Virtual Loading Time Analysis in Explicit Dynamic FEM Simulation of Hydraulic Forming of Pipe Fittings

Overview and Research Significance

This paper, published in China Mechanical Engineering (Vol. 18, Issue 8, 2007, pp. 904-906), addresses a critical numerical simulation issue in the finite element analysis (FEA) of hydraulic forming of pipe fittings. The authors from Shanghai Jiao Tong University and Baoshan Iron & Steel Co., Ltd. investigate the effect of virtual loading time on the accuracy and computational efficiency of explicit dynamic FEM simulations. The research was motivated by the practical need to accurately predict the forming process while maintaining reasonable computation times.

Hydraulic forming is a widely used process for manufacturing pipe fittings, particularly for forming complex geometries from tube blanks. The process involves pressurizing the interior of a tube blank that is constrained by a die, causing the tube to expand and take the shape of the die cavity. The process is inherently dynamic, with the hydraulic pressure applied over a finite time period, and the workpiece deforming in response to the pressure loading.

In explicit dynamic FEM, the time step is limited by the Courant condition, which requires that the time step be smaller than the time required for an acoustic wave to cross the smallest element. For typical hydraulic forming simulations, this results in very small time steps and a large number of time steps, leading to significant computational cost. The virtual loading time is a simulation technique used to accelerate the loading process without significantly affecting the results.

Technical Background

In explicit dynamic FEM, the equations of motion are solved using a central difference time integration scheme. The time step is determined by the smallest element size and the material wave speed, and is typically on the order of microseconds. For a hydraulic forming process that takes several seconds in reality, simulating the process at real time scale would require millions of time steps, which is computationally prohibitive.

The virtual loading time technique involves accelerating the loading process in the simulation by applying the hydraulic pressure over a shorter time period than would be used in the actual process. This reduces the total number of time steps and hence the computation time. However, if the virtual loading time is too short, the simulation may not capture the quasi-static behavior of the forming process, leading to inaccurate predictions of deformation, stress, and strain distributions.

The key question addressed by this paper is: what is the appropriate virtual loading time that balances simulation accuracy and computational efficiency? The authors approach this question by comparing simulation results with experimental data for different virtual loading times.

Methodology and Results

The study uses a circular cross-section part as the primary test case for hydraulic forming simulation. The simulation is performed using explicit dynamic FEM with different virtual loading times, and the results are compared with experimental measurements of the formed part geometry. The comparison allows the authors to identify the range of virtual loading times that produce accurate predictions.

The authors also analyze the energy distribution in the system for different virtual loading times. In explicit dynamic FEM, the total energy is composed of kinetic energy, internal (strain) energy, hourglass energy, and other energy components. For a quasi-static forming process, the kinetic energy should be negligible compared to the internal energy. If the virtual loading time is too short, the kinetic energy becomes significant, indicating that the simulation is not capturing the quasi-static behavior and the results may be inaccurate.

The energy analysis provides a physical explanation for the observed accuracy trends. When the virtual loading time is sufficiently long, the kinetic energy remains low throughout the simulation, and the results accurately represent the quasi-static forming process. When the virtual loading time is too short, the kinetic energy increases significantly, indicating dynamic effects that are not present in the actual process.

The authors establish a criterion for selecting the virtual loading time based on the kinetic energy ratio (kinetic energy divided by total energy). When this ratio remains below a certain threshold throughout the simulation, the results are considered accurate. This criterion provides a practical and physically meaningful guideline for virtual loading time selection.

Validation with Subframe Hydraulic Forming

The criterion established for the circular cross-section case is validated using a more complex geometry: a subframe part for a passenger car. The subframe part has a complex cross-section and geometry, making it a challenging test case for hydraulic forming simulation. The simulation results for different virtual loading times are compared with experimental data, and the criterion successfully predicts the accuracy of the simulation results.

This validation is important because it demonstrates that the criterion is not specific to simple geometries and can be applied to complex industrial components. The subframe part represents a realistic manufacturing scenario where hydraulic forming is used to produce complex structural components from tube blanks.

Engineering Practice Implications

For engineers performing hydraulic forming simulations, this paper provides several practical guidelines:

1. Virtual loading time selection: The kinetic energy ratio criterion provides a systematic method for selecting the virtual loading time. The criterion is simple to implement and does not require additional simulations or experimental data.
2. Computation time management: By selecting an appropriate virtual loading time, engineers can significantly reduce computation time without sacrificing accuracy. This is particularly important for process optimization studies where multiple simulations are required.
3. Quality prediction: Accurate simulation of the hydraulic forming process enables reliable prediction of part quality, including dimensional accuracy, wall thickness distribution, and springback. This information is essential for die design and process parameter optimization.

The paper also highlights the importance of energy monitoring in explicit dynamic FEM simulations. Monitoring the kinetic energy ratio during the simulation provides a real-time indicator of simulation accuracy and can alert the engineer to potential issues before the simulation is completed.

Critical Reflection

This paper addresses a practical and important issue in the simulation of hydraulic forming processes. The virtual loading time is a critical parameter that significantly affects both accuracy and efficiency, and the systematic approach proposed by the authors provides a valuable tool for simulation practitioners.

One limitation of the study is the focus on explicit dynamic FEM. Implicit dynamic or quasi-static FEM methods, which use larger time steps and do not require virtual loading time adjustments, may be more suitable for some hydraulic forming simulations. However, explicit methods have advantages in handling contact problems and large deformations, which are common in hydraulic forming. The choice between explicit and implicit methods should be made based on the specific simulation requirements and available computational resources.

The kinetic energy ratio criterion is a valuable addition to the simulation practitioner's toolkit. It provides a physically meaningful and computationally efficient method for assessing simulation accuracy, and its applicability to complex geometries has been demonstrated. For engineers working on hydraulic forming process development, this paper provides practical guidance for improving simulation accuracy and efficiency, which directly contributes to faster process development and better product quality.