Model order reduction in space and parameter dimension - towards damage-based modeling of polymorphic uncertainty in the context of robustness and reliability

在多态性不确定性的存在下，现实结构的鲁棒性和可靠性的评估涉及具有非常高的自由度以及参数的数值模拟。这些参数中的一些是确定的，因为它们是先验已知的。然而，第二组参数是不精确的，模糊的，不确定的或基于不完整的信息。在这一组中有许多参数，这些参数对损伤和失效行为有着至关重要的影响，因此对结构的承载能力有很大的影响。将其不确定性和可预测性纳入建模至关重要。该项目的主要目标是为此目的开发一个建模框架，该框架使用扩展模型降阶和分层张量近似，以及最近开发的数据驱动力学方法，以显着减少计算工作量。其思想是结合联合收割机适当的正交分解与分层张量近似和智能数据力学的自适应方法，以这样一种方式，不确定或模糊的感兴趣的量可以通过简单的功能评估计算。感兴趣的参数是例如不应超过的某些应力或变形水平。也可以设想要求损害水平低于某个规定值。虽然项目的重点是土木工程结构，但旨在使该方法相对普遍适用，并以这种方式在优先级程序1886内产生多种合作可能性。因此，模型简化和分层张量近似的新组合工具将对几种材料模型，各种类型的几何描述以及最后的不确定性和模糊性的不同概念开放。相对于后一方面的灵活性首先由以许多不同方式建立原始参数空间的可能性给出。其次，通过感兴趣的量的函数表示，大大简化了不确定性或模糊性的评估。

The evaluation of robustness and reliability of realistic structures in the presence of polymorphic uncertainty involves numerical simulations with a very high number of degrees-of-freedom as well as parameters. Some of these parameters are certain in the way that they are a priori known. However, the second group of parameters is imprecise, vague, uncertain or based on incomplete information. In this group are many parameters which crucially influence the damage and failure behavior and as such have a strong influence on the load bearing capacity of a structure. It is of pivotal importance to include their uncertainty and fuzziness into the modelling. The main goal of the project is to develop a modeling framework for this purpose which uses extended model order reduction and hierarchical tensor approximation, together with a recently developed method of data-driven mechanics to significantly reduce the computational effort. The idea is to combine an adaptive method of proper orthogonal decomposition with hierarchical tensor approximation and intelligent data mechanics in such a way that the uncertain or fuzzy quantities of interest can be computed by simple functional evaluation. Quantities of interest are e.g. certain stress or deformation levels which shall not be exceeded. It is also conceivable to require the damage level going below a certain prescribed value.Although the focus of the project is on civil engineering structures, it is intended to make the method relatively generally applicable and in this way generate multiple cooperation possibilities within the priority program 1886. Therefore, the new combined tool of model reduction and hierarchical tensor approximation shall be open to several material models, various types of geometry description and finally also different notions of uncertainty and fuzziness. The flexibility with respect to the latter aspect is firstly given by the possibility to set up the original parameter space in many different ways. Secondly the evaluation of uncertainty or fuzziness is majorly simplified by the functional representation of the quantities of interest.