Model Adaptivity in Elasticity

Licentiatavhandling, 2008

We consider model adaptivity in elasticity for dimensionally reduced forms, and shall treat different conceptual approaches. The basic idea, however, is to adaptively refine, not only the computational mesh, but also the underlying mathematical formulation. The intention is that the algorithm, provided with an hierarchy of models, should have the local complexity tailor-made for each problem, and thus become more efficient. Reduced forms of the 3D-elasticity theory are usually obtained via simplified deformation relations. A typical example is the Bernoulli and Timoshenko beam theories. We discuss Navier's equations of linear elasticity in a thin domain setting, and construct a model hierarchy based on increasingly higher polynomial approximations through the thickness of the domain, coupled with a Galerkin approach. We suggest a finite element method for an extension of the Kirchhoff-Love plate equation, which includes the effects of membrane stresses. The stresses are obtained from the solution of a plane-stress problem, and plays the role of underlying model. Since the modeling error actually is a discretization error, this is different from constructing a model hierarchy. The aim has been to establish efficient solution procedures alongside accurate error control. To succeed in this ambition, a posteriori error estimates are derived, which separate the discretization and modeling errors. Frameworks for adaptive algorithms are suggested, and accompanied by numerical results to exemplify their behavior.