May 2021 Volume 3

FORGING RESEARCH

To obtain the highest system frequency, the calculation of eigenvalues is usually inevitable. For a huge system, this procedure could be very difficult (if not impossible), even using iterative algorithms, and the results may not be accurate. For a nonlinear system, because its stiffness and element size are changing all the time, this calculation needs to be carried out in each iteration. Alternately, in many finite element programs, a simple and efficient estimation is used. Assuming the critical wavelength to be L w and the wave speed to be c , the total time for this wave to travel past a point is:

scaling can be fixed throughout the simulation, variable mass scaling is better suited because it automatically calculates the stable time increment and adds appropriate mass to the specified element group based on the user-defined minimum stable time increment. It should be noted that an overly large mass scaling factor can cause errors in the geometry of the workpiece because of the dynamic effect of the inertial force from the added mass. To ensure a smooth simulation, the mass scaling factor must be carefully selected so that no degradation occurs when the simulation approaches the final stage. The advantages of the explicit finite element formulation make this method well suited for incremental metal forming processes. 5.3 Major considerations in metal forming modeling Before any model can be built for a given metal forming modeling request, the very first questions an engineer should ask are what finite element program is to be used; what, if any, historical data from production or experiments exists; and if there is sufficient computer power. In each step of the model building, attention should be paid to ensure the input parameters are as precise as possible. Figure 16 shows four key components to be considered in this process, which are discussed in the following sections. Please note that there are many more things to be considered in building a highly effective and efficient finite element model, and be aware that all of these are interconnected. Figure 16. Considerations in metal forming modeling

(27)

If n time increments are necessary for the wave to travel, the time increment is:

(28)

(29)

where E is Young’s modulus and is the mass density. Therefore, the stable time increment can be defined as:

(30)

Usually, the time increment in a simulation can be in the order of 10 -7 , meaning that a simulation needs millions and millions of increments to finish. To prevent a high computational cost, the nature of the explicit method determines that a number of ways can be selected to accelerate the simulation, including using larger elements, reducing the Young’s modulus or increasing the mass density. Because using large elements severely deteriorates the simulation quality, the Young’s modulus of a material in the metal forming process at a high temperature is originally small and the elastic region is limited, the only option left is to increase the mass density, a process called mass scaling . It can be seen from Eq. (30) that artificially increasing the material mass density by a factor of n 2 increases the stable time increment by a factor of n or speeds up the simulation by n times if the number of elements is kept constant. Usually, only the mass density of the workpiece is scaled up. The criterion for selecting a proper mass scaling factor is that the simulation efficiency can be improved without degrading the simulation accuracy. While mass density

5.3.1 Workpiece The workpiece should be closely examined when the modeling request arrives because it is the centerpiece of the modeling investigation. The geometry of the workpiece and its complexity level determine the modeling space (2D axisymmetric, plane-stress, plane-strain or 3Dmodel), element types (one type or multiple types of elements), element sizes (the minimum and maximum as well as their ratio) and mesh configurations (finer mesh in the areas of high geometric variations). Starting with a coarse mesh is recommended to get the initial result quickly. A reasonable fine mesh is used until all modeling parameters and setup procedures are correct and verified.

FIA MAGAZINE | MAY 2021 77