The science of autonomy is the systematic development of fundamental knowledge about autonomous decision making and task completing in the form of testable autonomous methods, models and systems. In ocean applications, it involves varied disciplines that are not often connected. However, marine autonomy applications are rapidly growing, both in numbers and in complexity. This new paradigm in ocean science and operations motivates the need to carry out interdisciplinary research in the science of autonomy. This chapter reviews some recent results and research directions in time-optimal path planning and optimal adaptive sampling. The aim is to set a basis for a large number of vehicles forming heterogeneous and collaborative underwater swarms that are smart, i.e. knowledgeable about the predicted environment and their uncertainties, and about the predicted effects of autonomous sensing on future operations. The methodologies are generic and applicable to any swarm that moves and senses dynamic environmental fields. However, our focus is underwater path planning and adaptive sampling with a range of vehicles such as AUVs, gliders, ships or remote sensing platforms.

Schemes for the incompressible Navier-Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit-Explicit Runge-Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed–element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.

Abstract

In a variety of disciplines including atmospheric, oceanic, hydrologic and environmental sciences, large numerical simulations have become an essential tool for understanding the physical processes, synthesizing data, and for prediction. A key problem for modeling these dynamical systems is how to deal with uncertainties and error in the models’ representation of key physical processes. This talk will introduce some of the recent ensemble-based data assimilation approaches such as the use of an ensemble Kalman filter for Simultaneous State and Parameter Estimation (SSPE) in the treatment and quantification of model error and uncertainties. Applications of SSPE to a variety of phenomena ranging from the atmospheric boundary layer transport, air-sea fluxes and a physically based land-surface hydrologic model will be presented.

Biography

Prof. Zhang’s research interests include atmospheric dynamics and predictability, data assimilation, tropical cyclones, gravity waves and regional-scale climate. He earned his B.S. and M.S. in meteorology from Nanjing University, China in 1991 and 1994, respectively, and his Ph.D. in atmospheric science in 2000 from North Carolina State University. He has authored/co-authored over 150 peer reviewed journal publications that have a total of more than 3300 citations. He has received numerous awards for his research and service.

Wear is the process of material removal under abrasion. Multi-material systems are characterized by non-uniform wear-rates that are leading to non-planar profiles under polishing. Some recent iterative models have predicted with success the well established experimental fact, that under an initially uniform pressure load, multi-component wearing surfaces reach asymptotically a steady-state profile that continues to recess at a constant rate. In a first part, a continuous model is derived from the original iterative scheme, and extrapolated to a mathematical, abstract, but versatile framework. It is shown that the steady-state can be computed directly by solving a time-independent elliptic partial differential equation, providing a substantial computational gain against the first iterative scheme. In a second part, this formulation is used to apply modern shape optimization methods to optimize wear performance of bi-material systems. Several objectives for systems undergoing wear are identified and formalized with shape derivatives. As an example, the unit-cell of 2D multi-material composites is optimized using a level-set based method. A minimum feature size must be taken into account to avoid the convergence of minimizing design sequences toward composite materials. To address this issue, a variant of a the level-set topology optimization method is developed . Through the use of a single updating equation, this scheme conveniently enforces volume equality constraints, controls the complexity of design features with a perimeter penalization, and nucleates material inclusions with the use of the topological gradient. Keywords: topology and shape optimization, wear, multi-composite surface, level-set methods.

A stochastic optimization methodology is formulated for computing energy-optimal paths from among time-optimal paths of autonomous vehicles navigating in a dynamic flow field. To set up the energy optimization, the relative vehicle speed and headings are considered to be stochastic, and new stochastic Dynamically Orthogonal (DO) level-set equations that govern their stochastic time-optimal reachability fronts are derived. Their solution provides the distribution of time-optimal reachability fronts and corresponding distribution of time-optimal paths. An optimization is then performed on the vehicle’s energy-time joint distribution to select the energy-optimal paths for each arrival time, among all stochastic time-optimal paths for that arrival time. The accuracy and efficiency of the DO level-set equations for solving the governing stochastic level-set reachability fronts are quantitatively assessed, including comparisons with independent semi-analytical solutions. The stochastic DO level-set equations is then extended to account for uncertainties in the flow field. Time-optimal planning is completed in a wind-driven barotropic quasi-geostrophic stochastic double-gyre ocean circulation (these stochastic flow fields are simulated using our DO Navier Stokes equations). Energy-optimal missions are studied in wind-driven barotropic quasi-geostrophic double-gyre circulations, and in realistic data-assimilative re-analyses of multiscale coastal ocean flows. The latter re-analyses are obtained from multi-resolution 2-way nested primitive-equation simulations of tidal-to-mesoscale dynamics in the Middle Atlantic Bight and Shelbreak Front region. The effects of tidal currents, strong wind events, coastal jets, and shelfbreak fronts on the energy-optimal paths are illustrated and quantified. Results showcase the opportunities for longer-duration missions that intelligently utilize the ocean environment to save energy, rigorously integrating ocean forecasting with optimal control of autonomous vehicles.