| Abstract: |
| This talk introduces a mixed Finite Element Method aimed at approximating solutions to fourth-order variational problems with constraints.
We first address the biharmonic obstacle problem and propose an error convergence framework that offers an alternative to the established approaches by Ciarlet & Raviart and Ciarlet & Glowinski. Our method emphasises improved ease of numerical implementation, which may enhance practical usability.
Next, we investigate a two-dimensional variational problem involving linearly elastic shallow shells constrained within a specified half-space. We begin with cases where the middle surface has non-zero curvature and demonstrate that applying a mixed Finite Element Method with conforming elements requires a symmetry constraint on the gradient matrix of the dual variable. Notably, we find that this implementation cannot rely solely on Courant triangles, indicating a variation in approach based on the geometric features of the problem. This constitutes a counterexample to the statement that solutions of fourth order linear problems can be approximated by solely resorting to Courant triangles if one considers the mixed formulation of the original problem. We notice that this counterexample arises in connection with the lack of rigidity of linearly elastic shallow shells middle surface.
In cases where the middle surface is flat, we observe that the symmetry constraint is not necessary, allowing for the use of Courant triangles alone for solution approximation. This observation suggests that shell geometry can significantly influence the selection of Finite Element methods for discretization.
Our theoretical findings are supported by a series of supplementary numerical experiments, illustrating the practicality of the proposed methods. |
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