We construct a cut finite element method for the membrane elasticity problem
on an embedded mesh using tangential differential calculus. Both free membranes
and membranes coupled to 3D elasticity are considered. The discretization comes
from a Galerkin method using the restriction of 3D basis funtions (linear or
trilinear) to the surface representing the membrane. In the case of coupling to
3D elasticity, we view the membrane as giving additional stiffness
contributions to the standard stiffness matrix resulting from the
discretization of the three-dimensional continuum
We suggest a finite element method for computing minimal surfaces based on computing a discrete Laplace-Beltrami operator operating on the coordinates of the surface. The surface is a discrete representation of the zero level set of a distance function using linear tetrahedral finite elements, and the finite element discretization is done on the piecewise planar isosurface using the shape functions from the background three dimensional mesh used to represent the distance function. A recently suggested stabilization scheme is a crucial component in the method.
In this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces. In particular, we adopt a stabilization technique which allows for first order L 2convergence of the mean curvature vector and apply this stabilization technique also to the computation of continuous, recovered, normals using L 2 -projections of the piecewise constant face normals. Finally, we use our projected normals to define an adaptive mesh refinement approach to geometry resolution where we also employ spline techniques to reconstruct the surface before refinement. We compare or results to previously proposed approaches.
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