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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.
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:
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:
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.
The studies developed here deal with the second issue. In particular, we focused on the following mixed elements:
Main results It has been showed that in the context of the presented method:
Numerical tests Deformation of the mesh Motion of the mesh of a thin hinged bar in a fluidstructure interaction problem An accurate method for 2D thin FSI problems FSI of a thin hinged bar with P2bubble/P1: top speed; bottom
pressure field 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 fluidstructure 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 fluidstructure interaction problems. PhD thesis, Istituto Universitario degli Studi Superiori di Pavia, 2014 
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