Multiscale modeling based on a novel combination of direct data-driven methods with Fourier transform-based microstructure simulation

对具有复杂微观结构的材料进行多尺度计算可以为其在实际应用中提供有价值的信息，这在很大程度上是毋庸置疑的。然而，在实践中使用适当的计算方法受到若干挑战的阻碍：(A)计算时间过长，(B)除其他外，由于与不同的微观结构有关的不确定性，在微观一级对参数识别不足，以及(C)不同的尺度。本项目的目的是通过将数据驱动的非弹性材料建模方法与基于快速傅立叶变换(FFT)的有效微结构模拟相结合并进一步发展，从而显著减少这些缺点。将使用所谓的直接数据驱动(DD)方法，在该方法中，数据被传输到有限元(FE)模型，而不需要进一步的内插。在数据缺失的情况下，直接使用基于FFT的微结构模拟来生成这些数据。这导致了基于DD-FFT的双尺度算法，其中DD方法被合并到宏观级别的有限元模型中。本申请的目的是(1)进一步发展考虑不确定性的非弹性材料的直接数据驱动方法，(2)通过基于FFT的复杂材料微观结构水平的有效模拟来自适应地获取数据，(3)进一步开发基于FFT的特别有效的模拟的模型简化技术，以及(4)开发关于如何最佳地结合数据获取和数据驱动模拟的策略。因此，应该实现有效的双尺度数据驱动计算，这也可以解释微观结构中的不确定性。对于自适应数据采集，最好使用基于FFT的模拟来捕获微结构，因为该方法的图像处理性质允许以直接的方式研究不同的微结构。

It is largely undisputed that multi-scale calculations of materials with complex microstructures can provide valuable information for their use in practical applications. However, the use of suitable computational methods in practice is hampered by several challenges: (a) the excessively long computation times, (b) the insufficient identification of parameters at the microstructure level, caused, among other things, by uncertainties related to varying microstructures, and (c) different scales. The aim of the present project is to significantly reduce these disadvantages by a novel combination and further development of data-driven methods for inelastic material modeling with efficient microstructure simulations based on fast Fourier transforms (FFT). The so-called direct data-driven (DD) method is to be used, in which the data are transferred into a finite element (FE) model without further interpolation. In case of missing data, these are generated directly using FFT-based microstructure simulation. This results in a two-scale DD-FFT-based algorithm, where the DD method is incorporated into an FE model at the macro level. The present application intends to (1) further develop the direct data-driven methodology for inelastic materials considering uncertainties, (2) adaptively acquire data with efficient FFT-based simulations for complex inelastic materials at the microstructure level, (3) further develop model reduction techniques for particularly efficient FFT-based simulation, and (4) develop strategies on how to optimally combine data acquisition and data-driven simulation. Thus, an efficient two-scale data-driven computation should be enabled, which can also account for uncertainties in the microstructure. For adaptive data acquisition, the microstructure is best captured with FFT-based simulation, since the image-processing nature of the method allows varying microstructures to be studied in a straightforward manner.