Springback is a major problem in the deep drawing process. When the tools are released after the forming stage, the product springs back due to the action of internal stresses. In many cases the shape deviation is too large and springback compensation is needed: the tools of the deep drawing process are changed so, that the product becomes geometrically accurate after springback. In this paper, two different ways of geometric optimization are presented, the Smooth Displacement Adjustment (SDA) method and the Surface Controlled Overbending (SCO) method. Both methods use results from a finite elements deep drawing simulation for the optimization of the tool shape. The methods are demonstrated on an industrial product. The results are satisfactory, but it is shown that both methods still need to be improved and that the FE simulation needs to become more reliable to allow industrial application.

SUMMARYIn this paper we present two types of local error estimators for the primal ÿnite-element-method (FEM) by duality arguments. They are ÿrst derived from the (explicit) residual error estimation method (REM) and then-as a new contribution-from the (implicit) posterior equilibrium method (PEM) using improved boundary tractions, gained by local post-processing with local Neumann problems, with applications in elastic problems. For the displacements a local error estimator with an upper bound is derived and also a local estimator for stresses. Furthermore-for better numerical e ciency-the residua are projected energy-invariant onto reference elements, where the local Neumann problems have to be SURVEY OF ERROR ESTIMATORS FOR THE PRIMAL FINITE-ELEMENT-METHOD (FEM) FOR ELLIPTIC BOUNDARY VALUE PROBLEMSWe have to distinguish between error estimators (having upper and=or lower bounds) and error indicators (without bounding properties). Next, we have to distinguish between global error estimators (yielding at least an upper bound for the energy norm of the solution error within the whole domain) and local error estimators. Global error estimators also provide local convergence of the solution error for well-posed problems, but without bounds. The natural global (explicit) error estimator of primal FEM is the residual BabuÄ ska-Miller estimator in the energy norm [1; 2], computed locally on element level with the residua of equilibrium and the traction jumps to the neighbouring elements and summed up over the system. With local error

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