Iterative Space-Time Finite Element Method for Problem-Dependent Domains
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The metalworking process of blanking or piercing produces parts with smooth sheared edges from sheet metal, which can be used as a functional surface in further manufacturing processes [1]. Parts produced by blanking have along their sheared edge a die roll on one side and a burr on the other side. While the burr can be removed in a secondary process, the die roll cannot and it reduces the workable surface area of the edge. It is therefore of interest to choose process parameters such that the die roll is minimized in the final product. With an ever-increasing compute power, numerical simulations are an attractive tool to investigate factors influencing the die roll's shape. Eulerian descriptions [2] or phase-field methods can capture the topology change caused by the cut, but only represent the cut in a diffuse manner. Since we are interested in a precise representation of the roll, we may use a Lagrangian or Arbitrary-Lagrangian-Eulerian description. These descriptions however require remeshing around the topology change once the finished part is fully separated from the sheet. This is where we see potential for space-time methods in combination with a Lagrangian material description. The use of space-time finite elements allows us to include the topology change directly in the space-time mesh. An example of space-time finite elements, where the evolution of the domain over time is prescribed, was demonstrated by Danwitz et al. [3]. We extend their approach by making finding the time and location of the topology change a part of our problem formulation. This is done in an iterative procedure, where the space-time mesh of an initially guessed space-time domain is deformed using the elastic mesh update method [4] to comply with the new location of the topology change, thus, avoiding remeshing. We show a proof of concept of our novel approach and have a first look at an error analysis. References [1] Stanke et al., Journal of Physics: Conference Series. IOP Publishing, 2017 [2] Furlan et al., Proceedings in Applied Mathematics and Mechanics, 2023 [3] von Danwitz et al., International Journal for Numerical Methods in Engineering, 2023 [4] von Danwitz et al., International Journal for Numerical Methods in Fluids, 2021
