Data-Based Estimation of Critical Time Steps for Explicit Time Integration

Abstract

Finding the critical time step for conditionally stable time integration methods has been a decades-long problem. The apparently obvious option of directly computing it from a generalized eigenvalue analysis, identifying the largest eigenfrequency of the discrete system, is usually impractical because of its numerical expense and because a stiffness matrix is often unavailable in the context of explicit analysis. There exist two popular approaches to efficiently estimate the critical time step: A characteristic element length can be estimated based on heuristic formulas. The resulting estimate, however, cannot be guaranteed to be conservative. Another approach is to reformulate and simplify the underlying eigenvalue problem on the element level and to use certain inequalities to derive an upper bound for the largest eigenvalue. This is conservative but may show poor performance by significantly under-predicting the actual critical time step. Moreover, the necessary simplifications are usually specific to the investigated element formulation. Many works that develop time step estimators demonstrate their performance only for particular element configurations, making it difficult to compare the estimators. In this paper, data-driven approaches for time step estimation for 2d-elements that address several of the aforementioned problems are proposed. First, the set of all possible quadrilateral element geometries and its discrete representation are described. A detailed comparison of nine existing time step estimators based on more than ten million element configurations is presented. Additionally, the concept of an optimal safety factor function is introduced. This concept allows us to generate the optimal and conservative version of an existing estimator and thus solves two problems at the same time: It can be used to make non-conservative estimators conservative and to improve the performance of estimators that are conservative by construction. Finally, we formulate time step estimation as a function approximation problem. It allows us to derive customizable time step estimators solely based on data. Through two examples, we demonstrate that this data-driven approach yields time step estimators that outperform state-of-the-art estimators in terms of accuracy while also being efficient to evaluate.