The sheet metal forming process is an effective process widely used in industries for parts and bodies. There have been considerable efforts to optimize sheet metal forming processes and its progress has been remarkable with the aid of numerical simulation owing to amazingly increased computer capacity. Numerical simulation of sheet metal forming processes has become popular instead of the conventional trial-and-error or the expert-depending approach since the numerical analysis can easily deal with the nonlinear material properties in the plastic range, frictional conditions, complicated geometry of tools and material-die interaction. Recently, The finite element simulation has been actively applied to the initial and final design of vehicle structures to minimize design costs and time and automotive industries have made lots of efforts to reduce the weight of vehicle structures with increased safety to reduce fossil energy consumption as well as environment pollution. In most cases, forming simulation and crash simulation are usually carried out separately, although forming histories should be taken into account as the initial condition in the crash simulation for a robust design of the light vehicle structure. The design of vehicle structures to improve crash characteristics considering forming effects has been studied and attempted by several automotive companies, although it is not easy to apply for the actual structural design of vehicles due to the tremendous computing time to calculate forming histories.
In this research, an inverse finite element method is employed for more capability to estimate formability from the desired final shape as well as to calculate forming histories such as thickness variation and plastic strain distributions which have great influences on the energy absorption of parts with small amount of computation time. The finite element inverse method adopts Henky`s deformation theory, Hill`s anisotropic yield criterion and simplified boundary conditions. The bending effect is considered by the modified membrane theory. Since the inverse analysis constructs a mesh system for the final stage of forming, crash simulation can be directly performed after the forming stage without a smoothing or remeshing. Although finite element inverse analysis is a quite effective scheme for those purposes in view of design cost and calculation time, it is extremely difficult to calculate an initial guess for the part with the large aspect ratio or the large angle of inclination. In this research, a direct mesh mapping scheme is newly proposed to estimate initial guesses for the finite element inverse analysis of arbitrary three-dimensional parts. Sliding constraint surfaces are described by finite element patches and constructed from the designed final shape or directly from the mesh system of the die and punch set that is used in the direct finite element analysis. An initial guess and a final shape are both calculated by the direct mesh mapping scheme from the sliding constraint surfaces of selected models.
The present algorithm has been implemented in a finite element code and applied to several sheet metal forming processes to verify its validity and effectiveness. Multi-stage finite element inverse analysis is applied to multi-stage rectangular cup drawing processes with the large aspect ratio to calculate the initial and the intermediate shapes and the thickness strain distribution in each intermediate shape. The scheme can calculate deformed shapes and thickness strain distribution qualitatively for the initial tool design in the multi-stage deep drawing process. The design modification has been carried out to improve the uniform thickness distribution in each intermediate stage as well as the final stage. One-step inverse analysis is extended to the multi-step inverse analysis for the rectangular cup deep drawing process. The results are compared with the direct finite element method. The error induced by the one-step inverse analysis was reduced by the multi-step analysis, while reasonable analysis results still can be acquired with the one-step inverse analysis. The numerical example of an S-rail forming process demonstrates that the mapping scheme proposed is applicable to calculate an initial guess for a part with the large angle of inclination. The analysis result of the S-rail forming process is compared with that from a direct finite element analysis to evaluate the effectiveness of the mapping scheme proposed. In order to verify the validity and reliability of the inverse method in the crashworthiness simulation, crash analyses are simulated considering forming histories calculated by both the direct and inverse analysis. Analysis results demonstrate that energy absorption of structures is increased when simulation considers forming effects of thickness variation and work hardening. The total analysis time to estimate crashworthiness was significantly reduced when the inverse analysis is employed for the calculation of forming effects. Forming simulations for front frame members which have complicated geometry are also carried out to estimate formability and to calculate forming histories for the crash simulation without the die and punch set. Finally, crash simulation of front frame members is carried out considering forming effects from the inverse analysis to estimate crashworthiness. The results indicate that the inverse finite element can be applied to the optimum design of the light vehicle structures with enhanced reliabilities.
The numerical results fully demonstrate that the finite element inverse analysis with the proposed direct mesh mapping scheme is greatly useful to estimate formability and to calculate forming histories in small amount of calculation time even with the complex geometry.
