Partial Differential Equations Commons^™

Safe Control Design For Quadruped Locomotion In Unstructured Environments Using Linear Transfer Operators, Sriram Sundar Krishnamoorthy Shankara Narayanan

All Dissertations

Deploying quadruped robots in unstructured, obstacle-rich environments requires control and planning methods that remain safe and reliable despite complex terrain geometry, limited sensing, and inevitable modeling errors. This thesis develops operator-theoretic tools for safe control design of robotic systems using linear transfer operators, with a focus on quadruped locomotion in unstructured environments. The central goal is to develop a unified operator-theoretic framework for safe control design based on the Perron–Frobenius (P–F) and Koopman operators. In particular, the thesis leverages \emph{density functions} to develop safe navigation frameworks in the dual space of densities. In the operator-theoretic perspective, the P–F operator governs …

Accurate Temporal Integration Schemes For Nonlinear Adsorption Problems, Evan D. Butterworth

All Dissertations

We consider a nonlinear transport problem to model the chromatography process of high-capacity multimodal membranes. Robust and efficient algorithms that simulate these bioseparation processes are critical to developing therapeutics for various chronic illnesses and infectious diseases. However, much of the current methodology focuses on stabilization and linearization techniques, often implementing low-order time-discretizations and linearized adsorption, resulting in inefficiencies and inaccuracies in the numerical solution. Utilizing Rothe's method, we develop various time-discretization schemes coupled with the finite element method to solve the fully implicit problems. Stability and solvability results are presented for several methods. Through multiple high-level software implementations paired with …

Robust And Efficient Solvers For Physics-Based Pde’S, Elizabeth Hawkins

All Dissertations

This work was partially supported by the U.S. Department of Energy under award DE- SC0025292, by NSF grant DMS 2152623, and by NSF grant DMS 2011490.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Mathematical Multifaceted Integrated Capability Centers (MMICCs) program, under Field Work Proposal 22-025291 (Multifaceted Math- ematics for Predictive Digital Twins (M2dt)), Field Work Proposal 23-020467, and Computing and Information Sciences (CIS) investment area in the Laboratory Directed Research and Development program at Sandia National Laboratories. This written work is authored by an employee …

Domain Decomposition For Coupled Systems Of Fluid-Structure Interaction And Numerical Modeling For Thin Film Polymers, Amy De Castro

All Dissertations

We consider two primary areas of physical application in this work: fluid interaction systems with either linear elastic structures or with poroelastic structures, and thin film polymers, where the majority of the work focuses on the fluid-structure interaction systems.

In the first chapter, we present a strongly coupled partitioned method for fluid structure interaction (FSI) problems based on a monolithic formulation of the system which employs a Lagrange multiplier (LM). We prove that both the semi-discrete and fully discrete formulations are well-posed. To derive the partitioned scheme, a Schur complement equation, which implicitly expresses the Lagrange multiplier and the fluid …

Domain Decomposition Methods For Fluid-Structure Interaction Problems Involving Elastic, Porous, Or Poroelastic Structures, Hemanta Kunwar

All Dissertations

We introduce two global-in-time domain decomposition methods, namely the Steklov-Poincare method and Schwarz waveform relaxation (SWR) method using Robin transmission conditions (or the Robin method), for solving fluid-structure interaction systems involving elastic, porous, or poroelastic structure. These methods allow us to formulate the coupled system as a space-time interface problem and apply iterative algorithms directly to the evolutionary problem. Each time-dependent fluid and the structure subdomain problem is solved independently, which enables the use of different time discretization schemes and time step sizes in the subsystems. This leads to an efficient way of simulating time-dependent multiphysics phenomena. For the fluid-porous …

Null Space Removal In Finite Element Discretizations, Pengfei Jia

All Theses

Partial differential equations are frequently utilized in the mathematical formulation of physical problems. Boundary conditions need to be applied in order to obtain the unique solution to such problems. However, some types of boundary conditions do not lead to unique solutions because the continuous problem has a null space. In this thesis, we will discuss how to solve such problems effectively. We first review the foundation of all three problems and prove that Laplace problem, linear elasticity problem and Stokes problem can be well posed if we restrict the test and trial space in the continuous and discrete finite element …

Recovering Coefficients Of Second-Order Hyperbolic And Plate Equations Via Finite Measurements On The Boundary, Scott Randall Scruggs

All Dissertations

Abstract In this dissertation, we consider the inverse problem for a second-order hyperbolic equation of recovering n + 3 unknown coefficients defined on an open bounded domain with a smooth enough boundary. We also consider the inverse problem of recovering an unknown coefficient on the Euler- Bernoulli plate equation on a lower-order term again defined on an open bounded domain with a smooth enough boundary. For the second-order hyperbolic equation, we show that we can uniquely and (Lipschitz) stably recover all these coefficients from only using half of the corresponding boundary measurements of their solutions, and for the plate equation, …

Advancements In Fluid Simulation Through Enhanced Conservation Schemes, Sean Ingimarson

All Dissertations

To better understand and solve problems involving the natural phenomenon of fluid and air flows, one must understand the Navier-Stokes equations. Branching several different fields including engineering, chemistry, physics, etc., these are among the most important equations in mathematics. However, these equations do not have analytic solutions save for trivial solutions. Hence researchers have striven to make advancements in varieties of numerical models and simulations. With many variations of numerical models of the Navier-Stokes equations, many lose important physical meaningfulness. In particular, many finite element schemes do not conserve energy, momentum, or angular momentum. In this thesis, we will study …

A Conservative Numerical Scheme For The Multilayer Shallow Water Equations, Evan Butterworth

All Theses

An energy-conserving numerical scheme is developed for the multilayer shallow water equations (SWE’s). The scheme is derived through the Hamiltonian formulation of the inviscid shallow water flows related to the vorticity-divergence variables. Through the employment of the skew-symmetric Poisson bracket, the continuous system for the multilayer SWE’s is shown to preserve an infinite number of quantities, most notably the energy and enstrophy. An energy-preserving numerical scheme is then developed through the careful discretization of the Hamiltonian and the Poisson bracket, ensuring the skew-symmetry of the latter. This serves as the groundwork for developing additional schemes that preserve other conservation properties …