Reduced-Order Modeling for Optimization and Control of Complex Flows

Abstract

OBJECTIVESThe primary objectives of this research were to develop reduced-order modeling methodologies for control and optimization applications. This was achieved through three specific research goals. First of all, we focused on improving the accuracy of reduced-order model simulations by including closure terms (that can be implemented in practical applications). Secondly, we used sensitivity analysis to improve the accuracy of models over ranges of parameters. Finally, we developed a better methodology for using reduced-order modeling methods for feedback control calculations. We made significant advances in each of these three areas. ACCOMPLISHMENTS OverviewOur research program focused on reduced-order models for large-scale systems with an emphasis on fluids and application to control and optimization problems. In this summary, we will focus on three results that correspond to the three main tasks outlined in our original proposal. In the first section, we provide an overview of our progress introducing closure models for POD / Galerkin dynamical systems based on the advances in the turbulence modeling community. Results are presented for Smagorinsky closure models in two-dimensional and three-dimensional flows about circular cylinders. This includes those corresponding to a multi-level discretization approach to make this closure approach feasible. In the second section, we highlight the use of sensitivity analysis to improve the accuracy of reduced-order models over ranges in parameters. Each of these two approaches (closure and sensitivity) provide significant improvements to reduced-order models making them more attractive as surrogates in optimal design problems. In the third section, we present a novel approach to solving flow control problems by developing reduced-order models specifically for Lyapunov and Chandrasekhar equations (rather than designing controllers for reduced-order models). Closure for POD ModelsAccurate modeling of complex flows using the proper orthogonal decomposition (POD) requires closure terms that account for the discarded (truncated) basis functions. Prior to this research, closure models were either based on tuning model coefficients to match data or based on simplified turbulence models. The latter is due to the desire to build models that can be evaluated rapidly. However, as our preliminary studies have shown, introducing models based on modern turbulent flow practices leads to more accurate reduced-order models at a significantly higher computational cost. A survey of our closure modeling approaches appear in papers #7, 10, 11, 18, 24, 39, and 47. An enabling method for implementing our closure models is two-level (or multi-level) discretization of the nonlinear closure terms. The general approach is to use coarser meshes to discretize the nonlinear closure terms involving the lower order POD modes. We highlight the main results from this study using a two-dimensional flow past a cylinder at Reynolds number Re = 200 and a three-dimensional flow past a cylinder at Reynol...

Nonlinear model reduction based on the finite element method with interpolated coefficients: Semilinear parabolic equations

“…On the other hand, the efficiency is severely limited by the nonlinearity of the system . For treating the first issue, research has been done in two main directions: (i) to improve the POD basis functions ; (ii) to improve the ROM by modeling the effect of discarded POD basis . To our knowledge, the research for seeking a reliable reduced‐order modeling approach for general complex systems is still active, however, beyond the scopes of this article.…”