A two-stage surrogate model for Neo-Hookean problems based on adaptive proper orthogonal decomposition and hierarchical tensor approximation

Abstract The evaluation of robustness and reliability of realistic structures in the presence of uncertainty is numerically costly. This motivates model order reduction techniques like the proper orthogonal decomposition (POD), which gives an approximate model on the basis of a set of precomputations of a full model. This reduces the computational time. The reduction achieved by POD is usually not sufficient in the uncertainty quantification or optimization context where a large number of evaluations has to be carried out. In this context, it is also common that only a few quantities are of interest and a further reduction is possible. The second reduction may be represented by a mapping from a possibly high-dimensional parameter space onto each quantity of interest (QoI). In general, it is difficult to construct such a mapping from an unreduced model. Hence, in this paper a two-stage surrogate model that combines both reduction approaches is introduced. This surrogate model is tailored and applied to Neo-Hookean model problems with plasticity, hardening and damage. Here, the knowledge of the model allows an adaptive selection of the POD basis on the first stage. This idea is elaborated into a generalized framework called adaptive proper orthogonal decomposition (APOD). The second stage consists of the hierarchical tensor approximation (HTA) which is easily adjusted to the accuracy and computational cost of the first stage such that the two-stage surrogate model becomes an efficient option for uncertainty quantification and optimization for Neo-Hookean problems. Additionally, with both stages at hand, the HTA is utilized for a greedy snapshot search to improve the basis of APOD and POD. This is useful if no expert knowledge is available to guide the snapshot selection in the offline phase.