What are the current trends and developments in reduced order methods for FEA?

1 What are coupled problems?

Coupled problems are those that involve multiple physical domains or processes that interact with each other. For example, fluid-structure interaction (FSI) is a coupled problem that involves the mutual influence of a fluid and a solid structure, such as an aircraft wing, a blood vessel, or a wind turbine. Coupled problems are often challenging to model and simulate, because they require solving different sets of governing equations, with different spatial and temporal scales, and different boundary and interface conditions.

2 Why use ROMs for coupled problems?

ROMs are attractive for coupled problems because they can reduce the dimensionality and complexity of the original models, while preserving the essential features and dynamics of the system. ROMs can also facilitate the coupling of different models, by providing a common basis for communication and data transfer between them. ROMs can enable faster and cheaper simulations, which can be useful for parametric studies, optimization, uncertainty quantification, or real-time control.

3 How do ROMs work for coupled problems?

ROMs work by constructing a low-dimensional approximation of the solution space of the original model, based on a set of representative snapshots or modes. This approximation can cover the whole domain or just a subdomain, and it can use either a linear combination of modes or a nonlinear mapping function. One of the most common ROM techniques for coupled problems are Proper Orthogonal Decomposition (POD), which uses a Singular Value Decomposition (SVD) to extract dominant modes of variation from snapshots. Another is Dynamic Mode Decomposition (DMD), which uses linear regression to identify modes and frequencies that govern the temporal evolution of the system. Reduced Basis Methods (RBM) uses a greedy algorithm to select relevant modes from a parametric family of solutions. Lastly, Proper Generalized Decomposition (PGD) uses a separated representation to decompose the solution into a product of functions of each variable or parameter.

4 What are the challenges and limitations of ROMs for coupled problems?

ROMs for coupled problems come with a range of challenges and limitations, such as the accuracy and stability of the ROMs being dependent on the quality and quantity of the snapshots or modes used to construct the approximation. Coupling ROMs with different models can also lead to errors or inconsistencies due to the mismatch of scales, discretizations, or boundary conditions. Furthermore, constructing and updating ROMs may be expensive or impractical for certain problems, such as nonlinear, transient, or stochastic problems, or for problems with high-dimensional parameters or outputs. In addition, ROMs may lose their validity or efficiency over time due to changes in the system or environment.