Virtual Loading Time Analysis in Dynamic Explicit FEM Simulation of Pipe Fitting Hydroforming

Literature Overview

This paper by Yang Bing, Zhang Weigang, Lin Zhongqin, and Chen Xinping, published in China Mechanical Engineering (2007, Vol. 18, No. 8, pp. 904-906), addresses a critical numerical challenge in the finite element simulation of pipe fitting hydroforming: the selection of virtual loading time in dynamic explicit analysis. The authors, from Shanghai Jiao Tong University and Baosteel, investigate how virtual loading time affects both computational accuracy and efficiency, establish a criterion for optimal selection, and validate the criterion with a practical hydroforming simulation of an automotive subframe component.

Background: Dynamic Explicit FEM in Hydroforming Simulation

Hydroforming is a forming process in which internal fluid pressure (typically oil or water) is applied to a tube or pipe to plastically deform it against a die. The process is inherently quasi-static in nature, as the pressure is applied slowly and the material deforms gradually. However, simulating hydroforming with implicit FEM solvers is computationally expensive due to the large number of contact constraints and the need for iterative convergence at each time step.

Dynamic explicit FEM solvers (such as LS-DYNA or Abaqus/Explicit) offer a computationally efficient alternative by using a mass-proportional damping approach and explicit time integration. However, dynamic explicit solvers are designed for dynamic problems, and applying them to quasi-static processes requires careful control of inertial effects. The virtual loading time is the parameter that controls the rate at which the hydroforming pressure is applied in the simulation, and its selection directly affects the balance between computational accuracy and efficiency.

The Virtual Loading Time Problem

In dynamic explicit simulation of hydroforming, the actual hydroforming process (which may take minutes or hours) is compressed into a much shorter simulated time. The virtual loading time (T_virtual) is the duration over which the full hydroforming pressure is applied in the simulation. This creates a fundamental trade-off:

• Short virtual loading time: Fast computation but high inertial effects, leading to artificial oscillations, inaccurate stress states, and potential numerical instability.
• Long virtual loading time: Accurate quasi-static behavior but computationally expensive, potentially negating the efficiency advantage of the explicit solver.

The goal is to find the shortest virtual loading time that maintains acceptable accuracy while maximizing computational efficiency.

Methodology: Establishing the Selection Criterion

Energy-Based Analysis

The authors use the system energy distribution as the primary diagnostic tool for evaluating virtual loading time adequacy. In a dynamic explicit simulation, the total energy of the system is composed of:

For a quasi-static process, the kinetic energy should be negligible compared to the internal energy. The ratio of kinetic energy to internal energy (U_kin / U_int) serves as the primary accuracy indicator.

Criterion Development

The authors establish the following criterion for optimal virtual loading time selection:

1. Accuracy threshold: The kinetic energy ratio (U_kin / U_int) should remain below 5% throughout the simulation. This threshold ensures that inertial effects do not significantly distort the stress-strain state.
2. Efficiency maximization: Among all virtual loading times that satisfy the accuracy threshold, the shortest time is selected to maximize computational efficiency.
3. Stability verification: The simulation must remain numerically stable, with no excessive hourglass energy or artificial strain energy.

Circular Cross-Section Hydroforming Example

The criterion was first validated on a simple circular cross-section hydroforming problem, where the simulation results were compared with experimental measurements:

• The simulated cross-sectional profile (diameter, wall thickness, and shape) was compared with the experimentally measured profile.
• The optimal virtual loading time was determined as the shortest time that produced simulation results within acceptable agreement with experiments.
• The energy distribution analysis at different virtual loading times confirmed that the selected criterion effectively identifies the boundary between accurate and inaccurate simulations.

Practical Application: Automotive Subframe Hydroforming

The criterion was then applied to a practical hydroforming simulation of an automotive subframe component, which has a complex geometry with multiple bends, varying cross-sections, and intricate die configurations. The simulation demonstrated:

• Accuracy: The predicted deformation pattern, wall thickness distribution, and springback were in good agreement with experimental results.
• Efficiency: The selected virtual loading time achieved acceptable accuracy with significantly reduced computational time compared to longer loading times.
• Robustness: The energy-based criterion was effective in identifying the optimal loading time even for this complex geometry, confirming its generalizability.

Engineering Practice Implications

Simulation Setup Guidelines

For engineers performing hydroforming simulations using dynamic explicit solvers, the following guidelines emerge from this study:

1. Always monitor energy distribution: Track U_kin / U_int throughout the simulation and ensure it remains below 5%. If it exceeds this threshold, increase the virtual loading time.
2. Start with a conservative estimate: Begin with a virtual loading time that is clearly long enough (e.g., 10 times the minimum element transit time), then reduce it iteratively while monitoring energy ratios.
3. Account for mass scaling: If mass scaling is applied to improve computational efficiency, ensure that the scaled mass does not introduce excessive kinetic energy. The effective mass scaling ratio should be limited (typically below 100:1).
4. Validate against experiments: Whenever possible, compare simulation results with experimental data to verify that the selected virtual loading time produces accurate predictions.

Quality Control Considerations

The accuracy of hydroforming simulations directly impacts quality prediction:

• Wall thickness prediction: Inaccurate simulations may underpredict minimum wall thickness, leading to thin-wall failures in service.
• Springback prediction: Errors in the stress state due to excessive inertial effects can lead to incorrect springback predictions, resulting in dimensional non-conformance.
• Residual stress prediction: The residual stress state after hydroforming affects fatigue life and stress corrosion resistance. Accurate simulation is essential for predicting these long-term performance characteristics.

Key Questions and Reflections

The 5% kinetic energy threshold proposed by the authors is a practical starting point, but it may not be universally applicable. For processes involving high strain rate sensitivity or complex contact interactions, a lower threshold (such as 2-3%) may be necessary to ensure accuracy. Conversely, for simple geometries with low contact complexity, the threshold could potentially be relaxed.

Another consideration is the interaction between virtual loading time and other simulation parameters. The time step size in explicit FEM is determined by the minimum element size and material wave speed, and mass scaling modifies this relationship. The virtual loading time must be considered in conjunction with the time step size, mass scaling ratio, and damping parameters to ensure a consistent and accurate simulation setup.

The study also raises the question of whether implicit FEM solvers, with improved contact algorithms and adaptive meshing, might now offer a more straightforward approach for hydroforming simulation. The computational cost of implicit solvers has decreased with advances in parallel computing, and the accuracy advantages of implicit methods for quasi-static processes may outweigh the efficiency benefits of explicit methods for many applications.

Study Insights and Outlook

This paper provides a clear and practical framework for selecting virtual loading time in dynamic explicit hydroforming simulations. The energy-based criterion is intuitive, easy to implement, and validated against both simple and complex geometries. For engineers routinely performing hydroforming simulations, the key insight is that virtual loading time is not merely a computational convenience but a critical accuracy parameter that must be systematically evaluated through energy analysis. The criterion established in this study provides a reliable methodology for balancing accuracy and efficiency, ensuring that simulation results can be trusted for process optimization and quality prediction in pipe fitting hydroforming production.