Data-driven finite element computation with material uncertainty

Data-driven methodologies are attracting more and more attention. The “data-rich” world ushered in the 21st century with ever larger and more unwieldy data sets directed interest toward processing them. Gradually, data-driven computations are also gaining ground in the natural sciences. In numerical mechanics, such methodologies are used in particular for describing the material behavior. Normally, the behavior has to be characterized via experiments and then fitted into models. The second step can be omitted if one is able to use the data obtained by the experiments directly in the calculation. This methodology was presented in 2016 by Trenton Kirchdoerfer and Michael Ortiz for a finite element analysis and will be followed up in this work.
First, the so-called “data-driven finite element method” is investigated with respect to its different input parameters. It turns out that the method is robust concerning the numerical stiffness, but the amount and quality of data is essential for the quality of the solution. Then this methodology is investigated in terms of material uncertainties. For this purpose, data-driven computations are performed with data that are generated from stochastic fields. In order to limit the additional numerical effort of the data-driven computation, a method is presented, which by means of a multi-level computation is able to reduce the numerical effort. Instead of a simulation with the complete data set, several simulations with small, adaptive data sets are used. To avoid the complex experimental generation of three-dimensional data, a methodology is presented on how data from numerical calculations can be obtained. Here using the example of a representative volume element. Finally, two further application examples are presented: First, the datadriven methodology is applied to a diffusion problem instead of a mechanical problem. Second, a polymorphic uncertainty model is generated by adding a fuzzy variable as a further uncertainty to the model.