Data-based time step estimators for explicit time integration methods

• Applicant: Professor Dr.-Ing. Manfred Bischoff
• Subject Area: Applied Mechanics, Statics and Dynamics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 576498883

Determining the critical time step for conditionally stable explicit time integration methods in structural dynamics is a decades-old problem. Since common explicit algorithms are only conditionally stable, a limitation of the time step size is required to ensure numerical stability. However, the so-called critical time step is unknown a priori and can change at every time step for non-linear problems. An exact calculation is usually not practicable, so approximation methods are required for estimation. These must be sufficiently conservative to guarantee stability but at the same time accurate enough to avoid choosing an unnecessarily small time step. State-of-the-art estimators are usually based on heuristic formulas that only take into account the geometric properties of the elements. This often results in inaccurate and sometimes non-conservative estimates. The aim of the project is to develop data-based methods for efficient, conservative and accurate estimation of the critical time step in explicit finite element simulations. In the first step, sampling methods will be investigated and further developed to generate representative data sets for complex finite element types. In addition, different machine learning approaches are investigated to accurately map the complex relationship between element configuration and critical time step. The models are trained and validated on extensive data sets with the aim of developing conservative, efficient and accurate estimators. These methods are developed for different element types that are of particular importance for practical simulations. In addition, methods for time step estimation at the patch level will be developed to reduce an inappropriately large influence of individual small elements or irregularly shaped elements on the time step estimation and to improve accuracy. Finally, the consideration of geometric nonlinearities is investigated in order to improve the suitability of the estimation methods for strongly nonlinear applications. In summary, the project aims to significantly improve the accuracy and efficiency of time step estimation in finite element simulations by combining advanced sampling strategies, machine learning methods and the consideration of element patches and non-linearities. The expected results are a well-founded analysis of the influence of relevant parameters on the critical time step and their integration into new, data-based estimation methods derived from suitable machine learning methods. The added value of such estimators lies in reliably stable simulations with shorter computing times because the maximum permissible time step is utilized.

• DFG Programme: Research Grants
• Co-Investigator: Dr.-Ing. Malte Von Scheven