Integrated computational product and production engineering for multi-material lightweight structures

Nowadays, three-dimensional computations of deep drawing processes under consideration of a planar anisotropy have become a matter of course. With the help of additional adaptive mesh refinement algorithms, even complex sheet metal components can be efficiently simulated within acceptable computing times [20]. In early 2000s, the state of the art of process variables that can be calculated reliably using FEM were successively extended [21‐23]. This includes failure due to fracture, sheet thickness, strain distribution, wrinkling, blank-holder force, and strain-based damage. Therefore, it is possible to achieve improved tool designs and process times, as well as a high product quality. For determining the stress distribution in deep-drawn parts, the calculation of the elastic-plastic part of the forming process is necessary, since even small deviations of plastic deformations have a considerable influence on the springback [24, 25]. Springback and stress-based damage are still challenging issues in deep drawing simulations. Also, failures due to friction are difficult to predict. In this respect, existing friction models are constantly being advanced and improved [26]. With the growing demand for new materials like fiber-reinforced thermoplastics as well as their functional combination to metals, temperature effects due to forming at elevated temperatures have become an important factor as well [27].

Injection molding of thermoplastics is a highly efficient manufacturing process and therefore commonly used for plastic parts in large-scale production industries. The numerical analysis of the injection molding process is used for improving tooling and product quality [28, 29]. Process parameters can be identified finding optimal process settings for mold filling. Furthermore, cooling [30, 31] and runner systems [32, 33] can be optimized. In addition to the process and tool design, effects influencing the structural performance can also be analyzed. Those effects are e.g. the position of weld lines [34] or, in the case of FRP, the fiber orientation due to the plastic flow [35‐37]. In this work, we focus on short fiber-reinforced thermoplastics (SFRTP), where the obtained fiber orientation mainly influences the thermoelastic properties [38] and the failure [39]. For the structural analysis, the fiber orientation is mapped onto the structural mesh to consider the process-induced anisotropy [40, 41]. Furthermore, residual stresses due to the manufacturing process are considered in subsequent analyses for computing distortions of the structure [42, 43].

Conventional deep drawing steel DX56 was applied as a material, which is mainly used in automotive body design. The low-carbon deep drawing steel shows a purely ferritic microstructure that ensures a good slipping of grains to achieve a good formability. The sheet thickness of the material is h = 1.2 mm. To determine the mechanical properties and the flow curves that are required for the simulation, tensile tests were carried out at room temperature. Furthermore, three strain rates \(\dot \varphi =0.01 \textit {s}^{-1}\), 0.1s^− 1, and 1 s^− 1 were tested. These are generally used for deep drawing applications. In the tensile test, only true stress values up to the beginning of the necking can be evaluated accurately. However, higher strains can be achieved at deep drawing processes. Therefore, values are also necessary for the further progression of the plastic strain. Thus, the experimental results were extrapolated by means of different model approaches to consider advanced plastic work behavior, where for DX56 the model approach of Ghosh (cf. [83]): provides a good qualitative agreement. Therein k_f,Ghosh corresponds to the true stress, φ to the effective plastic strain and A, B, C, D to the derived material-specific parameters given in Table 1.

The resulting flow curves are presented in Fig. 8. The diagram shows a comparison between experimentally measured and extrapolated flow curves depending on different strain rates. It can be seen that with increasing strain rate the strain hardening rises. However, this influence decreases with higher strain rates. The remaining physical and mechanical properties, which have been used within the numerical study, are summarized in Table 2.

The simulation of the deep drawing process was created in LS-DYNA and an explicit time integration has been used. As material model MAT_24 is chosen describing an elasto-plastic material behavior with an arbitrary stress versus strain curve and strain rate dependency. In order to consider the strain rate dependence, a table function was used. The table defines for each strain rate value a flow curve giving the stress versus effective plastic strain for that rate [84]. By means of the presented properties, the model got parameterized. It already has been validated for DX56 within the context of [85]. The tool selection as well as the cutting of the sheet metal was chosen by means of the sheet thickness. Thus, the functional surfaces were derived from the existing CAD data and meshed as shell elements.

The general structure of the simulation model and the initial blank are shown in Fig. 9.

The displayed tools consist of blank holder, die, and punch (left). The sheet metal blank (right) is clamped between blank holder and die with a defined force F_bh. The mesh of the blank consists of 13,300 elements and 13,403 nodes. During the test procedure, the punch is deep drawing the blank with a constant speed v_p through the cavity of the die. The punch has a rectangular body with the dimensions 160 mm×80 mm. All tools are modeled as rigid bodies. For friction, the standard value for sheet metal forming of μ = 0.15 has been applied. In the parametric study, the process parameters blank holder force F_bh and punch speed v_p are varied according to Table 3.

Therefore, F_bh was varied starting from 50 kN in 10 kN increments to 200 kN at a constant punch speed of 20 mm/s. In the second step, v_p was varied in steps of 10 mm/s between 20 and 60 mm/s with a blank holder force F_bh = 100 kN. In order to reduce the computation time, the process time in the simulation was scaled down. To guarantee the accuracy of the results and to avoid influences on the process chain, all time-dependent variables were scaled, as well. Exemplary simulations without time scaling showed no differences in the results. For the process chain simulation in 4.4, the unscaled time is used to compute the corresponding cycle times. In total, 20 simulations were carried out. Each simulation was stopped after reaching the drawing depth of d = 40 mm, so that comparable parts are available.

First, the punch forces and the duration of the deep drawing process were evaluated as a function of the process parameters that are supposed to be applied in the process chain simulation. These data are used in Section 4.4 for the demand of machine components. Thus, the amount of energy required for forming can be estimated. Figure 10 shows the maximum punch force F_p,max achieved during deep drawing and the punch force in the last stage F_p,final for the various blank holder forces.

The diagram shows that with an increasing blank holder force, a higher maximum punch force is necessary to deep draw the blank at a constant speed of v_p = 20 mm/s. Due to higher holding pressure, less material flows, so the general resistance of the material during deep drawing increases. In the last stage, when the drawing depth of d = 40 mm is reached, the force F_p,final decreases as a result of the strong thinning of the cup wall.

Figure 11 shows how the maximum punch force changes at different punch speeds. With increasing punch speed, the process works faster to reach the prescribed drawing depth of d = 40 mm. Due to strain rate dependence, the maximum punch force rises with higher punch speed. Also, strong thinning of the cup wall takes places as the process accelerates.

For the data analysis in Section 4.4, the last stage of the deep drawing process was evaluated. From each simulation, the x-, y-, and z-coordinates of all nodes of the blank were plotted. Furthermore, for each element of the blank, the parameters effective plastic strain φ, von Mises stress σ_vM, and the thickness t were exported. Figure 12 shows exemplary results of the simulations for elevated blank holder forces.

In the surrogate modeling, all shown parameters are taken into account. The matrix can be extended at any time. In general, it can be seen that with an increasing blank holder force the blank is clamped more strongly. As a result, the flange is not drawn insufficiently. This has an influence on all shown parameters. The effective plastic strain φ and von Mises stress σ_vM at 50 kN mainly show maximum values in the corners of the cup. This is where the cup thins out the most. With increasing blank holder force, maximum values of plastic strain, stress, and shell thickness are moving in the direction of the cup wall. In the case of shell thickness, differences can be observed in the flange area, as well. As the blank holder force decreases, more material flows from the flange into the cup wall. Therefore, radial tension stress and tangential compression stress are increasing. This causes a thickening of the material in the edge area of the flange. At 200 kN, a thickening is prevented, which means that the shell thickness remains at its initial value of h = 1.2 mm. In order to get an overview of the distribution of the sheet thickness, the global maximum and minimum values were written out and shown in Fig. 13 as a function of the process parameters.

Depending on the punch speed, the maximum sheet thickness remains constant at t ≈ 1.3 mm. This is the area in the flange that increases in thickness. The minimum sheet thickness, on the other hand, decreases further with an increasing punch speed. For v_p > 30 mm/s, the strain rate dependence leads to significant thinning of the cup wall. A similar behavior can be observed as a function of the blank holder force. Both the maximum and the minimum shell thickness are decreasing with rising forces due to the reduction of radial and tangential stresses in the flange. The minimum shell thickness drops very sharply between F_bh = 110 kN and F_bh = 150 kN. For these parameter sets, the cup wall begins to thin more strongly because the pressure in the flange is sufficient to prevent the material from flowing any further.

The displayed model consists of two volume bodies representing the mold cavity (black) and the metallic insert part (gray). For both parts, tetrahedral elements are used, where the mold cavity is discretized with eight elements over the wall thickness to ensure a fine grid for the plastic flow analysis. The injection location (yellow) is placed in the center. The sheet metal insert has been modeled with a constant sheet thickness of 1.2 mm. The initial temperature is set constant to room temperature. The contact time between mold surface and insert is set 5 s before the injection starts. This ensures that the temperature of the metallic insert is high enough to prevent a fast solidification of the plastic melt when in contact with the insert. The mechanical and thermal properties of the insert are summarized in Table 2. For the analysis of the fluid flow, the Cross-WLF viscosity [87] model is used. Therein, the melt viscosity: is described by the shear rate \(\dot \gamma \), an critical stress level τ^∗ and a power law index n. The Newtonian limit: with A₂ = A₃ + D₃p, the glass transition temperature T^∗ = D₂ + D₃p and the pressure p is described by material specific coefficients A₁, A₃, D₁, D₂, and D₃. In this study, we use a polyamide 6 material with 30 mass-% glass fibers. In Fig. 15, the viscosity over shear rate is plotted for the listed parameters.

The Cross-WLF coefficients of the SFRTP used within the injection molding simulations are given in Table 4.

The mechanical behavior of the final structure is mainly determined by the fiber orientation a within the SFRTP component. In the numerical analysis, it is computed by the Folgar-Tucker model [88‐91] with a fiber interaction coefficient of c_a = 0.0156.

For the parametric study, the mold temperature T_Mold was varied in 10 °C increments between 70 and 100 °C. Within the DoE for the melt temperature T_Melt the discrete values 270 °C, 280 °C and 290 °C are used. Hence, a virtual experiment with 12 simulations for the overmolding was set up.

The parametric study described above yields 12 virtual injection molding results. In the variation of mold and melt temperature already information in injection pressure and time are obtained. In Fig. 16a the injection pressure for the different melt temperature is plotted over the mold temperature. The maximum injection pressure p_Inj of 10.4 MPa is computed at T_Mold = 70 °C and T_Melt = 270 °C. The lowest injection pressure is computed for the parameter combination T_Mold = 100 °C and T_Melt = 290 °C. In general, the injection pressure increase with increasing mold and melt temperature. Regarding Fig. 15, this is obvious since the higher the temperature of the melt the lower the viscosity is. With a higher mold temperature the cooling of melt arises slower which leads to less viscosity and a better flow. Consequently, higher temperatures support the fluid flow. However, in Fig. 16b, the diagrams show an increase of the injection time with increasing mold temperature.

Besides the energy and time related global properties of the process, detailed information on local field variables is obtained. In the case of SFRTP, the fiber orientation determines the mechanical properties of the structure and yields anisotropic behavior. Hence, for each simulation result, the tensor of fiber orientation a, exemplary depicted in Fig. 14b, is exported and used for the subsequent data analysis to achieve a fast and robust surrogate model for the fiber orientation.

Firstly, the sheet metal cup is deep drawn on a forming press. Afterwards, the plastic rib structure is integrated through fiber injection molding. For both processes, FEM simulations are performed, covering a specific DoE on process parameters (see Fig. 17). For deep drawing, the process parameters 50kN ≤ F_bh ≤ 200 kN and punch speed 20mm/s ≤ v_p ≤ 60 mm/s are analyzed. The FEM simulation delivers structure properties, like the deep drawn geometry, wall thickness, strain, and stress, dependent on the parameters. The process-specific surrogate model for deep drawing should reflect these properties with a low error rate. Subsequently, for the fiber injection molding process, the FEM simulates the fiber orientation of the rib structure. The fiber orientation is described by the symmetric tensor a comprising the six elements a_xx, a_yy, a_zz, a_xy, a_xz, and a_yz, which makes the FEM surrogate modeling a multi-output regression task as well. Similar to deep drawing, two process parameters are evaluated in terms of a DoE for fiber injection molding. Firstly, the melt temperature T_Melt, covering the temperature range of 270 to 290 °Cand secondly, the mold temperature T_Mold, having a temperature zone from 70 to 100 °C. Supplementary to the process parameters, the FEM mesh of each part is considered input feature for modeling. Within this use case, the mesh for deep drawing consists of 13,300 elements; the rib structure is modeled by 12,240 nodes.

For deriving a generalized process-specific FEM surrogate model, nonlinear modeling techniques in the form of multi-output regression methods are deployed for mapping the features, i.e., mesh and parameters, to the target variables, i.e., part properties. The ability for FEM surrogate modeling of four machine learning methods (simple regression tree, random forest, multilayer perceptron, and gradient boosted trees) is tested for each process variant. In order to achieve low error rates, respectively high scores for R², a model parameter optimization, i.e., hyper parameter optimization, is performed. Table 5 lists the varied model parameters, their variation range, and the amount of variation steps for each method. Furthermore, the modeling results are evaluated through a 6-fold cross-validation, due to the prevention of overfitting and the target of deriving a generalized model for untested points within the parameter space. Table 5 also shows the best found model parameters. Interestingly, in injection molding, the best parameters for the methods simple regression tree and gradient boosted trees are found at the outer limit of the variation range, e.g., amount of trees = 260 for gradient boosted trees. Hence, a further increase in R² could be expected when increasing the parameter limit.

Results for R² of each part property and the original FEM model compared with those of the FEM surrogate model as a 3D plot are shown in Fig. 18 for each process variant. The 3D plot of the deep drawn part, which shows the wall thickness for F_bh = 110 kN and v_p = 20 mm/s, reveals the well fit between the original FEM model (Fig. 18a) and its surrogate model built through a random forest (Fig. 18b). For example, the material withdrawal in the cup corner is precisely reproduced. Deviations between FEM and surrogate are located on the cups’ short edge. The comparison of R² over all four regression methods (Fig. 18c) shows that the plotted wall thickness and strain and stress are the least correlated part properties, still reaching good values for R² for simple regression tree and random forest. On the contrary, the part property deep drawn geometry, i.e., X-Final, Y-Final, and Z-Final, yields the highest scores for R². A reason for this can be found in the small deviations of the final deep drawn geometry across the whole DoE. As for deep drawing, an exemplary plot for fiber injection molding of the plastic rib structure for the original FEM yielded by Moldflow (Fig. 18d) and the FEM surrogate model built through a random forest approach (Fig. 18e) is drawn for the fiber direction element a_xx. The underlying injection molding process parameters are melt temperature T_Melt = 270 °C and mold temperature T_Mold = 70 °C. Both plots show just minor deviations in the color hue. The bar plot for R² (Fig. 18f) emphasizes the good modeling performance. Especially the random forest approach yields constant high scores for R² with a minimum of 0.909 for a_yz and a maximum score of 0.977 for a_xz. The methods simple regression tree and gradient decision tree delivered similar modeling performance, just the multilayer perceptron approach performed weak for injection molding. The obtained process-specific FEM surrogate models are feasible to be applied for untried process parameter combinations in scenario analysis delivering adequate information on part properties for process chain and factory simulation.

The process chain model starts with pre-cut steel sheets that enter the process chain at a hydraulic press for deep drawing. A buffer follows the press for the temporary storage of the formed sheets. The cutting step has not been modeled in detail. Rather, it is represented by a simple discrete event block. The reinforcing ribs are formed thereafter in an injection molding machine. The last process step is quality control, where reject parts are sorted out. Bottom-up machine models are connected together with buffers and a quality control station to form a process chain model. The modeling environment for both the machine modeling and process chain modeling is AnyLogic. The models were developed combining the principles of agent based and discrete event simulation. Machines and products act in this modeling approach as agents with individual parameters, behavior, and communication interfaces to other agents. The product agents pass through a discrete event process chain that utilizes the machines as resources. After the machines are ready for production after ramp-up, the product agents are created and enter the process chain. When a product agent enters a machine agent, the machine agents start processing the product according to the defined process-specific steps. During each step, the electrical power is calculated based on the process parameters (e.g., temperature or processing time), machine parameters (e.g., efficiency of machine components) and product parameters (e.g., weight, material). The quality of the produced part is defined in each process step according to a probability distribution. If a reject part is produced, a new product agent is generated and sent to the process chain. The output parameters, such as direct energy demand, waiting time, or reject/good part, are stored in the product agent. After processing ended for the whole production batch including the additional parts due to reject parts, the indirect energy demand per part is calculated. To this end, the total indirect energy demand of the machines and direct energy demand of reject parts are allocated equally on all good parts. The machine models and the process chain model are connected via a spreadsheet interface with the numerical process models.

Machine parameters were determined based on the machine pool at the OHLF. Process parameters were acquired based on a parametric study of the numerical process simulation of forming and injection molding. Tables 6 and 7 provide in this context an overview of the employed parameters and the respected intervals. Altogether, the parametric study provided twenty parameter sets for forming and twelve sets for injection molding. In the process chain simulation, all parameter set combinations were tested out; altogether 240 simulation runs were carried out. Remaining process parameters (that could not be retrieved from the process simulation) were defined according to typical process windows from literature and expert interviews.

The case study simulated the manufacturing of 100 parts. The quality rate of both machines was assumed to be 98%, thus addressing also inefficiencies by reject parts that indirectly lead to an increase of the indirect energy demand of products. The goal was to illustrate the interdependencies between process parameters, which are decided on in the process simulation, and the product energy intensity as well as lead time.

Figure 19 compares the minimum and maximum values of the cycle times on both machines. In an ideal case, the cycle times are even or close to even at each process step. This is here however not the case since we choose a strong criterion for the cooling time. The cycle time of injection molding exceeds the cycle time of the press for all parameter sets. The higher cycle time on the injection molding machine causes a time bottleneck and drives the lead time of the production upwards. The case study illustrates however effectively that process time reduction on single-process steps does not necessarily lead to a reduced lead time. Instead, the whole process chain needs to be considered. In this case, the process time for injection molding should be minimized. The process time for deep drawing could even be longer without affecting the lead time negatively.

The cycle time difference between the two processes is represented in the composition of the lead time. Waiting time occurs while a product is stored temporarily in the buffer between the press and injection molding machine. The comparison of average processing times with average waiting times (see Fig. 20) illustrates the impact of uneven cycle times. The figure illustrates the shares of processing and waiting time for the minimum, average, and maximum lead time. In all three cases, the lead time is largely composed of waiting time with a share of up to 90%. For the present use case, this indicates that cooling times seem to be too large and can most probably be reduced in order to reduce the waiting time. However, in order to decide whether the lead time is too high and needs to be reduced, it needs to be compared with the customer tact time. For the calculation of the customer tact time, further information regarding the working hours, working days, and customer demand is needed. Therefore, it has been excluded in this case study. Based on the customer tact time and the current lead time, further decisions can be made, e.g., whether a new machine should be installed to increase the output.

Figure 21 illustrates the highest, lowest, and average product energy intensity for all 240 simulation runs, broken down into direct and indirect energy demands. While direct energy demands are considered as value adding, indirect energy demands represent the share of non-value adding energy demand. The direct energy demand of one product represents the sum of each machine’s energy demand in the processing state that is related to one good part. The direct energy demand of reject parts is assigned to the indirect energy demand. The indirect energy demand per part further includes the energy demand of the machines in the ramp-up and standby stages. As explained above, the buffer between the machines permits a high utilization without noteworthy waiting times of the machines. The indirect energy demand is consequently diminishing in comparison with the direct energy demand. Regarding the direct energy demand, the parameter variation on the injection molding machine shows an over 200% difference between the best and worst cases. In contrast to that, the difference on the hydraulic press stays below 1%. This may be due to the higher range of process windows that were applied on the injection molding machine.

The initial concern of the case study on the process chain level was to investigate the interdependencies between the lead time and product energy intensity. This should help product, process, and factory planners to design a product and production system that fits best economic and environmental targets. Figure 22 illustrates the relation between the cycle times of the process steps and the resulting direct energy demand on the press and injection molding machine. It confirms the abovementioned small variation of the direct energy demand on the press. The results indicate that the energy demand per part increases only slightly with a higher process time.

The injection molding machine shows contrast to that a much higher effect. A higher cycle time seems to lead to a higher direct energy demand. This is a favorable effect, as the cycle time of the injection molding machine needs to be minimized in order to reduce the difference between the two process steps. This way, not only the lead time can be reduced, but also the energy demand of the injection molding machine. It should be noted that not all process parameter sets with a lower cycle time lead to a reduction in the energy demand. In order to explore the contribution of the input parameters to the energy intensity in detail, further parameter studies are needed.

In conclusion, the process chain simulation enhanced with the integration of numeric process simulation results can support planning processes in production engineering. It can contribute to understanding the effects of process parameter variation on the whole process chain and explore different scenarios regarding lead times and energy demands.