Geometric integration applied to moving mesh methods and degenerate Lagrangians

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this thesis we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. The proposed methods are readily applicable to (weakly) non-degenerate field theories---numerical results for the Sine-Gordon equation are presented.

In an attempt to extend our approach to degenerate field theories, in the last part of this thesis we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the 'Hamiltonian' equations of motion can be formulated as an index 1 differential-algebraic system. We then proceed to construct variational Runge-Kutta methods and analyze their properties. The general properties of Runge-Kutta methods depend on the 'velocity' part of the Lagrangian. If the 'velocity' part is also linear in the position coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the 'velocity' part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We also apply our methods to several models and present the results of our numerical experiments.

Part I. On the breaking of nonlinear dispersive waves. Part II. Variational principles in continuum mechanics

A model equation for water waves has been suggested by Whitham to study, qualitatively at least, the different kinds of breaking. This is an integro-differential equation which combines a typical nonlinear convection term with an integral for the dispersive effects and is of independent mathematical interest. For an approximate kernel of the form e^(-b|x|) it is shown first that solitary waves have a maximum height with sharp crests and secondly that waves which are sufficiently asymmetric break into "bores." The second part applies to a wide class of bounded kernels, but the kernel giving the correct dispersion effects of water waves has a square root singularity and the present argument does not go through. Nevertheless the possibility of the two kinds of breaking in such integro-differential equations is demonstrated.

Difficulties arise in finding variational principles for continuum mechanics problems in the Eulerian (field) description. The reason is found to be that continuum equations in the original field variables lack a mathematical "self-adjointness" property which is necessary for Euler equations. This is a feature of the Eulerian description and occurs in non-dissipative problems which have variational principles for their Lagrangian description. To overcome this difficulty a "potential representation" approach is used which consists of transforming to new (Eulerian) variables whose equations are self-adjoint. The transformations to the velocity potential or stream function in fluids or the scaler and vector potentials in electromagnetism often lead to variational principles in this way. As yet no general procedure is available for finding suitable transformations. Existing variational principles for the inviscid fluid equations in the Eulerian description are reviewed and some ideas on the form of the appropriate transformations and Lagrangians for fluid problems are obtained. These ideas are developed in a series of examples which include finding variational principles for Rossby waves and for the internal waves of a stratified fluid.

Estimation and inference for grasping and manipulation tasks using vision and kinesthetic sensors

This thesis presents a novel framework for state estimation in the context of robotic grasping and manipulation. The overall estimation approach is based on fusing various visual cues for manipulator tracking, namely appearance and feature-based, shape-based, and silhouette-based visual cues. Similarly, a framework is developed to fuse the above visual cues, but also kinesthetic cues such as force-torque and tactile measurements, for in-hand object pose estimation. The cues are extracted from multiple sensor modalities and are fused in a variety of Kalman filters.

A hybrid estimator is developed to estimate both a continuous state (robot and object states) and discrete states, called contact modes, which specify how each finger contacts a particular object surface. A static multiple model estimator is used to compute and maintain this mode probability. The thesis also develops an estimation framework for estimating model parameters associated with object grasping. Dual and joint state-parameter estimation is explored for parameter estimation of a grasped object's mass and center of mass. Experimental results demonstrate simultaneous object localization and center of mass estimation.

Dual-arm estimation is developed for two arm robotic manipulation tasks. Two types of filters are explored; the first is an augmented filter that contains both arms in the state vector while the second runs two filters in parallel, one for each arm. These two frameworks and their performance is compared in a dual-arm task of removing a wheel from a hub.

This thesis also presents a new method for action selection involving touch. This next best touch method selects an available action for interacting with an object that will gain the most information. The algorithm employs information theory to compute an information gain metric that is based on a probabilistic belief suitable for the task. An estimation framework is used to maintain this belief over time. Kinesthetic measurements such as contact and tactile measurements are used to update the state belief after every interactive action. Simulation and experimental results are demonstrated using next best touch for object localization, specifically a door handle on a door. The next best touch theory is extended for model parameter determination. Since many objects within a particular object category share the same rough shape, principle component analysis may be used to parametrize the object mesh models. These parameters can be estimated using the action selection technique that selects the touching action which best both localizes and estimates these parameters. Simulation results are then presented involving localizing and determining a parameter of a screwdriver.

Lastly, the next best touch theory is further extended to model classes. Instead of estimating parameters, object class determination is incorporated into the information gain metric calculation. The best touching action is selected in order to best discern between the possible model classes. Simulation results are presented to validate the theory.

Dirac spectra, summation formulae, and the spectral action

Noncommutative geometry is a source of particle physics models with matter Lagrangians coupled to gravity. One may associate to any noncommutative space (A, H, D) its spectral action, which is defined in terms of the Dirac spectrum of its Dirac operator D. When viewing a spin manifold as a noncommutative space, D is the usual Dirac operator. In this paper, we give nonperturbative computations of the spectral action for quotients of SU(2), Bieberbach manifolds, and SU(3) equipped with a variety of geometries. Along the way we will compute several Dirac spectra and refer to applications of this computation.

Interplay of martensitic phase transformation and plastic slip in polycrystals

Inspired by key experimental and analytical results regarding Shape Memory Alloys (SMAs), we propose a modelling framework to explore the interplay between martensitic phase transformations and plastic slip in polycrystalline materials, with an eye towards computational efficiency. The resulting framework uses a convexified potential for the internal energy density to capture the stored energy associated with transformation at the meso-scale, and introduces kinetic potentials to govern the evolution of transformation and plastic slip. The framework is novel in the way it treats plasticity on par with transformation.

We implement the framework in the setting of anti-plane shear, using a staggered implicit/explict update: we first use a Fast-Fourier Transform (FFT) solver based on an Augmented Lagrangian formulation to implicitly solve for the full-field displacements of a simulated polycrystal, then explicitly update the volume fraction of martensite and plastic slip using their respective stick-slip type kinetic laws. We observe that, even in this simple setting with an idealized material comprising four martensitic variants and four slip systems, the model recovers a rich variety of SMA type behaviors. We use this model to gain insight into the isothermal behavior of stress-stabilized martensite, looking at the effects of the relative plastic yield strength, the memory of deformation history under non-proportional loading, and several others.

We extend the framework to the generalized 3-D setting, for which the convexified potential is a lower bound on the actual internal energy, and show that the fully implicit discrete time formulation of the framework is governed by a variational principle for mechanical equilibrium. We further propose an extension of the method to finite deformations via an exponential mapping. We implement the generalized framework using an existing Optimal Transport Mesh-free (OTM) solver. We then model the $\alpha$--$\gamma$ and $\alpha$--$\varepsilon$ transformations in pure iron, with an initial attempt in the latter to account for twinning in the parent phase. We demonstrate the scalability of the framework to large scale computing by simulating Taylor impact experiments, observing nearly linear (ideal) speed-up through 256 MPI tasks. Finally, we present preliminary results of a simulated Split-Hopkinson Pressure Bar (SHPB) experiment using the $\alpha$--$\varepsilon$ model.