Analysis of an innovative sampling strategy based on k-means clustering algorithm for POD and POD-DEIM reduced order models of a 2-D reaction-diffusion system

In this work, a model-order reduction methodology based on proper orthogonal decomposition (POD) and Galërkin projection is presented and applied to the simulation of the self-ignition of a stockpile of solid fuel. Self-ignition is a phenomenon associated with steep changes in space and time, yielding high gradients of state variables which demand grid refinement and, thus, increase of the computational burden. To cope with this difficulty, first, a full order model (FOM), generated by finite-difference discretisation of the PDEs constituting the differential model, is employed to generate reference solutions. Two different POD-based formulations are proposed: the classical POD-Galërkin is employed to generate reduced order models (ROM), then discrete empirical interpolation method (DEIM) is employed to deal with nonlinearities in a more efficient manner. These reduction techniques are further supplemented with an innovative sampling approach based on k-means clustering. The resulting agile ROM is validated against the FOM. Both model-order reduction strategies, particularly the POD-DEIM model, reproduce the FOM solutions with high accuracy and much lower computational cost: The results of the application of a combination of the DEIM algorithm and k-means clustering show that the computational time for the calculation of one solution reduces up to 1020 times, while remaining able to reproduce all bifurcation points found with the FOM, thus demonstrating quantitative and qualitative agreement.