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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

Wu, XiaofengReduced Deformable Body Simulation with Richer Dynamics
Doctor of Philosophy, The Ohio State University, 2016, Computer Science and Engineering
Deformable body simulation can provide visually interesting results that have wide applications in both entertainment industries and scientific fields. However, it is computationally demanding when simulating detailed models with high degrees of freedom (DoFs). Subspace simulation is known for its ability to significantly accelerate the simulation by constraining the deformation of the model to lie within a prescribed low-dimensional space so that the high-DoFs dynamic system is reduced to a much lower one. State-of-the-art subspace simulation technique only allows around 100 simulation bases to be used for real-time applications. This limitation causes many interesting deformations results to be lost or even leads to deformation artifacts. In this dissertation, we focus on developing new techniques that empower subspace simulation to capture richer deformation dynamics. First, we propose a unified approach for simulating reduced multi-domain objects, where each domain of the object is simulated in its own subspace. The key challenge in implementing this method is how to handle the coupling among multiple deformable do- mains, so that the overall effect is free of gap or locking issues. We present a new domain decomposition framework that connects two disjoint domains through coupling elements. Under this framework, we present a unified simulation system that solves subspace deformations and rigid motions of all of the domains by a single linear solve. Since the coupling elements are part of the deformable body, their elastic properties are the same as the rest of the body and our system does not need stiffness parameter tuning. To quickly evaluate the reduced elastic forces and their Jacobian matrices caused by the coupling elements, we further develop two cubature optimization schemes using uniform and non-uniform cubature weights. Our experiment shows that the whole system can efficiently handle large and complex scenes, many of which cannot be easily simulated by previous techniques without limitations. Second, we designed a novel single-domain subspace solver that is superior to previous methods in terms of theoretical time complexity, actual running time and GPU implementation efficiency. Inspired by the recently proposed Projective Dynamics (PD) framework that makes use of a constant approximate Hessian for simulation, we proposed a new sub- space solver that utilizes BFGS’s self-correcting property to directly approximate the in- verse of the Hessian. Thus, GPU-unfriendly linear solve can be avoided. Moreover, our method does not require the expensive run-time evaluation of Hessian, which we identify as the bottleneck of existing subspace solver and is not GPU implementation friendly either. We designed experiments to show that our subspace solver can achieve several orders- of-magnitude speedup and much better scalability w.r.t. number of simulation bases over existing method.

Committee:

Huamin Wang (Advisor); Yusu Wang (Committee Member); Han-Wei Shen (Committee Member)

Subjects:

Computer Science

Keywords:

deformable body simulation;model reduction;subspace simulation;domain decomposition;

Huston, Genevieve A.Practical model reduction for large flexible structures using residue comparison techniques
Master of Science (MS), Ohio University, 1991, Electrical Engineering & Computer Science (Engineering and Technology)
Practical model reduction for large flexible structures using residue comparison techniques

Committee:

Jerrel Mitchell (Advisor)

Keywords:

Practical Model Reduction; Large Flexible Structures; Residue Comparison Techniques