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Anisotropic triangles in an immersed finite element approach

Presentation: Immersed methods

The fundamental idea of immersed methods in having the mesh independent of the domain of definition of the problem. The term immersed has to be taken in a broad sense, i.e., it includes methods such as the immersed boundary method, the fictitious domain, embedded and unfitted methods.
The immersed method is thus useful for the following types of problems with:

  • complex geometries

  • interface problems

  • fluid-structure interaction

  • ...

One of the major issues of immersed methods is accuracy. Indeed, problems tackled by immersed methods are likely to show singularities: across the interface, or the solid, etc. The loss of accuracy is in general due to the non conformity of the mesh with the location of the singularity.

Furthermore, in many immersed problems essential boundary conditions (BCs) must be enforced across the interface. Two approaches are possible:

  • strongly, in the finite element space

  • weakly, with a Lagrange multiplier, via a penalty method, etc.

Moreover, enforcing essential BCs in weak sense when the interface does not fit/conform the interface is not a trivial matter, and it is still a dynamic research topic. For these reasons, the proposed method satisfies:

  1. Conformity of the mesh with the interface, but only locally, i.e. at the element level. More precisely, only the elements crossed by the interface are remeshed such that only few elements are modified when the interface moves.

  2. Strong enforcement of the essential BCs.

A notable feature of the proposed approach is the anisotropy of the remeshed elements. This approach is named here: a locally anisotropic remeshing strategy. In what follows we only consider triangles.

A locally anisotropic remeshing strategy

It is well known that anisotropic triangles are allowed, as long as their largest angle is bounded away from π. We point out that this condition is not necessary for triangle.
Nevertheless, the use of anisotropic elements implies at least:

  1. conditioning issues;

  2. and for mixed elements: the inf-sup constant maybe degenerate to zero as the element is distorted.

The studies developed here deal with the second issue. In particular, we focused on the following mixed elements:

  • P2/P0

  • P2-bubble/P1-discontinuous: (the Crouzeix-Raviart element)

  • P2/P1: the Hood-Taylor element

  • P2-bubble/P1

Main results

It has been showed that in the context of the presented method:

  1. P2/P0: unstable, despite Apel (2004)

  2. P2-bubble/P1-discontinuous: unstable to the best knowledge of the author this result was unknown (see Lefieux PhD)

  3. P2/P1: unstable but in very rare occasions see Auricchio 2014

  4. P2-bubble/P1: stable see Auricchio 2014

Numerical tests

Deformation of the mesh

Motion of the mesh of a thin hinged bar in a fluid-structure interaction problem

An accurate method for 2D thin FSI problems

FSI of a thin hinged bar with P2-bubble/P1: top speed; bottom pressure field
This problem is taken from:
P. Causin, N. D. Santos, J.-F. Gerbeau, C. Guiver, and P. Métier. An embedded surface method for valve simulation. application to stenotic aortic valve estimation. In ESAIM: PROCEEDINGS, volume 14, pages 48–62, 2005.

Bibliography

F. Auricchio, F. Brezzi, A. Lefieux, and A. Reali. An “immersed” finite element method based on a locally anisotropic remeshing for the incompressible stokes problem. Computer Methods In Applied Mechanics and Engineering, 2014. DOI:10.1016/j.cma.2014.10.001.

F. Auricchio, A. Lefieux, A. Reali, and A. Veneziani. A locally anisotropic fluid-structure interaction remeshing strategy for thin structures with application to a hinged rigid leaflet. International Journal for Numerical Methods in Engineering, 2015. Submitted

A. Lefieux. On the use of anisotropic triangles in an immersed finite element approach with application to fluid-structure interaction problems. PhD thesis, Istituto Universitario degli Studi Superiori di Pavia, 2014

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