| Implementation 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 and nodal positions.; This new scheme has been implemented with a rigid-viscoplastic three-dimensional FEM program using general surface descriptions based on B-splines and linear interpolation. Analytic representation 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% (unoptimized) with respect to analytic surface descriptions.; In order to describe an arbitrarily-shaped tool surface, a B-spline scheme and a piecewise linear scheme are developed. The tool surface is described in 2 steps: data preparation and data fitting. In the data preparation step, raw surface points come from a CAD/CAM system, digitizer, or blueprint, are converted to equally spaced data. Data fitting involves the calculation of point characteristics (height, slopes, and curvatures) at equally spaced grid points and then at points interrogated by the same finite element method.; For simulating the stretch/draw forming operation of two-dimensional, general-shape, plane-strain sections, a sectional analysis program, SHEET-S, was modified to use the mesh-based formulation and the generalized tool description method. Real-world stamping problems were then tested. Numerical results were compared with the measured data. Excellent agreements between FEM results and measurements were found in the stretch/draw solutions. |
