Search Results (1 - 2 of 2 Results)

Sort By  
Sort Dir
 
Results per page  

Hall, Brenton TaylorUsing the Non-Uniform Dynamic Mode Decomposition to Reduce the Storage Required for PDE Simulations
Master of Mathematical Sciences, The Ohio State University, 2017, Mathematical Sciences
Partial Differential Equation simulations can produce large amounts of data that are very slow to transfer. There have been many model reduction techniques that have been proposed and utilized over the past three decades. Two popular techniques Proper Orthogonal Decomposition and Dynamic Mode Decomposition have some hindrances. Non-Uniform Dynamic Mode Decomposition (NU-DMD), which was introduced in 2015 by Gueniat et al., that overcomes some of these hindrances. In this thesis, the NU-DMD's mathematics are explained in detail, and three versions of the NU-DMD's algorithm are outlined. Furthermore, different numerical experiments were performed on the NU-DMD to ascertain its behavior with repect to errors, memory usage, and computational efficiency. It was shown that the NU-DMD could reduce an advection-diffusion simulation to 6.0075% of its original memory storage size. The NU-DMD was also applied to a computational fluid dynamics simulation of a NASA single-stage compressor rotor, which resulted in a reduced model of the simulation (using only three of the five simulation variables) that used only about 4.67% of the full simulation's storage with an overall average percent error of 8.90%. It was concluded that the NU-DMD, if used appropriately, could be used to possibly reduce a model that uses 400GB of memory to a model that uses as little as 18.67GB with less than 9% error. Further conclusions were made about how to best implement the NU-DMD.

Committee:

Ching-Shan Chou (Advisor); Jen-Ping Chen (Committee Member)

Subjects:

Aerospace Engineering; Applied Mathematics; Computer Science; Mathematics; Mechanical Engineering

Keywords:

Fluid Dynamics; Fluid Flow; Model Reduction; Partial Differential Equations; reducing memory; Dynamic Mode Decomposition; Decomposition; memory; Non-Uniform Dynamic Mode Decomposition

Deshmukh, RohitModel Order Reduction of Incompressible Turbulent Flows
Doctor of Philosophy, The Ohio State University, 2016, Aero/Astro Engineering
Galerkin projection is a commonly used reduced order modeling approach; however, stability and accuracy of the resulting reduced order models are highly dependent on the modal decomposition technique used. In particular, deriving stable and accurate reduced order models from highly turbulent flow fields is challenging due to the presence of multi-scale phenomenon that cannot be ignored and are not well captured using the ubiquitous Proper Orthogonal Decomposition (POD). A truncated set of proper orthogonal modes is biased towards energy dominant, large-scale structures and results in over-prediction of kinetic energy from the corresponding reduced order model. The accumulation of energy during time-integration of a reduced order model may even cause instabilities. A modal decomposition technique that captures both the energy dominant structures and energy dissipating small scale structures is desired in order to achieve a power balance. The goal of this dissertation is to address the stability and accuracy issues by developing and examining alternative basis identification techniques. In particular, two modal decomposition methods are explored namely, sparse coding and Dynamic Mode Decomposition (DMD). Compared to Proper Orthogonal Decomposition, which seeks to truncate the basis spanning an observed data set into a small set of dominant modes, sparse coding is used to identify a compact representation that span all scales of the observed data. Dynamic mode decomposition seeks to identity bases that capture the underlying dynamics of a full order system. Each of the modal decomposition techniques (POD, Sparse, and DMD) are demonstrated for two canonical problems of an incompressible flow inside a two-dimensional lid-driven cavity and past a stationary cylinder. The constructed reduced order models are compared against the high-fidelity solutions. The sparse coding based reduced order models were found to outperform those developed using the dynamic mode and proper orthogonal decompositions. Furthermore, energy component analyses of high-fidelity and reduced order solutions indicate that the sparse models capture the rate of energy production and dissipation with greater accuracy compared to the dynamic mode and proper orthogonal decomposition based approaches. Significant computational speedups in the fluid flow predictions are obtained using the computed reduced order models as compared to the high-fidelity solvers.

Committee:

Jack McNamara (Advisor); Datta Gaitonde (Committee Member); Ryan Gosse (Committee Member); Joseph Hollkamp (Committee Member); Mohammad Samimy (Committee Member)

Subjects:

Aerospace Engineering

Keywords:

Turbulent flows; reduced order modeling; Navier-Stokes equations; nonlinear dynamics; Galerkin projection; modal decomposition; proper orthogonal decomposition; dynamic mode decomposition; sparse coding