Both future air traffic management programs SESAR and NextGen foresee trajectory based operations as one of the major enablers for more efficient handling of air traffic. Furthermore, both initiatives claim to support user preferred trajectories. Using a common airspace it is unlikely to get a conflict-free scenario from independently optimized preferred trajectories. This paper investigates if the concept of user preferred trajectories is feasible, and what constraints are necessary to reach a conflict-free global scenario. Restricting aircraft’s trajectories a priori by forcing them to fly on pre-defined en-route segments, respect semi-circular routes, or apply given departure and arrival times reduces the achievable efficiency, but helps to avoid conflicts and keep the concept simple. On the other hand, trying to combine independently optimized trajectories to a full traffic scenario needs a lot of adaptations to achieve conflict-free operations. Even though the good answer to a problem lies often in-between the extreme solutions, this paper investigates the second approach. Based on a real day of actual traffic over Europe, an optimized sample is generated applying user’s preferred trajectory for each flight. Based on an individually optimized traffic scenario yielding nearly 29,000 conflicts, strategic resolution is performed generating least possible penalties for involved aircraft. The main technique of solving conflicts applied is shifting whole flights in time by a few minutes. Direct and recursive algorithms are presented; a focus is put on solving airport-related conflicts first. The efficiency of the original optimized scenario stays untouched, as lateral, vertical, and speed profiles remain identical; flights are only rescheduled to a slightly earlier or later departure (and arrival) slot in order to minimize conflicts. More than 94% of all conflicts can be solved by shifting whole flights in time respecting a maximum offset of 10 minutes. In a subsequent step, lateral and vertical maneuvers are facilitated; resulting in a resolution for more than 97% of all conflicts.

In the present work, we extend the theoretical and numerical discussion of the well-known Laplace–Beltrami operator by equipping the underlying manifolds with additional structure provided by vector bundles. Focusing on the particular class of flat complex line bundles, we examine a whole family of Laplacians including the Laplace–Beltrami operator as a special case. To demonstrate that our proposed approach is numerically feasible, we describe a robust and efficient finite-element discretization, supplementing the theoretical discussion with first numerical spectral decompositions of those Laplacians.

Our method is based on the concept of introducing complex phase discontinuities into the finite element basis functions across a set of homology generators of the given manifold. More precisely, given an m-dimensional manifold M and a set of n generators that span the relative homology group H_m-1(M, ∂ M), we have the freedom to choose n phase shifts, one for each generator, resulting in a n-dimensional family of Laplacians with associated spectra and eigenfunctions. The spectra and absolute magnitudes of the eigenfunctions are not influenced by the exact location of the paths, depending only on their corresponding homology classes.

Employing our discretization technique, we provide and discuss several interesting computational examples highlighting special properties of the resulting spectral decompositions. We examine the spectrum, the eigenfunctions and their zero sets which depend continuously on the choice of phase shifts.

The well-known Laplace-Beltrami operator, established as a basic tool in shape processing, builds on a long history of mathematical investigations that have induced several numerical models for computational purposes. However, the Laplace-Beltrami operator is only one special case of many possible generalizations that have been researched theoretically. Thereby it is natural to supplement some of those extensions with concrete computational frameworks. In this work we study a particularly interesting class of extended Laplacians acting on sections of flat line bundles over compact Riemannian manifolds. Numerical computations for these operators have recently been accomplished on two-dimensional surfaces. Using the notions of line bundles and differential forms, we follow up on that work giving a more general theoretical and computational account of the underlying ideas and their relationships. Building on this we describe how the modified Laplacians and the corresponding computations can be extended to three-dimensional Riemannian manifolds, yielding a method that is able to deal robustly with volumetric objects of intricate shape and topology. We investigate and visualize the two-dimensional zero sets of the first eigenfunctions of the modified Laplacians, yielding an approach for constructing characteristic well-behaving, particularly robust homology generators invariant under isometric deformation. The latter include nicely embedded Seifert surfaces and their non-orientable counterparts for knot complements.