Airport Taxi Planning: Lagrangian Decomposition

The airport taxi planning (TP) module is a decision tool intended to guide airport surface management operations. TP is defined by a flow network optimization model that represents flight ground movements and improves aircraft taxiing routes and schedules during periods of aircraft congestion. TP is not intended to operate as a stand‐alone tool for airport operations management: on the contrary, it must be used in conjunction with existing departing and arriving traffic tools and overseen by the taxi planner of the airport, also known as the aircraft ground controller. TP must be flexible in order to accommodate changing inputs while maintaining consistent routes and schedules already delivered from past executions. Within this dynamic environment, the execution time of TP may not exceed a few minutes. Classic methods for solving binary multi‐commodity flow networks with side constraints are not efficient enough; therefore, a Lagrangian decomposition methodology has been adapted to solve it. We demonstrate TP Lagrangian decomposition using actual data from the Madrid‐Barajas Airport

Lagrangian skeleton on the Gulf of Mexico oil spill

The understanding of the circulation of ocean currents, the exchange of CO2 between atmosphere and oceans, and the influence of the oceans on the distribution of heat on a global scale is key to our ability to predict and assess the future evolution of climate [1, 2]. Global climate change is affecting sea breathing through mechanisms not yet understood.

A Lagrangian descriptor applied to the Kuroshio current

The understanding of the circulation of ocean currents, the exchange of CO2 between atmosphere and oceans, and the in uence of the oceans on the distribution of heat on a global scale is key to our ability to predict and assess the future evolution of climate.

Involutivity of the Hamilton-Cartan equations of a second-order Lagrangian admitting a first-order Hamiltonian formalism

Involutivity of the Hamilton-Cartan equations of a second-order Lagrangian admitting a first-order Hamiltonian formalism

A global Lagrangian descriptor applied to the Kuroshio current

A global Lagrangian descriptor applied to the Kuroshio current

Lagrangian structure on the oil

The understanding of the circulation of ocean currents, the exchange of CO2 between atmosphere and oceans, and the influence of the oceans on the distribution of heat and momentum on a global scale is key to our ability to predict and assess the future evolution of climate.

The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current

This article reviews several recently developed Lagrangian tools and shows how their com- bined use succeeds in obtaining a detailed description of purely advective transport events in general aperiodic flows. In particular, because of the climate impact of ocean transport processes, we illustrate a 2D application on altimeter data sets over the area of the Kuroshio Current, although the proposed techniques are general and applicable to arbitrary time depen- dent aperiodic flows. The first challenge for describing transport in aperiodical time dependent flows is obtaining a representation of the phase portrait where the most relevant dynamical features may be identified. This representation is accomplished by using global Lagrangian descriptors that when applied for instance to the altimeter data sets retrieve over the ocean surface a phase portrait where the geometry of interconnected dynamical systems is visible. The phase portrait picture is essential because it evinces which transport routes are acting on the whole flow. Once these routes are roughly recognised it is possible to complete a detailed description by the direct computation of the finite time stable and unstable manifolds of special hyperbolic trajectories that act as organising centres of the flow.

A Semi-Lagrangian Particle Level Set Finite Element Method for Interface Problems

We present a quasi-monotone semi-Lagrangian particle level set (QMSL-PLS) method for moving interfaces. The QMSL method is a blend of first order monotone and second order semi-Lagrangian methods. The QMSL-PLS method is easy to implement, efficient, and well adapted for unstructured, either simplicial or hexahedral, meshes. We prove that it is unconditionally stable in the maximum discrete norm, � · �h,∞, and the error analysis shows that when the level set solution u(t) is in the Sobolev space Wr+1,∞(D), r ≥ 0, the convergence in the maximum norm is of the form (KT/Δt)min(1,Δt � v �h,∞ /h)((1 − α)hp + hq), p = min(2, r + 1), and q = min(3, r + 1),where v is a velocity. This means that at high CFL numbers, that is, when Δt > h, the error is O( (1−α)hp+hq) Δt ), whereas at CFL numbers less than 1, the error is O((1 − α)hp−1 + hq−1)). We have tested our method with satisfactory results in benchmark problems such as the Zalesak’s slotted disk, the single vortex flow, and the rising bubble.

Lagrangian descriptors: a method for revealing phase space structures of general time dependent dynamical systems

Lagrangian descriptors are a recent technique which reveals geometrical structures in phase space and which are valid for aperiodically time dependent dynamical systems. We discuss a general methodology for constructing them and we discuss a "heuristic argument" that explains why this method is successful. We support this argument by explicit calculations on a benchmark problem. Several other benchmark examples are considered that allow us to assess the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field ("time averages"). In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods.

Lagrangian tools and the assessment of the predictive capacity of geophysical data sets

We examine, with recently developed Lagrangian tools, altimeter data and numerical simulations obtained from the HYCOM model in the Gulf of Mexico. Our data correspond to the months just after the Deepwater Horizon oil spill in the year 2010. Our Lagrangian analysis provides a skeleton that allows the interpretation of transport routes over the ocean surface. The transport routes are further verified by the simultaneous study of the evolution of several drifters launched during those months in the Gulf of Mexico. We find that there exist Lagrangian structures that justify the dynamics of the drifters, although the agreement depends on the quality of the data. We discuss the impact of the Lagrangian tools on the assessment of the predictive capacity of these data sets.

Analysis of Meshfree Methods for Lagrangian Fluid-Structure Interaction

In this dissertation a new numerical method for solving Fluid-Structure Interaction (FSI) problems in a Lagrangian framework is developed, where solids of different constitutive laws can suffer very large deformations and fluids are considered to be newtonian and incompressible. For that, we first introduce a meshless discretization based on local maximum-entropy interpolants. This allows to discretize a spatial domain with no need of tessellation, avoiding the mesh limitations. Later, the Stokes flow problem is studied. The Galerkin meshless method based on a max-ent scheme for this problem suffers from instabilities, and therefore stabilization techniques are discussed and analyzed. An unconditionally stable method is finally formulated based on a Douglas-Wang stabilization. Then, a Langrangian expression for fluid mechanics is derived. This allows us to establish a common framework for fluid and solid domains, such that interaction can be naturally accounted. The resulting equations are also in the need of stabilization, what is corrected with an analogous technique as for the Stokes problem. The fully Lagrangian framework for fluid/solid interaction is completed with simple point-to-point and point-to-surface contact algorithms. The method is finally validated, and some numerical examples show the potential scope of applications.

Numerical issues in Lagrangian tracking and topological evolution of fluid particles in wallbounded turbulent flows

The determination of the local Lagrangian evolution of the flow topology in wall-bounded turbulence, and of the Lagrangian evolution associated with entrainment across the turbulent / non-turbulent interface into a turbulent boundary layer, require accurate tracking of a fluid particle and its local velocity gradients. This paper addresses the implementation of fluid-particle tracking in both a turbulent boundary layer direct numerical simulation and in a fully developed channel flow simulation. Determination of the sub-grid particle velocity is performed using both cubic B-spline, four-point Hermite spline and higher-order Hermite spline interpolation. Both wall-bounded flows show similar oscillations in the Lagrangian tracers of both velocity and velocity gradients, corresponding to the movement of particles across the boundaries of computational cells. While these oscillation in the particle velocity are relatively small and have negligible effect on the particle trajectories for time-steps of the order of CFL = 0.1, they appear to be the cause of significant oscillations in the evolution of the invariants of the velocity gradient tensor.

First-order equivalent to Einstein-Hilbert Lagrangian

A first-order Lagrangian L ∇ variationally equivalent to the second-order Einstein- Hilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by L ∇ is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to ∇ .