NASTRAN solutions of problems described by simultaneous parabolic differential equations

NASTRAN solution techniques are shown for a numerical analysis of a class of coupled vector flow processes described by simultaneous parabolic differential equations. To define one physical problem type where equations of this form arise, the differential equations describing the coupled transfers of heat and mass in mechanical equilibrium with negligible mass average velocity are presented and discussed. Also shown are the equations describing seepage when both electrokinetic and hydrodynamic forces occur. Based on a variational statement of the general problem type, the concepts of scalar transfer elements and parallel element systems are introduced. It is shown that adoptation of these concepts allows the direct use of NASTRAN's existing Laplace type elements for uncoupled flow (the heat transfer elements) for treating multicomponent coupled transfer. Sample problems are included which demonstrate the application of these techniques for both steady-state and transient problems

Random-mass Dirac fermions in an imaginary vector potential: Delocalization transition and localization length

One dimensional system of Dirac fermions with a random-varying mass is studied by the transfer-matrix methods which we developed recently. We investigate the effects of nonlocal correlation of the spatial-varying Dirac mass on the delocalization transition. Especially we numerically calculate both the "typical" and "mean" localization lengths as a function of energy and the correlation length of the random mass. To this end we introduce an imaginary vector potential as suggested by Hatano and Nelson and solve the eigenvalue problem. Numerical calculations are in good agreement with the results of the analytical calculations.Comment: 4 page

Multi-objective Optimization of Orbit Transfer Trajectory Using Imperialist Competitive Algorithm

This paper proposes a systematic direct approach to carry out effective multi-objective optimization of space orbit transfer with high-level thrust acceleration. The objective is to provide a transfer trajectory with acceptable accuracy in all orbital parameters while minimizing spacecraft fuel consumption. With direct control parameterization, in which the steering angles of thrust vector are interpolated through a finite number of nodes, the optimal control problem is converted into the parameter optimization problem to be solved by nonlinear programming. Besides the thrust vector direction angles, the thrust magnitude is also considered as variable and unknown along with initial conditions. Since the deviation of thrust vector in spacecraft is limited in reality, mathematical modeling of thrust vector direction is carried out in order to satisfy constraints in maximum deviation of thrust vector direction angles. In this modeling, the polynomial function of each steering angle is defined by interpolation of a curve based on finite number of points in a specific range with a nominal center point with uniform distribution. This kind of definition involves additional parameters to the optimization problem which results the capability of search method in satisfying constraint on the variation of thrust direction angles. Thrust profile is also modeled based on polynomial functions of time with respect to solid and liquid propellant rockets. Imperialist competitive algorithm is used in order to find optimal coefficients of polynomial for thrust vector and the optimal initial states within the transfer. Results are mainly affected by the degree of polynomials involved in mathematical modeling of steering angles and thrust profile which results different optimal initial states where the transfer begins. It is shown that the proposed method is fairly beneficial in the viewpoint of optimality and convergence. The optimality of the technique is shown by comparing the finite thrust optimization with the impulsive analysis. Comparison shows that the accuracy is acceptable with respect to fair precision in orbital elements and minimum fuel mass. Also, the convergence of the optimization algorithm is investigated by comparing the solution of the problem with other optimization techniques such as Genetic Algorithm. Results confirms the practicality of Imperialist Competitive Algorithm in finding optimum variation of thrust vector which results best transfer accuracy along with minimizing fuel consumption

Hybrid low-thrust transfers to eight-shaped orbits for polar observation

In this paper, transfers from low Earth orbit (LEO) to so-called eight-shaped orbits at the collinear libration points in the circular restricted three-body problem are investigated. The potential of these orbits (both natural and sail displaced) for high-latitude observation and telecommunication has recently been established. The transfer is modelled by distinguishing between a near-Earth phase and an interplanetary phase. The near-Earth phase is first assumed to be executed with the Soyuz Fregat upper-stage, which brings the spacecraft from LEO to a highly elliptic orbit. From there, the interplanetary phase is initiated which uses low-thrust propulsion to inject the spacecraft into one of the eight-shaped orbit’s manifolds. Both solar electric propulsion (SEP), solar sailing and hybridised SEP and solar sailing are considered for this phase. The objective is to maximise the mass delivered to the eight-shaped orbit starting from a realistic Soyuz launch vehicle performance into LEO. Optimal trajectories are obtained by solving the optimal control problem in the interplanetary phase with a direct pseudospectral method. The results show that (over the full range of propulsion techniques) 1564 to 1603 kg can be injected into a natural eight-shaped orbit. Within this relatively small range, hybrid propulsion performs best in terms of mass delivered to the eight-shaped orbit, while SEP enables the fastest transfer times. With the interplanetary phase optimised, the upper-stage near-Earth phase is replaced by a multi-revolution low-thrust spiral. Locally optimal control laws for the SEP thruster and solar sail are derived to minimise the time of flight in the spiral. Both pure SEP and hybrid spiral show a significant reduction in the mass required in LEO to deliver the spacecraft to the eight-shaped orbits. While hybrid propulsion did not stand out for the use of an upper-stage near-Earth phase, it does for the use of a low-thrust spiral as it significantly reduces the spiral time with respect to the pure SEP case

A Homotopy-Based Method for Optimization of Hybrid High-Low Thrust Trajectories

Space missions require increasingly more efficient trajectories to provide payload transport and mission goals by means of lowest fuel consumption, a strategic mission design key-point. Recent works demonstrated that the combined (or hybrid) use of chemical and electrical propulsion can give important advantages in terms of fuel consumption, without losing the ability to reach other mission objectives: as an example the Hohmann Spiral Transfer, applied in the case of a transfer to GEO orbit, demonstrated a fuel mass saving between 5-10% of the spacecraft wet mass, whilst satisfying a pre-set boundary constraint for the time of flight. Nevertheless, methods specifically developed for optimizing space trajectories considering the use of hybrid high-low thrust propulsion systems have not been extensively developed, basically because of the intrinsic complexity in the solution of optimal problem equations with existent numerical methods. The study undertaken and presented in this paper develops a numerical strategy for the optimization of hybrid high-low thrust space trajectories. An indirect optimization method has been developed, which makes use of a homotopic approach for numerical convergence improvement. The adoption of a homotopic approach provides a relaxation to the optimal problem, transforming it into a simplest problem to solve in which the optimal problem presents smoother equations and the shooting function acquires an increased convergence radius: the original optimal problem is then reached through a homotopy parameter continuation. Moreover, the use of homotopy can make possible to include a high thrust impulse (treated as velocity discontinuity) to the low thrust optimal control obtained from the indirect method. The impulse magnitude, location and direction are obtained following from a numerical continuation in order to minimize the problem cost function. The initial study carried out in this paper is finally correlated with particular test cases, in order to validate the work developed and to start investigating in which cases the effectiveness of hybrid-thrust propulsion subsists

Displaced geostationary orbit design using hybrid sail propulsion

Because of an increase in the number of geostationary spacecraft and the limits imposed by east–west spacing requirements, the geostationary orbit is becoming congested. To increase its capacity, this paper proposes to create new geostationary slots by displacing the geostationary orbit either out of or in the equatorial plane by means of hybrid solar sail and solar electric propulsion. To minimize propellant consumption, optimal steering laws for the solar sail and solar-electric-propulsion thrust vectors are derived and the performance in terms of mission lifetime is assessed. For comparison, similar analyses are performed for conventional propulsion, including impulsive and pure solar electric propulsion. It is shown that hybrid sails outperform these propulsion techniques and that out-of-plane displacements outperform in-plane displacements. The out-of-plane case is therefore further investigated in a spacecraft mass budget to determine the payload mass capacity. Finally, two transfers that enable a further improvement of the performance of hybrid sails for the out-of-plane case are optimized using a direct pseudospectral method: a seasonal transit between orbits displaced above and below the equatorial plane and a transit to a parking orbit when geostationary coverage is not needed. Both transfers are shown to require only a modest propellant budget, outweighing the improvements they can establish

Optimisation of Low-Thrust and Hybrid Earth-Moon Transfers

This paper presents an optimization procedure to generate fast and low-∆v Earth-Moon transfer trajectories, by exploiting the multi-body dynamics of the Sun-Earth-Moon system. Ideal (first-guess) trajectories are generated at first, using two coupled planar circular restricted three-body problems, one representing the Earth-Moon system, and one representing the Sun-Earth. The trajectories consist of a first ballistic arc in the Sun-Earth system, and a second ballistic arc in the Earth-Moon system. The two are connected at a patching point at one end (with an instantaneous ∆v), and they are bounded at Earth and Moon respectively at the other end. Families of these trajectories are found by means of an evolutionary optimization method. Subsequently, they are used as first-guess for solving an optimal control problem, in which the full three-dimensional 4-body problem is introduced and the patching point is set free. The objective of the optimisation is to reduce the total ∆v, and the time of flight, together with introducing the constraints on the transfer boundary conditions and of the considered propulsion technology. Sets of different optimal trajectories are presented, which represents trade-off options between ∆v and time of flight. These optimal transfers include conventional solar-electric low-thrust and hybrid chemical/solar-electric high/low-thrust, envisaging future spacecraft that can carry both systems. A final comparison is made between the optimal transfers found and only chemical high-thrust optimal solutions retrieved from literature

Displaced geostationary orbits using hybrid low-thrust propulsion

In this paper, displaced geostationary orbits using hybrid low-thrust propulsion, a complementary combination of Solar Electric Propulsion (SEP) and solar sailing, are investigated to increase the capacity of the geostationary ring that is starting to get congested. The SEP propellant consumption is minimized in order to maximize the mission lifetime by deriving semi-analytical formulae for the optimal steering laws for the SEP and solar sail accelerations. By considering the spacecraft mass budget, the performance is also expressed in terms of payload mass capacity. The analyses are performed for both the use of SEP and hybrid sail control to allow for a comparison. It is found that hybrid sail control outperforms the pure SEP case both in terms of payload capacity and mission lifetime for all displacements considered. Hybrid sails enable payloads of 250-450 kg to be maintained in a 35 km displaced orbit for 10-15 years. Finally, two transfers that allow for an improvement in the performance of hybrid sail control are optimized for the SEP propellant consumption by solving an optimal control problem using a direct pseudo-spectral method. The first type of transfer enables a transit between orbits displaced above and below the equatorial plane, while the second type of transfer enables „customized service‟ in which the spacecraft is transferred to a Keplerian parking orbit when coverage is not needed. While the latter requires a modest propellant budget, the first type of transfer comes at the cost of a negligible SEP propellant consumption

Trajectory and spacecraft design for a pole-sitter mission

This paper provides a detailed mission analysis and systems design of a pole-sitter mission. It considers a spacecraft that is continuously above either the North or South Pole and, as such, can provide real-time, continuous and hemispherical coverage of the polar regions. Two different propulsion strategies are proposed, which result in a near-term pole-sitter mission using solar electric propulsion and a far-term pole-sitter mission where the electric thruster is hybridized with a solar sail. For both propulsion strategies, minimum propellant pole-sitter orbits are designed. Optimal transfers from Earth to the pole-sitter are designed assuming Soyuz and Ariane 5 launch options, and a controller is shown to be able to maintain the trajectory under unexpected conditions such as injection errors. A detailed mass budget analysis allows for a trade-off between mission lifetime and payload mass capacity, and candidate payloads for a range of applications are investigated. It results that a payload of about 100 kg can operate for approximately 4 years with the solar-electric spacecraft, while the hybrid propulsion technology enables extending the missions up to 7 years. Transfers between north and south pole-sitter orbits are also considered to observe either pole when illuminated by the Sun