fully-implicit dynamics, and variational formulation

Error estimates for projection-based dynamic augmented Lagrangian boundary condition enforcement, with application to fluid–structure interaction

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