Best: bézier-enhanced shell triangle, a new rotation-free thin shell finite element
- Autores: Pere Andreu Ubach de Fuentes
- Directores de la Tesis: Eugenio Oñate Ibáñez de Navarra (dir. tes.), J. García Espinosa (codir. tes.)
- Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2020
- Idioma: español
- Materias:
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- Resumen
A new thin shell finite element is presented. This new element doesn’ t have rotational degrees of freedom. Instead, in order to overcome the C1 continuity requirement across elements, the author resorts to enhance the geometric description of the flat triangles of a mesh made out of linear triangles, by means of Bernstein polynomials and triangular Bernstein-Bézier patches.
The author estimates the surface normals at the nodes of a mesh of triangles, in order to use them to define the Bernstein-Bézier patches. Ubach, Estruch and García-Espinosa performed a comprehensive statistical comparison of different weighting factors. The conclusion of that work is that the inverse of the area of the circumscribed circle to the triangle and the internal angle of the triangle at the node considered, should be used as weighting factor. Using this new weighting factor, we reduce by about 10% the root mean square error in the estimation of normals of randomly generated surfaces with respect to the previous best weighting factor found in the literature.
The author uses the information of the normal vectors at the nodes and the triangular Bernstein-Bézier patches to build cubic Bézier triangles. These cubic Bézier triangles are surface interpolants; C1 continuous at the nodes and C0 continuous across the edges. Owing to this approach, the new element is called Bézier-enhanced shell triangle (BEST).
The BEST element takes advantage of all the nodes’ connectivities in each triangle of the mesh. The computation of the normal vectors at the nodes doesn’ t depend on the number of triangles surrounding each node of the mesh. The BEST element is independent from the mesh topology.
A new paradigm is presented consisting on the reconstruction of the geometry of a cubic triangular element. This geometric reconstruction exploits the properties of cubic B-spline functions (cubic Bézier triangle). This way, the author builds a conforming continuum-based shell finite element.
A cubic Bézier triangle has 30 parameters (3 coordinates for each of the 10 control points). Therefore it needs to apply 30 independent conditions. 15 of these conditions are given directly by the positions of the 3 vertices of the triangle and the orientations of the normal vectors at the 3 vertices.
8 of the remaining conditions are imposed introducing energy minimization considerations. These energy minimization considerations serve also to define a well-posed element. The author defines 3 different reduced problems for the 3 different shell deformation modes: bending deformation, membrane (in-plane extension) deformation and in-plane shear (drilling rotation) deformation.
The only degrees of freedom of the BEST element are the vertices’ coordinates (9 variables). The remaining 21 parameters are solved internally. In order to fix the values of these 21 internal parameters, each BEST element solves 9 systems of linear equations of rank 3.
The BEST element is successfully applied to the analysis of thin shells in linear and geometrically non-linear regimes using an implicit method. The non-linearity is solved using a Total Lagrangian formulation.
The author succeeds at pre-integrating through-the-thickness efficiently and accurately. The through-the-thickness integrals are evaluated just once: at the reference configuration. There are just 14 through-the-thickness scalar integrals to perform for each Gauss point.
The numerical examples results show that the BEST element has the potential to achieve cubic convergence. Although they also cast doubts on the possibility of reproducing this result for a wide range of problems. For in-plane shear dominated problems, the formulation used in this thesis only achieves linear convergence. For membrane oriented tests with curvature, the convergence is quadratic.
The BEST element exhibits membrane locking behavior. The author suggests exploiting further the drilling rotations kinematics in order to solve membrane locking.
