Competitive 0 and π states in S/F/S trilayers: Multimode approach
Summary (5 min read)
Introduction
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- This results in the dispersion of the impact locations characterizing the ammunition precision error.
II. Maneuver concept
- The choice of deploying a spoiler during the projectile flight has been particularly motivated by the work of Patel et al. [13] and Simon [14].
- In their work, Patel et al. demonstrated the effectiveness of micro-spoiler in the capability of modifying the behavior of a projectile for an angle of attack ranging from 0 to 18◦, which is the effective range of the present uncontrolled projectile.
- A unique deployment instant is chosen for the spoiler.
- During the projectile flight and until the spoiler deployment, information about the trajectory perturbation would be gathered using an external radar system.
- Since the spoiler position is time-invariant, the projectile reaches an equilibrium after a short period of time.
III. Surrogate models
- As estimated from the work of Dietrich [1], the accurate computation of aerodynamic coefficients along the trajectory is incompatible with the optimization framework presented in this study.
- A short description of the models used in this study is presented here.
- According to Jones et al. [19], a good initial sampling size counts 10 samples in every dimension which is usually not reachable on high-dimensional problems.
- From a practical point of view, only the spreading of 3p points will be considered to establish a kriging response surface in the following.
- The optimization of the points distribution in the design space is achieved considering the Morris and Mitchell criterion [20] which maximizes the minimal distance between the sampling sites.
1. Exploitation and exploration of the Kriging response surface
- To combine the local and global search of an optimum, Jones et al. [19] proposed the computation of the Expected Improvement (EI) which is the mathematical definition of the probability that an unsampled point performs better than the already best known sampled point of the function.
- The EI proposes a trade-off between the exploration and the exploitation of the surrogate model.
- Indeed, values associated to the EI score are high in the regions of the design space where the function values are significant but are also high in the unknown regions of the model where the error associated to the model s(x) is large .
2. Constraints handling from the Kriging model
- The addition of constraint functions to the initial problem definition leads to the reduction of parameters combinations susceptible to minimize the objective function.
- They are modelled following the definition of the Probability of Feasibility (PF) introduced by Schonlau [22] and applied successfully to a surrogate-based optimization by Parr et al. [23].
- In the same way as for the EI computation clarified in the preceding paragraph, the PF takes advantages of the kriging modeling of a function to estimate if constraints are likely to be respected by comparing the estimation of a function to the constraint limit, see eq. (2). P[F(x)] = 1 s(x) √ 2π ∫ ∞ 0 e−[(G(x)−gmin)−ĝ(x)].
- G(x) is a random variable and s(x) is constraint standard deviation of the Kriging model.
C. Artificial Neural Networks
- A multi-layer Perceptron is specifically used in this study to model the evolution of the spoiler aerodynamic coefficients (∆CA, ∆CN and ∆Cm) as a function of the aerodynamic conditions (Mach number M∞ and angle of attack α).
- The training database is composed of more than 1300 CFD evaluations of the spoiler.
- To avoid an over-fitting of the MLP, resulting in the degradation of the aerodynamic coefficients estimations, the training iterations stop when the coefficient of determination does no improve anymore on a subset of the training database (10% of the evaluations in the present case).
- The MLP models of each coefficient were implemented using the Python module Scikit-learn [24].
- It has to be noticed that the best estimations using MLP are obtained when the inputs and outputs are scaled in the same interval of range [−1; 1].
A. Simulations overview
- A MLP is used to model the variations in the aerodynamic coefficients of the spoiler configurations.
- The database is composed of RANS-based CFD evaluations.
- The contribution of the spoiler is isolated by simulating the flow around both controlled and uncontrolled projectile.
- The uncontrolled projectile has been studied prior to the present work, for instance by Simon et al. [25] and Zeidler et al. [26].
- The aerodynamic modifications induced by the spoiler deployment are described in the following.
1. Meshing strategy
- The 155 mm caliber (cal.) projectile is composed of a 5.6 cal-long body constituted of a 3 cal-long truncated nose.
- The computational domain external boundary extends at a distance of 50 caliber around the projectile.
- The spoiler setup on the projectile body is ensured via the chimera method which consists in merging two grids into a unique one as shown in fig.
- A cell-size ratio close to 1 at the interfaces between the background and the patch grids has been imposed.
2. elsA code
- The CFD solver used in this study to compute the aerodynamic coefficients of the different projectile configurations is the ONERA’s solver elsA based on a cell-centered finite volume approach for solving the Navier-Stokes equations on structured multiblock grids [27].
- The computations constituting the initial aerodynamic coefficients database were realized using the Spalart-Allmaras (SA) one-equation turbulence model.
- Consistently the same turbulence model is used in the computations enhancing this database.
- It is known that SA model performs poorly in the presence of massively detached flows.
- A comparison with other turbulence models (k −ω and EARSM not presented here) shows maximum discrepancies of the order of 10% for drag coefficient evaluations which is considered acceptable in the present case, since the authors are more interested in relative performance.
B. Flow around the controlled projectile
- The flight conditions encountered by the uncontrolled projectile are ranging from supersonic Mach number (at the muzzle exit) to high subsonic, the angle of attack remaining below 6◦.
- The deployment of the spoiler can occur at any point of the trajectory thus the uncontrolled and controlled projectile aerodynamic models are constructed in the same range of flight parameters.
- A generic spoiler is located in the middle of the projectile boat-tail to highlight the flow modifications induced by its presence.
- The detached areas are showed for both projectile configurations at M = 0.9 and α = 0.
- Moreover, due to the curvature of the streamlines an additional shock appears in the wake which can be designated as recompression shock, for these transonic freestream conditions, which is not present in the case of the uncontrolled projectile.
V. Spoiler optimization
- The optimization process developed within this study is illustrated by the flowchart presented in fig.
- Within the 88% of the uncontrolled projectile course, information is gathered on the external perturbations occurring during the flight.
- A sequential enrichment of the kriging response surfaces based on the selection of optimal design using a genetic algorithm is implemented to converge toward the optimum set of parameters achieving the required course correction, paragraph V.D.
- The trajectories optimization loop, referred to as inner loop in the following, defines which configuration of the controlled projectile is considered as optimum.
- This enrichment is the second adaptive sampling.
A. Aerodynamic coefficients estimations
- The advantage of using a MLP model against a kriging model here has been discussed by the authors in [28].
- Additionally, the distribution of the CFD evaluations over the parameters space does not follow an optimal design of experiment which eventually leads to an over-fitting of the kriging model.
- The validation of the MLP response surface training is illustrated in fig.
B. Trajectories initial sampling
- An initial database of controlled projectiles is spread over the p = 4 dimensional design domain (3 geometrical parameters Xs/D, Hs/D, θs and the roll position φs) using the optimized LHS methodology.
- The number of samples is directly determined from the rule of thumb stated earlier as Ninit = 34 = 81 configurations.
- The trajectories are computed using the 6-DOF flight mechanics code Balco [29] and presented in fig.
- A wide variety of corrections (around the uncontrolled projectile mean impact point at [1;1]) is proposed in this purely exploratory phase of the process.
- The lateral deviations and range modifications produced by the different spoilers are used to initialize the kriging response surfaces modeling the objective and constraint functions that are 15 described in the next paragraph.
C. Definition of the optimization problem
- Considering the distribution of the uncontrolled projectile impact points introduced in section II, the objective function is defined as the minimization of the distance between the controlled projectile impact point and the closest point belonging to the 3σ ellipse, eq. (4).
- It is then assumed that, if the spoiler geometry fulfills this specific case, it will allow producing every course correction, as illustrated in fig.
- An increase of the projectile range is imposed through eq. (5), illustrated in fig.
- This constraint translates inot eq. (7) in which the distance between the controlled projectile impact point and the uncontrolled projectile mean impact point is compared to the distance between the uncontrolled projectile mean impact point and the closest point belonging to the 3σ ellipse.
D. Bi-objective optimization
- An adaptive sampling is developed inside the inner loop in fig.
- 6, based on the kriging estimations of the objective and constraint functions.
- A bi-objective optimization is achieved with the aim of maximizing EI and PF through an evolutionary process simulated with the NSGA-II algorithm developed by Deb et al. [30].
- 10, can be selected to enrich the database.
- The maximization of EI and PF separately leads to the selection of better candidates than in the case of a single-objective optimization.
E. Course correction results
- The adaptive sampling described in the previous section is applied to the projectile course correction problem.
- The impact points are colored by the spoiler frontal surface associated to the trajectory modification.
- This means that in the 4-dimensional design space, the configurations producing the required 2-D course correction are located on an hypersurface depending on Hs/D × θs .
- The choice of any spoiler configuration belonging to this particular hypersurface determines the magnitude of the force induced by the spoiler.
- The longitudinal position of the spoiler Xs/D influences the magnitude of the pitching moment created by the spoiler which is directly linked to the lateral deviation produced by a spinning projectile while the roll position φs determines toward which direction the force induced by the spoiler is applied.
VI. Flight Mechanics study of the optimum configuration
- The optimum configuration determined during the process enables the course correction of the projectile.
- Moreover, the choice of a specific spoiler instant of deployment (12% of the trajectory before the impact in the studied case) in combination with a specific frontal surface value leads to the identification of designs able to produce the required 2-D course correction.
- In the present course correction problem, this value is lower than the maximum allowed by the parameters range, as indicated by the parameters combination in table 14a.
- The following iterations of the outer loop are used to increase the accuracy of the MLP database until a convergence state is reached.
- 13 where the evolution of the optimum projectile coefficients are presented.
VII. Conclusions
- A surrogate-based algorithm was developed with the aim of optimizing a spoiler geometry deployed during the flight of a spin-stabilized projectile.
- The selection of an optimum configuration is achieved through several adaptive sampling of the different response surfaces.
- Kriging response surfaces are enriched in the inner loop of the optimization problem to model the objective and constraint functions while the MLP database is enhanced with additional CFD evaluations of the optimum spoiler aerodynamic coefficients.
- The controlled projectile is able to reach an increased range and a reduced lateral deviation compared to the uncontrolled projectile trajectory.
- In the following of this study, multi-fidelity surrogate models could be used to integrate unsteady computations of the optimum configuration in the databases.
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Frequently Asked Questions (15)
Q2. What are the future works in this paper?
One of the interesting problems would be to extend the MMA to the nonequilibrium case by using the KeldyshUsadel Green ’ s function approach [ 84 ]. It is also interesting to study more complex phases in S/F multilayers in the MMA, extending the results of Ref. [ 63 ] obtained in the SMA.
Q3. What is the critical temperature in S/F/S trilayers?
In S/F/S trilayers, only the state with highest Tc is realized at certain d f , i.e., when increasing d f the dashed red line appears above the solid black line, the 0-π transition occurs, and the structure switches to the π phase state.
Q4. What is the effect of the inverse proximity effect on the critical temperature of the S/F?
With decrease of the S-layer thickness ds in S/F/S trilayers, the critical temperature is suppressed due to the inverse proximity effect, which becomes more profound in the case104502-6of small ds.
Q5. What is the critical temperature of a S/F interface?
For fully transparent S/F interfaces, γb = 0, the critical temperature appears at d f ∼ ξh (we note that in their case ξh = 0.54ξ f , since h = 6.8πTcs), reaches a maximum at a particular d f , and with further increase in d f eventually drops to zero.
Q6. What are the possible extensions of this work?
Other possible extensions will include spin-orbit coupling effects in equilibrium [72] and nonequilibrium cases [90] and considering Tc in S/F/S junctions in the presence of an equilibrium supercurrent [91].
Q7. What is the critical temperature of a ferromagnetic layer?
In this case, the critical temperature Tc vanishes when the π phase state becomes energetically unfavorable in a certain interval of d f , and at d f ξ f the Tc eventually tends to zero.
Q8. what is the usadel equation for the ferromagnetic layer?
2s πTcs d2Fs dx2− |ωn|Fs + = 0. (1) In the F layer (−d f /2 < x < d f /2), the Usadel equation can be written asξ 2f πTcs d2Ff dx2− (|ωn| + ih sgn ωn)Ff = 0.
Q9. What is the Usadel Eq. for F+s?
(13) According to the Usadel Eqs. (1)–(3), there is a symmetry relation F (−ωn) = F ∗(ωn), which implies that F+ is a real while F− is a purely imaginary function.
Q10. What is the resistance of the S/F boundary?
At the borders of the S layers with a vacuum, the authors naturally havedFs(±ds ± d f /2) dx= 0. (6) The solution of the Usadel equation in the F layer depends on the phase state of the structure.
Q11. what is the boundary condition for the s/f/s structure?
The self-consistency Eq. (14) and boundary conditions Eqs. (15), together with the Usadel equation for F+s ,ξ 2s πTcs d2F+s dx2− ωnF+s + 2 = 0, (17) will be used for finding the critical temperature of the S/F/S structure both in 0 and π phase states.
Q12. How can the authors explain the reentrant behavior of the phase state?
This situation corresponds to an S/F/S structure enclosed in a ring, where the π phase shift can be fixed by applying the magnetic flux quantum for any d f .
Q13. What is the importance of using the MMA in a wide range of parameters?
Thus the authors confirm the importance of using the MMA in a wide range of parameters in the case of S/F/S trilayers, where 0-π phase transitions are possible.
Q14. what is the boundary condition for f+s?
Using the boundary condition Eq. (10) the authors arrive at the effective boundary conditions for F+s at the boundaries of the right S layer,ξs dF+s (d f /2)dx = W 0,π (ωn)F+s (d f /2), (15a)dF+s (ds + d f /2) dx= 0, (15b) where the authors used the notationsW 0,π (ωn) = γ As( γb + Re B0,πf ) + γ As∣∣γb + B0,πf ∣∣2 + γ (γb + Re B0,πf ) , As = ksξs tanh(ksds), ks = 1ξs√ ωnπTcs .
Q15. How do the authors calculate the critical temperature in S/F/S trilayers?
To provide complete behavior of the critical temperature in S/F/S trilayers, the authors calculate Tc(d f ) dependencies in both 0 and π phase states by using the MMA and show them on the same plot, see Fig.