Adaptive finite element methods for plate bending problems

Doctoral thesis, 2011

The major theme of the thesis is the development of goal-oriented model adaptive continuous-discontinuous Galerkin (c/dG) finite element methods (FEM), for the numerical solution of the Kirchhoff and Mindlin-Reissner (MR) plate models. Hierarchical modeling for linear elasticity on thin domains (beam-like) in two spatial dimensions is also considered, as a natural extension of the Bernoulli and Timoshenko beam theories. The basic idea behind model adaptivity is to refine, not only the computational mesh, but the underlying physical model as well. Consequently different mathematical formulations - usually partial differential equations - may be discretized on the element level. Our algorithms use duality-based a posteriori error estimates, which separate the discretization and modeling errors into an additive split (allows for independent reduction the error contributions). The error representation formulas are linear functionals of the error, which is often more relevant in engineering applications. In standard FEM the continuity constraints can make it difficult to construct the approximating spaces on unstructured meshes. When solving the plate formulations, continuous quadratic polynomials are used for the lateral displacements, and first-order discontinuous polynomials for the rotation vector, whose inter-element continuity is imposed weakly by Nitsche's method. The bilinear form is coercive if a computable penalty parameter is large enough (and small enough to avoid locking). The discretization of the MR model converges to the Kirchhoff model as the thickness of the plate tends to zero. This makes the introduced c/dG FEM particularly interesting in the context of model adaptivity, and as such it constitutes the main result of the thesis.