دانلود رایگان مقاله مدل سازی المان برش محدود پوسته خطی

Abstract

We construct a cut finite element method for the membrane elasticity problem on an embedded mesh using tangential differential calculus, i.e., with the equilibrium equations pointwise projected onto the tangent plane of the surface to create a pointwise planar problem in the tangential direction. Both free membranes and membranes coupled to 3D elasticity are considered. The discretization of the membrane comes from a Galerkin method using the restriction of 3D basis functions (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.

5. Concluding remarks

In this paper we have introduced an FE model of curved membranes using higher dimensional shape functions that are restricted to (the approximation of) the membrane surface. This allows for rapid insertion of arbitrarily shaped membranes into already existing 3D FE models, to be used for example for optimization purposes. We have shown numerically that the cut element approach to membranes gives errors comparable to triangulated membranes, using the same degree of approximation, and we have proposed a stabilization method which provides stability to the solutionas well as giving the right conditioning of the discrete system, allowing for arbitrarily small cuts in the 3D mesh. The novelties of our approach are: (1) applying for the first time the cut element approach of [6] to models of thin elastic structures, and (2) modeling membranes embedded in a 3D bulk material with arbitrary orientation of the membrane relative to the meshing of the bulk. In future work, we will consider more realistic models of embedded membranes.