A challenge to engineers: The Babuška-Paradox

Frank Duderstadt (dudersta@wias-berlin.de)
Wolfgang Dreyer


Abstract

The bending problem of a plate supported by a curvilinear trustring,
where it may rotate freely, cannot be approximated by bending problem,
where the curvilinear trustring is substituted by an approximating
polygonal trustring. The Babuška-paradox proves that the deviation of
the polygonal solution from the curvilinear solution increases with the
increase of its nodes. In other words, the substitution of the
curvilinear curve by a polygon increases the failure of the polygon
problem to approximate the curvilinear solution.

The described phenomenon happens if the finite elements solution of the
original described bending problem is done with linear elements,
because here the curvilinear trustring becomes a polygon. If the
boundary conditions of the trustring are formulated along the
approximating polygon, then the deviation between the approximation and
the original problem increases with refinement of the finite element
mesh.

The example of a bending test for single crystal gallium arsenide wafer
serves to prove that a finite element treatment of the problem is
possible so that the Babuška-paradox does not appear. In particular, we
show that there is a strategy to obtain an approximate solution that
converges to the solution of the original problem with an increase of
the refinement of the finite element mesh.


Authors' address:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS)
Mohrenstr. 39
10117 Berlin
Germany