Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Sunday, January 26th, 1997

Anisotropic Elements in Linear Elasticity

We want to solve the equations of linear elasticity on long, but flat domains. Due to the degeneracy of the constant in Korn's inequality a straight forward finite element method with anisotropic elements gives bad approximation (locking).

The standard techniques to handle these beams and plates are models from structural mechanics. The 2D (3D) displacement functions are first approximated by functions of a lower dimensional function space, e.q. functions fulfilling the Kirchhoff hypotheses. Then the reduced model is discretized by finite elements.

In this talk, a non-conforming discretization method of the original linear elasticity problem is introduced. The discrete solution is equivalent to the discrete solution of the structural mechanics problem. Therefore no locking occurs. The implementation of the non-conforming method seems to be much simpler, especially when different models are coupled.

The convergence rate of multigrid solvers with point-Gauss-Seidel smoothers depends on the geometric parameters. A block-Gauss-Seidel smoother leads to a robust multigrid method.

Numerical examples demonstrate the accuracy of the discretization and the robustness of the solver.