Model Reduction for a Pulsed Detonation Combuster via Shifted Proper Orthogonal Decomposition

Abstract

We propose a new algorithm to compute a shifted proper orthogonal decomposition (sPOD) for systems dominated by multiple transport velocities. The sPOD is a recently proposed mode decomposition technique which overcomes the poor performance of classical methods like the proper orthogonal decomposition (POD) for transport-dominated phenomena. This is achieved by identifying the transport directions and velocities and by shifting the modes in space to track the transports. Our new algorithm carries out a residual minimization in which the main computational cost arises from solving a nonlinear optimization problem scaling with the snapshot dimension. We apply the algorithm to snapshot data from the simulation of a pulsed detonation combuster and observe that very few sPOD modes are sufficient to obtain a good approximation. For the same accuracy, the common POD needs ten times as many modes and, in contrast to the sPOD modes, the POD modes do not reflect the moving front profiles properly.

Efficient Wildland Fire Simulation via Nonlinear Model Order Reduction

“…A theoretical analysis of this optimization problem can be found in ( [22], Section 4) and [54], whereas corresponding numerical algorithms for the special case that T is a discretized shift operator have been proposed in [28][29][30], known as sPOD. For further details, we refer to the forthcoming numerical examples presented in Sections 5.3 and 5.4.…”

“…If the system at hand features both transport and non-transported phenomena, it may be reasonable to add POD modes to the approximation ansatz; see, for instance, [30]. This can be achieved by adding an additional transformation matrix that equals the identity matrix and setting the corresponding path variable to zero.…”

“…The method we use to compute the basis functions according to the approximation ansatz ( 16) is outlined in Sections 5.3.1 and 5.4.1, respectively. For transforming the basis functions, we use the constant extrapolation shift operator, which has been applied in [30] for constructing low-dimensional descriptions of reaction fronts in the context of combustion. Let us emphasize that the supply mass fraction attains different (approximately) constant values before and after the reaction front, such that a shift operator with periodic boundary conditions would lead to unphysical supply mass fraction modes, which do not properly capture the shape of the respective reaction front.…”

“…In contrast, techniques that are based on solving the minimization problem (15) or related minimization problems, cf. [29,30], involve expensive iterative solvers for an optimization problem, whose number of parameters scales with the dimension of the spatial and temporal discretization.…”

“…This showcases the potential of MOR methods tailored to transport phenomena in real-time wildland fire simulations. Our framework builds on an efficient (offline) decomposition [28][29][30] of already computed or measured solution data and constructs the reduced-order model (ROM) via residual minimization as detailed in [19,22]. Due to the nonlinearity of the projection method, the resulting ROM still depends on the original dimension, which conspires against an efficient simulation.…”