Compatible description of tool surfaces and FEM meshes for analysing sheet forming operations

Abstract

SUMMARYImplementation of the 3-D finite element method (FEM) of sheet forming operations has proceeded slowly. In particular, describing the arbitrary tool surfaces in an accurate and smooth manner, with well behaved first and second derivatives, has been a major obstacle. A new geometric method, suitable for representing any rigid tool surface, has been developed and shown to be superior to the usual schemes. The new method relies on the described tool surface for position data only. The spatial derivatives are obtained by considering only the sheet mesh nodal positions. This new scheme has been implemented with a rigid-viscoplastic FEM program using general surface descriptions based on B-splines and linear interpolation. Analytic representations of a hemispherical and rounded square punch were also compared. The comparisons show that the proposed method offers several advantages over other general formulations:1. The new representation is the proper one for formulating the equilibrium condition. Alternate forms can produce spurious results. 2. The proposed method introduces inherently smooth surface derivatives that improve numerical stability, even with crude surface resolution. Suitable test problems show convergence with the new method far past the divergence points of the simulations using the standard general surface description. 3. The method is reasonably efficient, with a time penalty of approximately 30-50 per cent (unoptimized) with respect to analytic surface descriptions.

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