A Trace Finite Element Method for Vector-Laplacians on Surfaces

Groß, Sven; Jankuhn, Thomas; Olshanskii, Maxim A.; Reusken, Arnold

doi:10.1137/17m1146038

Cited by 31 publications

(45 citation statements)

References 25 publications

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“…They may be labeled "active" elements and nodes, all others are neglected. The shape functions of the active nodes are constructed by (often isoparamet- The Trace FEM is a fictitious domain method (FDM) for PDEs on manifolds [33,32,25,23]. As in any FDM, there is no boundary-conforming mesh but a background mesh, herein further complicated by the fact that background mesh and manifold have different dimensions.…”

Section: Trace Femmentioning

confidence: 99%

“…Fortunately, different stabilization approaches exist to cure both issues and we refer to the overview given in [32] for the Trace FEM. Herein, we use the "normal derivative volume stabilization", introduced for scalar-valued problems in [24,7] and in [25] for vector-valued problems. This stabilization technique enables higher-order accurate results, does not change the sparsity pattern of the stiffness matrix, and only first derivatives are needed.…”

Section: Trace Femmentioning

confidence: 99%

“…Concerning the numerical approximation of ropes and membranes based on the Surface and Trace FEM as discussed herein, it is mentioned that the Trace FEM just as any other FDM simplifies the meshing of geometries significantly. However, additional effort is required for (i) the integration of the weak form, wherefore suitable integration points on the manifolds have to be provided [18,34,19,21], (ii) the application of boundary conditions which is more involved because the boundary is within the background elements and instead of prescribing nodal values one may have to use Nitsche's method [6,3,26,41,40] or other approaches for enforcing constraints, and (iii) stabilization terms which are necessary to address the existence of shape functions with small supports on the manifolds and to find unique solutions with the background mesh although the BVP is only defined on the manifold [32,24,7,25]. In spite of the increased effort for implementing the Trace FEM, it may have significant advantages over the Surface FEM.…”

Section: Introductionmentioning

confidence: 99%

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A unified finite strain theory for membranes and ropes

Fries

Schöllhammer

2020

Computer Methods in Applied Mechanics and Engineering

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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.

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Section: Trace Femmentioning

confidence: 99%

Section: Trace Femmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A unified finite strain theory for membranes and ropes

Fries

Schöllhammer

2020

Computer Methods in Applied Mechanics and Engineering

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“…Following Groß et al, the variational formulation of (2) is given by: Find u ∈ V τ such that

\begin{align} a false(u, v false) + m false(u, v false) = f false(v false) \forall v \in V_{bold-italicτ}, \end{align}

where

\begin{align} a false(u, v false) : = \int_{normalΓ} ε_{normalΓ} false(u false) : ε_{normalΓ} false(v false) normald x, m false(u, v false) : = \int_{normalΓ} u \cdot v normald x, f false(v false) : = \int_{normalΓ} f \cdot v normald x . \end{align}

…”

Section: Model Problems On the Surfaceunclassified

“…Further, it is already observed that—in contrast to the scalar problem—an isoparametric discretization is not sufficient to preserve optimal order L 2 ‐errors. In Jankuhn et al, similar approaches are considered and analyzed in the context of TraceFEM discretizations. Hansbo et al extended their approach to Darcy problems on surfaces using [ H 1 (Γ)] 3 ‐conforming (low‐order) surface FEM.…”

Section: Introductionmentioning

confidence: 99%

Divergence‐free tangential finite element methods for incompressible flows on surfaces

Lederer

Lehrenfeld

Schöberl

2020

Numerical Meth Engineering

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In this work we consider the numerical solution of incompressible flows on two-dimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning H 1 -conformity allows us to construct finite elements which are-due to an application of the Piola transformation-exactly tangential. To reintroduce continuity (in a weak sense)we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach, H(div Γ )-conforming finite elements can be used to obtain exactly divergence-free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy. K E Y W O R D Sdivergence-conforming finite elements, incompressible Navier-Stokes equations, Piola transformation, surface PDEs, tangential vector field Int J Numer Methods Eng. 2020;121:2503-2533.wileyonlinelibrary.com/journal/nme

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A higher‐order Trace finite element method for shells

Schöllhammer

Fries

2020

Int J Numer Methods Eng

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A higher-order fictitious domain method (FDM) for Reissner-Mindlin shells is proposed which uses a three-dimensional background mesh for the discretization. The midsurface of the shell is immersed into the higher-order background mesh and the geometry is implied by level-set functions. The mechanical model is based on the tangential differential calculus which extends the classical models based on curvilinear coordinates to implicit geometries. The shell model is described by partial differential equations on manifolds and the resulting FDM may typically be called Trace FEM. The three standard key aspects of FDMs have to be addressed in the Trace FEM as well to allow for a higher-order accurate method: (i) numerical integration in the cut background elements, (ii) stabilization of awkward cut situations and elimination of linear dependencies, and (iii) enforcement of boundary conditions using Nitsche's method. The numerical results confirm that higher-order accurate results are enabled by the proposed method provided that the solutions are sufficiently smooth.

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A Trace Finite Element Method for Vector-Laplacians on Surfaces

Cited by 31 publications

References 25 publications

A unified finite strain theory for membranes and ropes

A unified finite strain theory for membranes and ropes

Divergence‐free tangential finite element methods for incompressible flows on surfaces

A higher‐order Trace finite element method for shells

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