POD-Galerkin based ROM for fluid flow with moving boundaries and the model adaptation in parametric space

Abstract

In this study, a global Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order model (ROM) is proposed. It is extended from usual fixed-domain problems to more general fluid-solid systems with moving boundaries/interfaces. The idea of the extension is similar to the immersed boundary method in numerical simulations which uses embedded forcing terms to represent boundary motions and domain changes. This immersed boundary method allows a globally defined fixed domain including both fluid and solid, where POD-Galerkin projection can be directly applied. However, such a modified approach cannot get away with the unsteadiness of boundary terms which appear as time-dependent coefficients in the new Galerkin model. These coefficients need to be pre-computed for prescribed periodic motion, or worse, to be computed at each time step for non-prescribed (e.g. with fluid-structure interaction) or non-periodic situations. Though computational time for each unsteady coefficient is smaller than the coefficients in a typical Galerkin model, because the associated integration is only in the close neighborhood of moving boundaries. The time cost is still much higher than a typical Galerkin model with constant coefficients. This extra expense for moving-boundary treatment eventually undermines the value of using ROMs. An aggressive approach is to decompose the moving boundary/domain to orthogonal modes and derive another low-order model with fixed coefficients for boundary motion. With this domain decomposition, an approach including two coupled low-order models both with fixed coefficients is proposed. Therefore, the new global ROM with decomposed approach is more efficient. Though the model with the domain decomposition is less accurate at the boundary, it is a fair trade-off for the benefit on saving computational cost. The study further shows, however, that the most time-consuming integration in both approaches, which come from the unsteady motion, has almost negligible impact on the overall dynamics. Dropping these time-consuming terms reduces the computation cost by at least one order while having no obvious effect on model accuracy. Based on this global POD-Galerkin based ROM with forcing term, an improved ROM which can handle the parametric variation of body motions in a certain range is also presented. This study shows that these forcing terms not only represent the moving of the boundary, but also decouple the moving parameters from the computation of model coefficients. The decoupling of control parameters provides the convenience to adapt the model for the prediction on states under variation of control parameters. An improved ROM including a shit mode seems promising in model adaptation for typical problems in a fixed domain. However, the benefit from adding a shit mode to model diminishes when the method is applied to moving-boundary problems. Instead, a combined model, which integrates data from a different set of parameters to generate the POD modes, provides a stable and accurate ROM in a certain range of parametric space for moving-boundary problems. By introducing more data from a different set of parameters, the error of the new model can be further reduced. This shows that the combined model can be trained by introducing more and more information. With the idea of the combined model, the improved global ROM with forcing terms shows impressive capability to predict problems with different unknown moving parameters, and can be used in future parametric control and optimization problems.