Theoretical foundations for Spectral Submanifold computation.
In this section, we provide an introduction to the theory of SSM-Theory.
- To get acquainted with the concept of invariant manifolds in general, we first provide a brief introduction to the history of invariant manifold theory.
- In a next step, the mathematical and physical setting for Spectral Submanifolds (SSMs) is presented. In SSM-Theory we give an overview of the relevant mathematical results, which guarantee the existence of the SSMs, thereby ensuring that mathematical rigour serves as the basis for all computations performed thereafter.
- To stress the importance of invariance for the modelling subspace of reduced order models (ROM) in general, we consequently use a small example to exhibit how SSM-Theory achieves what other approaches fail to guarantee: providing an exact ROM for the full dynamical system, precisely due to the modelling space (the SSM) being invariant.
- Next we introduce the computational methodology employed for the computation of SSMs and the reduced dynamics on them in SSM-Computation.
- Consequently we take a look at the spectrum of the linear part of the underlying dynamical systems. This serves for better understanding the conditions which are necessary for the SSMs to exist, their importance for model reduction and how to handle the case of resonances in the spectrum.
- In semi-intrusive computation and non-intrusive computation we explain how the algorithm can be executed without explicit knowledge about the nonlinear forces governing the system.