Model order reduction in space and parameter dimension – Damage based modelling of polymorph uncertainty in connection with robustness and reliability
A continuation of the topic is currently taking place in the second project phase.
Summary of the first funding phase
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 to reduce the computational effort significantly.
Adaptive proper orthogonal decomposition (APOD): Classical model reduction methods have been known for over 50 years. They have also been applied in the field of continuum mechanics for more than 20 years. For the projection-based proper orthogonal decomposition (POD), simulations are first calculated for certain parameter combinations and general information about the behavior of the problem are extracted. Typically, many problems have characteristic deformation states, which are similar even with slight variations of individual parameters. These characteristic deformation states are typically called "modes" in the literature or "basic functions" in mathematics. For further calculations of very similar problems these modes can be used to reduce the calculation time, especially the time for solving the system of equations. Here results can be obtained which are almost identical to corresponding reference solutions. Especially for nearly linear problems the method is extremely efficient. For problems that consider plasticity and damage, the efficiency or accuracy of the POD suffers drastically. In order to apply the model order reduction for such nonlinear problems, an adaptive proper orthogonal decomposition was developed in the first funding phase of the project. It selects adaptively from a range of existing modes so that the most suitable modes are used for each state. The accuracy of the results was significantly improved compared to POD with similar computational effort.
Hierarchical tensor approximation (HTA): In order to consider different influences of the parameters and to set up corresponding density functions, the objective functions must be integrated over a high-dimensional space. Since the objective functions are usually unknown, it is often inevitable to generate a very large number of samples. Since this is still too expensive even with model order reduction methods, a low-rank approximation was obtained to generate many required samples. In addition, the low-rank approximation was used to generate information which helped to improve the choice of precalculations and therefore to improve accuracy of the results of the reduced computations.
Essential project findings
- Development of adaptive proper orthogonal decomposition (APOD)
- Application of APOD for problems including plasticity, damage and failure
- Training of hierarchical tensor approximation with reduced order models
- Use of hierarchical tensor approximation to improve model order reduction techniques
- Use of the combination of APOD and HTA to perform polymorphic uncertainty quantification
- Creation and testing of a surrogate model by means of Gaussian processes for a problem with plasticity, damage and random fields, application of polymorphic uncertainty quantification using this surrogate model
- Publication 1