Rothe–Legendre pseudospectral method for a semilinear pseudoparabolic equation with nonclassical boundary condition

A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.

Multi-Objective and Multi-Phase 4D Trajectory Optimization for Climate Mitigation-Oriented Flight Planning

Aviation contribution to global warming and anthropogenic climate change is increasing every year. To reverse this trend, it is crucial to identify greener alternatives to current aviation technologies and paradigms. Research in aircraft operations can provide a swift response to new environmental requirements, being easier to exploit on current fleets. This paper presents the development of a multi-objective and multi-phase 4D trajectory optimization tool to be integrated within a Flight Management System of a commercial aircraft capable of performing 4D trajectory tracking in a Free Route Airspace context. The optimization algorithm is based on a Chebyshev pseudospectral method, adapted to perform a multi-objective optimization with the two objectives being the Direct Operating Cost and the climate cost of a climb-cruise-descent trajectory. The climate cost function applies the Global Warming Potential metric to derive a comprehensive cost index that includes the climate forcing produced by CO2 and non-CO2 emissions, and by the formation of aircraft-induced clouds. The output of the optimization tool is a set of Pareto-optimal 4D trajectories among which the aircraft operator can choose the best solution that satisfies both its economic and environmental goals.

Trajectory Planning for Emergency Landing of VTOL Fixed-Wing Unmanned Aerial Vehicles

In recent years, inspired by technological progress and the outstanding performance of Unmanned Aerial Vehicles (UAVs) in several local wars, the UAV industry has witnessed explosive development, widely used in communication relay, logistics, surveying and mapping, patrol, surveillance, and other fields. Vertical Take-Off and Landing fixed-wing UAV has both the advantages of vertical take-off and landing of rotorcraft and the advantages of long endurance of fixed-wing UAV, which broadened its application field and is the most popular UAV at present. Recently, fixed-wing UAV failure analysis highlights that cruise engine shutdown is the most common reason for emergency landing, which is also a governing factor for Vertical Take-Off and Landing (VTOL) fixed-wing UAV failures. Nevertheless, the emergency landing trajectory of the latter UAV type after engine shutdown is different from that of the conventional fixed-wing UAVs due to the VTOL power system. Hence, spurred by the requirement of a safe emergency landing trajectory for VTOL fixed-wing UAVs, this paper develops an architecture capable of safe emergency landing for such platforms. The suggested method develops a particle dynamics model of the VTOL UAV and analyzes its aerodynamic characteristics utilizing Computational Fluid Dynamics (CFD) results. The UAV’s trajectory is divided into three parts for enhanced planning. For the guidance stage, the initial position and heading angle are arbitrary. Hence, the Dubins shortest cross-range and the fastest descent trajectory are adopted to steer the UAV above the landing window quickly. The spiral stage comprises a conical and cylindrical part combined with a spiral descent trajectory of variable radius for energy management and landing course alignment. Given the limited energy storage of VTOL power systems, the landing stage exploits an optimal control trajectory problem solved by a Gaussian pseudospectral method, involving trajectory conventional landing planning, unpowered landing, distance optimal landing, and wind-resistant landing. All trajectories meet the dynamics constraints, terminal constraints, and sliding performance constraints and cover both 2-dimensional and 3-dimensional trajectories. A large number of simulation experiments demonstrate that the proposed trajectories manage broad applicability and strong feasibility for VTOL fixed-wing UAVs.

Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions

This work deals with the pseudospectral method to solve the Sturm–Liouville eigenvalue problems with Caputo fractional derivative using Chebyshev cardinal functions. The method is based on reducing the problem to a weakly singular Volterra integro-differential equation. Then, using the matrices obtained from the representation of the fractional integration operator and derivative operator based on Chebyshev cardinal functions, the equation becomes an algebraic system. To get the eigenvalues, we find the roots of the characteristics polynomial of the coefficients matrix. We have proved the convergence of the proposed method. To illustrate the ability and accuracy of the method, some numerical examples are presented.

Hermite-Chebyshev pseudospectral method for inhomogeneous superconducting strip problems and magnetic flux pump modeling

Numerical simulation of superconducting devices is a powerful tool for understanding the principles of their work and improving their design. Usually, such simulations are based on a finite element method but, recently, a different approach, based on the spectral technique, has been presented for very efficient solution of several applied superconductivity problems described by one-dimensional integro-differential equations or a system of such equations. Here we propose a new pseudospectral method for two-dimensional magnetization and transport current superconducting strip problems with an arbitrary current-voltage relation, spatially inhomogeneous strips, and strips in a nonuniform applied field. The method is based on the bivariate expansions in Chebyshev polynomials and Hermite functions. It can be used for numerical modeling magnetic flux pumps of different types and investigating AC losses in coated conductors with local defects. Using a realistic two-dimensional version of the superconducting dynamo benchmark problem as an example, we showed that our new method is a competitive alternative to finite element methods.