Browsing by Subject "Numerical methods"
Now showing 1 - 3 of 3
Results Per Page
Sort Options
Item Application of wavelets in few-body problems.(2012-08) Hewawasam, KuraviThis study is an application of wavelet numerical techniques in solving a non-perturbative Yukawa Hamiltonian in light-front quantum field theory. Once the problem is stated in the form of an integral equation, a wavelet basis of a particular scale is used to discretize the problem into a dense matrix. Wavelets are a class of functions with special properties. Daubachies wavelets are a subset of wavelets defined to have vanishing lower order moments, enabling Daubachies 2 and 3 wavelet bases to exactly represent polynomials of degree up to two. These properties make them useful as a basis set for various numerical methods. It was observed that a kernel containing structure in fine scales requires a fine scaling function basis to converge closer to analytical results. Once the kernel matrix is obtained, the wavelet transform followed by an absolute thresholding filters the dense kernel matrix to a sparse matrix. The sparse matrix eigenvalue problem was then solved and compared with the original eigenvalue problem. It was observed that as long as the problem is discretized with a scale fine enough to resolve the features of the kernel, higher levels of filtering would still reproduce eigenvalues that agree with the unfiltered problem.Item Multiple time scale algorithms for gene network simulations.(2009-11) Sotiropoulos, VassiliosThe objective of this dissertation is the development and implementation of multiple time scale stochastic models necessary for analysis, design and construction of novel synthetic biological systems, such as gene networks. At the dawn of the 21 st century, scientists and engineers turned into engineering new biological systems. Synthetic biology emerged as a distinct discipline, combining biology and engineering towards the design and construction of new biological parts, devices and systems with useful applications. This ambitious endeavor would not have been possible, were it not for the recent, impressive discoveries in biology and the equally remarkable advances in biotechnology. Indeed, we can now literally "cut" and "paste" DNA at will. The impact in everyday life may be significant, with wide-ranging applications: from medicine, where gene regulatory networks can be used for gene therapy applications, to the production of biopolymers, to the removal of environmental pollutants, and to clean energy alternatives. Even though wet lab experiments have provided ample proof of concept, the challenge facing the scientific and engineering communities is how to rationally design novel biological systems. An answer lies with mathematical models and sophisticated algorithms. It is the same philosophy used to design many of the modern marvels of technology, such as airplanes. Analogously, sophisticated computer-aided design (CAD) algorithms, alongside with a minimal number of experiments, could be the standard in constructing novel biological systems, devices, even entire organisms, alleviating the need for expensive trial and error approaches. There are primarily three types of challenges in developing new CAD tools for synthetic biological systems, such as gene networks. First, the number of molecular components in biological systems is overwhelming. Second, all living microorganisms are impacted by thermal noise and on occasion behave randomly. Third, the time scales at which many of the biological phenomena occur can differ by many orders of magnitude, resulting in stiff mathematical descriptions. The aim of this thesis is the CAD of synthetic gene networks addressing these challenges. For that to be accomplished an important step is the development of multiple time scale methods for the efficient and accurate integration of stiff chemical Langevin equations. These describe the dynamics of many biological systems. Methods developed also include the description of noise through classical mathematical descriptions instead of the more demanding stochastic formulation. Algorithms developed as part of this dissertation are incorporated into CAD software tools built by our group. In the last part of this dissertation we discuss how such tools are employed for CAD of novel synthetic gene networks.Item Parallel numerical methods and data-driven analysis techniques for turbulent fluid-structure interaction(2021-06) Anantharamu, SreevatsaThis dissertation develops numerical methods and parallel codes to simulate turbulent fluid-structure interaction, and data-driven methods to understand the cause of this interaction. The unsteady pressure and shear-stress fluctuations within a turbulent flow can lead to structural vibration. These vibrations can radiate sound which can cause excessive noise. Large turbulent fluid loads can lead to large structural deformation. Such deformation can cause excessive stresses within the solid, damaging it. The tools developed in this thesis help predict this interaction and analyze the interaction's cause. For accurate simulation of turbulent flows in deforming geometries mappable to a unit cube, a new finite volume method is developed. This method discretizes the domain with quadratic hexahedral control volumes. It yields second-order accurate solution even in the presence of extremely skewed control volumes. Such control volumes can arise when the fluid mesh adapts to the deforming fluid-solid interface. A new cell-centered gradient approximation is developed using the Piola transform. This approximation yields second-order accurate gradients irrespective of the boundary condition. The commonly used Green-Gauss approximation can yield first-order accurate approximation in the presence of Dirichlet or Neumann boundary conditions. To simulate the structural deformation, an in-house parallel finite-element solver, MPCUGLES-SOLID, is developed. This solver can compute the response of compressible linear elastic materials (for e.g., steel and aluminum) and incompressible linear viscoelastic materials (for e.g., synthetic rubber and PDMS). For efficient solution of the spatially discretized problem, the former material requires the continuous Galerkin finite element method, while the latter requires the mixed finite-element method. To simplify code development of both these methods, we develop their unified implementation using specially designed data structures. A new method is developed to couple the finite volume fluid and the finite element solid solvers. This method allows for the concurrent execution of the two solvers. This concurrent execution is essential for the coupled solver's good parallel performance, especially for turbulent FSI problems. A new data-driven method is developed to study the wall-pressure fluctuations' sources in a turbulent channel flow. This method answers the questions -- for each frequency, how do turbulent fluid sources at different distances from the wall contribute to the wall-pressure fluctuation power spectral density (PSD)? To answer this question, the method combines the channel DNS data with the fluid's pressure fluctuation Poisson equation and spectral POD. The previous data-driven method is extended to study the fluid sources that contribute to the excitation of a plate in turbulent channel flow. This extended method answers the question -- for each frequency, how do turbulent fluid sources situated at different distances from the wall contribute to the plate-averaged displacement PSD? To answer this, the method combines the plate's modal decomposition with the channel DNS data, the fluid pressure fluctuation's Poisson equation and spectral POD. Finally, a new DMD algorithm, FOA based DMD, is developed to extract features from a general time-evolving large data. This method is streaming and can process extremely large data sets in parallel. Our algorithm can perform DMD of 201 snapshots of 240 million size in 3 seconds on 16,000 processors. The algorithm shows ideal strong scaling. Our new DMD algorithm's and a few existing DMD algorithm's finite-precision arithmetic error is analyzed. This error is shown to be proportional to (snapshot condition number)^p * O(machine epsilon), where the power āpā depends on the DMD algorithm. For most DMD algorithms, p is one, while for some algorithms, p is two. Therefore, for a given data set, the latter DMD algorithms amplify this error more than the former algorithms.