Browsing by Author "Bangerth, Wolfgang, advisor"
Now showing 1 - 3 of 3
Results Per Page
Sort Options
Item Open Access Computational feasibility of simultaneous analysis and design in interior point topology optimization(Colorado State University. Libraries, 2023) O'Connor, Justin, author; Bangerth, Wolfgang, advisor; Weinberger, Chris, committee member; Shipman, Patrick, committee member; Liu, James, committee member; Weinberger, Chris, committee memberTopology optimization is a class of algorithms designed to optimize a design or structure to accomplish some goal. It is part of a process of computer generated design that allows engineers to design better products faster. One such algorithm that has piqued the imagination of developers is called Simultaneous Analysis and Design (SAND), especially in the context of Interior Point Optimization (IPO). This method is known to generate extremely optimal designs, and is good at avoiding local minima. However, this method is not used in practice, due to its computational cost. This thesis examines the SAND IPO method, and develops an effective algorithm to generate a design using it. I begin by discussing nonlinear optimization algorithms, selecting pieces that work together for this problem, to generate a cohesive algorithm for the whole process. Inside this developed algorithm, as with most nonlinear optimization algorithms, the most ex- pensive part is a linear solve. In my case, it is a linear solve of a block system. I develop and implement a multi-tier preconditioning approach to solve this system in a reasonable amount of time. Finally, I present a large topology optimization problem presented in three dimensions that has been solved using IPO and SAND, demonstrating the usability of the implemented algorithm.Item Open Access Numerical solution of the Black-Scholes equation using finite element methods(Colorado State University. Libraries, 2023) Anderson, Tyler, author; Bangerth, Wolfgang, advisor; Aristoff, David, committee member; Wang, Tianyang, committee memberThe Black-Scholes model is a well known model for pricing financial options. This model takes the form of a partial differential equation (PDE) that, surprisingly, is deterministic. In the special case where the option only has one single underlying asset, what is called the one dimensional version of the Black-Scholes model, there exists an analytical solution. In higher dimensions, however, there is no such analytical solution. This higher dimensional version refers to what is called a Basket-Case Option. This means that to get a solution to this Basket-Case Option PDE, one must employ numerical methods. This thesis will first discuss the stochastic calculus theory necessary to derive the Black-Scholes model, then will explain in detail the time and space discretization used to solve the PDE using a Finite Element Method (FEM). Finally, this thesis will explain some of the results and convergence of this numerical solution.Item Open Access Quantification and application of uncertainty in the formation of nanoparticles(Colorado State University. Libraries, 2023) Long, Danny, author; Bangerth, Wolfgang, advisor; Shipman, Patrick, committee member; Liu, Jiangguo, committee member; Finke, Richard, committee memberNanoparticles are essential across many scientific applications, but their properties are size-dependent. Despite the usefulness of producing monodisperse particle size distributions, it still remains a challenge to fully understand – and hence be able to control – nanoparticle formation reactions due to limitations in what can be observed experimentally. This thesis transfers mathematical, statistical, and computational techniques to this area of nanoparticle chemistry to substantially bolster the sophistication of the quantitative analysis used to better understand nanoparticle systems. First, more efficient software is developed to simulate the reactions. Then, parameter estimation is performed in a robust manner through Bayesian inference, where I demonstrate the ability to parameterize nonlinear ordinary differential equations in such a way that I can fit the observed data and quantify the uncertainty in the parameter estimates. From Bayesian inference, I build three additional analysis frameworks. (1) Model selection through a Bayesian framework; (2) optimizing the yield of the nanoparticle-forming reactions while accounting for uncertainty; and (3) optimizing future measurements to collect data providing the most new information. The culmination of this thesis provides a quantitative framework to analyze arbitrary nanoparticle systems to complement and fill in the gaps of the current experimental techniques.