Professor Erwin Stein
IBNM, Gottfried Wilhelm Leibniz Universität Hannover
Error-controlled combined model and FE-discretization adaptivity via dual data in structural mechanics
Abstract:
Error-controlled hp-adaptive finite element methods are presented for goal-oriented combined model and discretization error estimates and mesh adaptivity for BVPs of elliptic PDEs for structural problems in non-linear and linear elasticity, like 2D plates as the "coarse" model and the 3D C1 point continuum.The methodology is aiming at combined verification and validation, i.e. required accuracy of quantities of interest for discretization and mathematical modeling during the loading process of a structure, especially influenced by boundary layers and various other disturbances of strain and stress fields. As a consequence, the "coarse" and the fine model are defined in varying subdomains.
A major problem is the necessary prolongation of FE solutions of the "coarse" 2D model to the corresponding solution space of the fine 3D model in order to admit a proper definition of the modeling error. By this, the orthogonality property of the used implicit error estimates (with improved element interface tractions by solving local Neumann problems) gets lost. However, discretization error estimates of the "coarse" model have not to be prolongated to the fine model due to the fact that goal-oriented total error estimates with associated dual problems are used, yielding linearity and Betti-Maxwell symmetry property, [1]. Implicit error estimates based on equilibrated residua with improved element interface tractions as well as of averaging type are used.
FE meshes of the dual problems are chosen much coarser than those of the related primal problems in order to improve the computational efficiency without noteworthy loss of accuracy.
Examples are presented for crack propagation and for plates in bending with boundary layers.
Reference: [1] Stein, E., Rüter, M., Ohnimus, S.: Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity, CMAME, in print: 32 pages
| Datum: | 02.02.2007 | Zeit: | 14.15 | Ort: | FU Berlin, Institut für Mathematik II, Arnimallee 6, 14195 Berlin. | Raum: | 032 im Erdgeschoss |