Higher‐order surface FEM for incompr. Navier‐Stokes flows on manifolds

Fries, Thomas‐Peter

doi:10.1002/pamm.201800012

Proc Appl Math and Mech

2018

DOI: 10.1002/pamm.201800012

|View full text |Cite

Higher‐order surface FEM for incompr. Navier‐Stokes flows on manifolds

Thomas‐Peter Fries

Abstract: Stationary and instationary Stokes and Navier-Stokes flows are considered on curved surfaces in three dimensions. A stabilized, mixed, higher-order surface FEM with individual element orders is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Benchmark test cases are proposed and higher-order convergence rates are confirmed.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2020

2021

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

A unified finite strain theory for membranes and ropes

Fries

Schöllhammer

2020

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

The finite strain theory is reformulated in the frame of the Tangential Differential Calculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, membranes and three-dimensional continua are treated with one set of governing equations. Secondly, the reformulated boundary value problem applies to parametrized and implicit geometries. Therefore, the formulation is more general than classical ones as it does not rely on parametrizations implying curvilinear coordinate systems and the concept of co-and contravariant base vectors. This leads to the third unification: TDC-based models are applicable to two fundamentally different numerical approaches. On the one hand, one may use the classical Surface FEM where the geometry is discretized by curved one-dimensional elements for ropes and two-dimensional surface elements for membranes. On the other hand, it also applies to recent Trace FEM approaches where the geometry is immersed in a higher-dimensional background mesh. Then, the shape functions of the background mesh are evaluated on the trace of the immersed geometry and used for the approximation. As such, the Trace FEM is a arXiv:1909.12640v1 [cs.CE] 27 Sep 2019 fictitious domain method for partial differential equations on manifolds. The numerical results show that the proposed finite strain theory yields higher-order convergence rates independent of the numerical methodology, the dimension of the manifold, and the geometric representation type.

show abstract

A unified finite strain theory for membranes and ropes

Fries

Schöllhammer

2020

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

show abstract

Inf-sup stability of the trace $\mathbf {P}_2$–$P_1$ Taylor–Hood elements for surface PDEs

2021

View full text Add to dashboard Cite

The paper studies a geometrically unfitted finite element method (FEM), known as trace FEM or cut FEM, for the numerical solution of the Stokes system posed on a closed smooth surface. A trace FEM based on standard Taylor-Hood (continuous P 2 -P 1 ) bulk elements is proposed. A so-called volume normal derivative stabilization, known from the literature on trace FEM, is an essential ingredient of this method. The key result proved in the paper is an inf-sup stability of the trace P 2 -P 1 finite element pair, with the stability constant uniformly bounded with respect to the discretization parameter and the position of the surface in the bulk mesh. Optimal order convergence of a consistent variant of the finite element method follows from this new stability result and interpolation properties of the trace FEM. Properties of the method are illustrated with numerical examples.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Higher‐order surface FEM for incompr. Navier‐Stokes flows on manifolds

Cited by 2 publications

References 4 publications

A unified finite strain theory for membranes and ropes

A unified finite strain theory for membranes and ropes

Inf-sup stability of the trace $\mathbf {P}_2$–$P_1$ Taylor–Hood elements for surface PDEs

Contact Info

Product

Resources

About