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. Based on partial differential equations, the methodology rigorously leverages the level–set equation that governs time–optimal reachability fronts for a given relative vehicle speed function. To set up the energy optimization, the relative vehicle speed is considered to be stochastic and new stochastic Dynamically Orthogonal (DO) level–set equations 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. Numerical schemes to solve the reduced stochastic DO level–set equations are obtained and accuracy and efficiency considerations are discussed. These reduced equations are first shown to be efficient at solving the governing stochastic level-sets, in part by comparisons with direct Monte Carlo simulations.To validate the methodology and illustrate its overall accuracy, comparisons with `semi–analytical’ energy–optimal path solutions are then completed. In particular, we consider the energy–optimal crossing of a canonical steady front and set up its `semi–analytical’ solution using a dual energy–time nested nonlinear optimization scheme. We then showcase the inner workings and nuances of the energy–optimal path planning, considering different mission scenarios. Finally, we study and discuss results of energy-optimal missions in a strong dynamic double–gyre flow field.

Abstract

In this talk, we will discuss the recent development of a class of hybridized DG methods for implicit large eddy simulation (ILES) of compressible flows. This class of DG methods encompass the hybridizable DG (HDG) method, the embedded DG (EDG) method, as well as new hybridized DG methods resulting from the marriage of the HDG method and the EDG method. While the HDG method is more accurate and robust that the EDG method, the latter is significantly less expensive than the former. This motives us to combine HDG and EDG to obtain new hybridized DG methods that enjoy the advantages of both HDG and EDG. However, this approach presents challenging issues in terms of domain decomposition preconditioners and parallelization because the resulting linear system has complicated sparsity structure. We will discuss our domain decomposition preconditioner and strategy to address some of the issues and leave other issues for future work. In addition, we will talk about various choices of the stabilization tensor and their influence on both nonlinear and linear convergence. Finally, we present ILES results and validate them against experimental data and other simulation data.

This is the joint work with Pablo Fernandez and Jaime Peraire.

Biography

Dr. Nguyen’s current research is focused on efficient methods for simulation of multiscale and multi-physics phenomena across disciplines and on uncertainty quantification techniques for inverse/design problems in engineering. He received his BE degree with first class honors in Aeronautical Engineering from HCMC, University of Technology in 2001, and his Ph.D. degree in High Performance Computation for Engineered Systems from National University of Singapore in 2005. Dr. Nguyen is the author and co-author of more than 25 research articles. He has presented his work in several major conferences, invited talks, and workshops.

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.