JOURNAL OF SPACECRAFT AND ROCKETS Vol. 49, No. 4, July–August 2012
Stagewise Multidisciplinary Design Optimization Formulation for Optimal Design of Expendable Launch Vehicles Mathieu Balesdent∗ Centre National d’Etudes Spatiales, 75612 Paris, France Nicolas Bérend† ONERA–The French Aerospace Lab, 91123 Palaiseau, France and Philippe Dépincé‡ IRCCyN, 44321 Nantes, France DOI: 10.2514/1.52507 Optimal design of launch vehicles is a complex process that gathers a series of disciplines. The classical method used to solve such problems consists in decomposing the problem into the different disciplines and in associating a global optimizer and disciplinary analyzers (multidiscipline feasible method, most used in launch vehicle design). This paper presents a new multidisciplinary design optimization method based on a transverse decomposition of the design process adapted to the multistage launch vehicle architecture. The proposed bilevel method splits up the optimization process into different flight phases and performs the different stage optimizations either sequentially or concurrently. Thus, the proposed approach transforms the global multidisciplinary design optimization problem into the coordination of elementary multidisciplinary design optimization problems and moves the problem complexity from the system level to the subsystem level. Three formulations of this method are proposed and compared with the multidiscipline feasible method on a multistage launch vehicle design problem. The proposed method allows the dimension of the search domain and the number of constraints at the system level to be reduced. In that way, this approach makes the use of heuristic methods such as the genetic algorithms more efficient in solving the large-scale highly nonlinear launch vehicle design problem.
x y
Nomenclature C D Dne f g H h Isp Mf Mp Md Mu nf Pc Pe q Rm r s T W
u v
= = = = = = = = = = = = = = = = = = = = = =
normalization coefficient on coupling constraints stage diameter, m nozzle exit diameter, n objective function inequality constraints normalization coefficient on equality constraints equality constraints specific impulse, s fairing mass, kg propellant mass, kg dry mass, kg payload mass, kg axial load factor chamber pressure, Pa nozzle exit pressure, Pa mass flow rate, kg:s1 mixture ratio radius, m state vector thrust-to-weight ratio control law velocity, m:s1
= = = = =
design variables coupling variables angle of attack, deg flight-path angle, deg pitch angle, deg
I. Introduction
M
ULTIDISCIPLINARY design optimization (MDO) of launch vehicles is a complex process that has to handle many disciplines (e.g., propulsion, aerodynamics, trajectory, etc.). The most common MDO method in launch vehicle design (LVD) consists in associating disciplinary analyzers and a global optimizer (multidiscipline feasible, or MDF method [1–4]). This formulation leads to a very large search space dimension (especially when the trajectory and design variables are handled at a same level), and consequently presents some problems of robustness to the initialization and an expensive calculation cost. To exploit the couplings existing between the optimization of the design variables and the optimal control law calculation, the method presented in this paper proposes a transverse decomposition of the optimization process into the different stages. Each stage gathers all the disciplines and is optimized independently. The global trajectory is decomposed into elementary flight phases which are optimized during the stage optimizations. The method consists of two levels of optimization. The system level is a global optimizer which coordinates the optimization process. The subsystem level is composed of the different stage optimizers which aim to optimize their respective configuration, with respect to the instructions given by the optimizer at the system level. The decomposition into the flight phases has already been applied to optimize the trajectory of launch vehicles [5–8]. The existing methods in literature consist in optimizing separately the different flight phases to split up the original optimization problem into smaller ones. These methods have been applied to optimize either the trajectory only or the trajectory and a small number of design variables (most often the propellant masses). This decomposition has
Presented as Paper 2010-9324 at the 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort Worth Texas, 13–15 September 2010; received 22 September 2010; revision received 12 January 2012; accepted for publication 12 January 2012. Copyright © 2012 by Onera & Cnes. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0022-4650/12 and $10.00 in correspondence with the CCC. ∗ Research Engineer;
[email protected]. Student Member AIAA. † Project Manager;
[email protected]. ‡ Professor, Systems Engineering Products, Performance and Perceptions Department;
[email protected]. 720
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been used to optimize the trajectory of expendable [5] and reusable launch vehicles [6–8]. In [5], a decomposition process is proposed to solve the all-up trajectory optimization problem concerning a booster and an upper stage to deliver a maximal payload into a target orbit. The optimizations of the booster and the upper stage trajectories are performed separately and are coordinated through the definition of a parking orbit between the two subsystems. The optimization strategy is an iterative process between the optimizations of two subsystem trajectories and a system-level coordinator, and sensitivity calculations are transmitted from the subsystem to the system levels to improve the optimization process. The results of this study show that the solution obtained by the proposed decomposition strategy is valid and can be a good guess for a traditional optimization process considering simultaneously all the optimization variables, which may present an important calculation time (and may fail) when the initialization is not in the vicinity of an optimum. In [6], the optimization of branching trajectories of a reusable twostage-to-orbit launch vehicle is solved. The optimization process handles the trajectory variables and the propellant masses of the different stages. The proposed strategy involves a system-level optimizer to coordinate the orbiter and booster flight phases through variables defining a staging point. This study allows to show the capability of the flight phase decomposition to optimize the trajectory and a few mass variables. Nevertheless, the problem treated in this paper does not perform a complete multidisciplinary optimization and does not consider all the design variables and disciplines other than the trajectory and the propellant mass definition (no disciplinary couplings). In [7], fixed point iteration (FPI), collaborative optimization (CO), and two formulations of individual discipline feasible (IDF) are compared in the optimization of branching trajectories of a fully reusable two-stage launch vehicle. The coordination of the different branches is performed through the handling of a staging point at the system level. This paper shows that the use of MDO methods associated to the flight phase decomposition allows to transform the non hierarchical initial problem to a hierarchical one and find a better optimum than using the classical engineering method (manual iteration which involves a loop between the sequential optimizations of the different branches). Nevertheless, CO appears to be more computationally expensive than the other used methods for this problem (Kistler K-1 problem). In [8], a FPI, a hybrid MDF which presents a multidisciplinary analysis and a suboptimization at the subsystem level, and a collaborative approach which optimizes independently the different stages at a subsystem level and coordinates them at a system level via a staging point, are compared in the trajectory optimization problem by using direct and indirect methods. As in [7], a FPI method that considers the optimization problem as a loop between the optimizations of the different flight phases is compared with the MDF method. Unfortunately, the results obtained for the collaborative approach are not shown. Nevertheless, this study shows that using the MDF method presents some benefits with respect to FPI which presents a lack of consistency of the results (no convergence towards the identified optimum whereas the MDO method allows to converge in a few system-level iterations). All these studies are mainly focused on the trajectory optimization, involving the trajectory variables (with occasionally a few design parameters concerning the masses, e.g., the propellant mass) and do not consider all the disciplinary design variables and the induced disciplinary couplings. The trajectory optimization of expendable launch vehicles presents similarities with reusable launch vehicle (RLV) trajectory optimization. In particular, the trajectory can also be decomposed into the different flight phases which correspond to the flights of the different stages. In this paper, we propose the extension of this method to the complete multidisciplinary design optimization of expendable launch vehicles, considering all the disciplines and the induced disciplinary couplings. The methods described in this paper are focused on the need to explore a large search space without requiring any initialization from the user and a priori fine knowledge on the variables search domains.
The paper is organized as follows : In Sec. II, we will describe several LVD methods. In this section, we will present the stagewise decomposition formulation. Section III will describe the application case. In this section, we will present the formulation of the MDO problem and its solving using the MDF and proposed methods. In Sec. IV, we will study the reduction of the problem dimension and detail the advantages of the proposed formulations. Finally, in Sec. V, we will compare the MDF and stagewise methods in the same application case and using the same optimization algorithms.
II. Formulation and Design Methods A.
MDO Formulation
A general MDO problem can be formulated as follows: Minimize fx; y With respect to Subject to
(1)
x gx; y 0
(2)
hx; y 0
(3)
cx; y 0
(4)
with: 1) f: objective function (gross-liftoff-weight), 2) g: inequality constraints (disciplinary constraints), 3) h: equality constraints (specifications of the mission), 4) c: coupling constraints (consistency of the coupling between the subsystems), 5) x: design variables; x fxsh ; x k g, where xsh are the variables which are shared between the different subsystems (global variables) and x k denotes the variables which are specific to the kth subsystem (local variables), and 6) y: coupling variables. B. Classical Design Methods 1. Engineering Method Used to Solve the LVD Problem
The LVD is a particular MDO problem because it combines the optimizations of design variables (determining the configuration of the launch vehicles, i.e., variables of propulsion, weights and sizing, aerodynamics, etc.) and a control law (determining the trajectory). The classical engineering method to solve this problem consists of the sequential design variables and trajectory optimizations (Fig. 1). It generally consists of a loop between the design variables optimization and the control law calculation (FPI [9]). This process may take a lot of time (even for relatively simple processes), may not converge (if the function defining the FPI is not a contraction mapping), and does not allow to find a compromise between the different disciplines in case of antagonistic disciplinary objectives (i.e., involving tradeoff search). 2.
Classical MDO methods
The classical MDO method used in the LVD consists in decomposing the design process into the different disciplines and using a MDF method [2–4]. Several other methods have then been developed (e.g., all at once (AAO) [1,10], then CO [11] and bilevel integrated system synthesis (BLISS) [12]) and tested in LVD but their use is less common. A detailed review of the application of MDO methods in LVD can be found in [13]. The MDF method consists of a multidisciplinary analysis (MDA) and a global optimizer (Fig. 2). The MDA is a process which calls disciplinary analyzers and ensures the consistency of the process with regard to the disciplinary feasibility and the couplings. A global optimizer is used to handle the optimization variables x and to satisfy constraints. When the disciplines are coupled, the MDA generally consists of a loop between the disciplines. The MDA is performed at each iteration of the global optimizer and handles the coupling variables y to satisfy the coupling constraints c. In the MDF method, the optimizer has to handle a large number of variables, especially when trajectory variables are considered at the same level as the design variables [3].
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Propulsion
NO END
YES
Weights/Sizing
level) and a global optimizer (at the system level) coordinates the different optimizations of the subsystem level while optimizing the objective function. Coupling variables are typically added to the optimization problem to ensure the consistency between the different disciplines.
Aerodynamics
2.
Convergence? Trajectory
Fig. 1 Classical engineering method.
OPTIMIZER Objective : Minimize GLOW Constraints : Orbit reached Angle of attack < 4° Dnei 0 For the ith stage Given xsh , y, fk;k>i Minimize fi xsh ; x i ; y Mstagei Mustagei Mdstagei Mpstagei Mustagei
(56)
With respect to Optimization at the Subsystem Level: At the subsystem level, all the equality constraints h have been removed and are included in the objective function. Indeed, each stage aims at minimizing a quadratic sum between the real values and the instructions given by the systemlevel optimizer. In this formulation, the couplings concerning the different stage masses are also considered in the objective function: Given xsh ; y Minimize fi xsh ; x i ; y
rfi rttfi 2
vfi vttfi
2
(57)
ci : nfmax maxnfmax ; nfmax 0
(58)
Subject to
gi1 : 0:4
R V fi ttfi 2 Mpi Mdi Mustagei Mustagei1 2 (46) M With respect to T ; Rmi ; Pci ; Pei ; i x i Di ; Mpi ; W i
T x i Di ; Mpi ; ; Rmi ; Pci ; Pei ; i W i
(47)
Pei 0 Par
(59)
gi2 : Dnei 0:8Di 0
(60)
gi3 : jimax 1 j 4 deg 0
(61)
gi4 : jimax 2 j 15 deg 0
(62)
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Table 2 Synthesis of the optimization problems complexity MDF First form. Second form. Number of variables for the three-stage configuration Number of additional variables when adding a stage Number of constraints for the three-stage configuration Number of additional constraints when adding a stage
hi1 : rfi rttfi 0
(63)
hi2 : vfi vttfi 0
(64)
hi3 : fi ttfi 0
(65)
13 6 4 1
14 6 5 1
11 5 0 0
characteristics of the LVD problem to compare theoretically the different methods and to expose the advantages of the proposed formulations. The numbers in parentheses are the current values taken by the variables for the considered study case. Let us consider: 1) ne : number of stages (3), 2) ns : dimension of the state vector (5: altitude, velocity, flight-path angle, longitude, and mass), 3) nc : dimension of the control law variables vector (1: pitch angle), 4) npp : number of control law crossing points per stage (4), 5) npsh : number of design variables per stage which play a part in the flight phases of several stages (1: diameter of the stage), and 6) npk : number of design variables per stage which only play a part in the kth stage flight phase (5: mixture ratio, chamber pressure, nozzle exit pressure, propellant mass, and initial thrust-to-weight ratio).
i i 1, and Di is considered in Eq. (57) only for i 1.
IV. Analysis of Problem Dimensionality In this section, we study analytically the dimension of search spaces and the number of constraints with respect to some
a) MDF
31 10 14 3
Third form.
b) First formulation
c) Second formulation
Fig. 7
Dimension of search space
50 40 30 20 10
0
2
3
Number of stages MDF
Fig. 8
Number of constraints
20
Number of constraints
Dimension of search space
d) Third formulation The different MDO methods (initial formulations).
First formulation
4
15
10
5
0
2
3
4
Number of stages Second formulation
Third formulation
Search domain and number of constraints at the system level with respect to the number of stages.
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Qualitative comparison of the different formulations (-: worse than MDF, : better than MDF, : about equal to MDF)
Table 3
No requirement of equality constraints at the system level No requirement of equality constraints at the subsystem level Dimension of the coupling vector Convergence velocity expected Consistency between the subsystems at each iteration Independence of the subsystem optimizations
Design variables for the reference optimal design
Disciplines
Variable
Stage 1
Stage 2
Stage 3
Weights Propulsion
Mp, t Rm Pc, bar Ps, bar
Geometry Trajectory GLOW
D, m nfmax
78.3 3.98 120 0.42 1.82 3.61
26.8 4.12 80 0.05 1.31 3.61 4.34 136.68 t
10.4 4.27 60 0.005 0.49 3.61
T W
The search space dimensions are given by: nMDF ne :npk nc :npp npsh 1
(66)
nfirst formulation ne 1:ns 2 ne 1:nc ne 1:npsh 1 ne 1
(67)
nsecond formulation ne 1:ns 2 ne 1:nc ne 1:npsh 1 ne
(68)
nthird formulation ne 1:ns 2 ne 1:nc ne 1:npsh 1 0
(69)
The numbers of constraints at the system level are given by: ncMDF 3:ne ns 2 2
Third form.
- -
-
(70)
ncfirst formulation ne 1
(71)
ncsecond formulation ne
(72)
ncthird formulation 0
(73)
8
Second form.
By decomposing the optimization problem, the search domain of each optimization process at the system and subsystem levels is considerably reduced that allows to perform a global optimization in comparison with MDF in which the single optimizer has to handle a great number of variables. Indeed, in the MDF method, the dimension of the optimizer search domain is 31, while the systemlevel optimizers of the first, second and third stagewise decomposition methods have to handle, respectively, 13, 14, and 11 variables. In the same manner, the MDF optimizer problem is very constrained (8 inequality and 6 equality constraints). The stagewise decomposition method allows the distribution of the constraints on the system and the subsystem levels. Table 2 summarizes the number of constraints and variables handled by the MDF and the stagewise decomposition system-level optimizers. In the MDF method, the use of heuristic algorithm implies a significant calculation time because the dimension of the search domain is large (e.g., for the use of a genetic algorithm, each chromosome will have 31 components). The evolutions of the search domain dimensions and the numbers of constraints with respect to the number of stages are given in Fig. 8. This figure points out that the more stages the optimization problem involves, the better are the stagewise decomposition formulations with respect to the MDF method. Globally, the third formulation is the one which presents the best characteristics. Indeed, this method does not require any constraint at the system level, involves the lowest number of coupling variables (i.e., the lowest search domain) and ensures the coupling consistency between the different subsystems at each iteration. This characteristic allows the user to have consistent solutions even if the algorithm at the system level has not converged yet, which is not the case with using the first and second formulations. Table 3 qualitatively summarizes the different characteristics of the proposed formulations.
V. A.
Results and Comparison
Reference Optimal Design
The values of the design variables for the optimal design (obtained with the third method in phase II) are given in Table 4. This design is obtained by using the third method and with finely tuning the initialization and the search domains. This optimum will be used as reference to compare the different formulations in the following
Evolution of the penalization coefficient
Evolution of the penalized objective
Penalization coefficient
10
6
10
Infeasible domain
4
10
Feasible domain 2
10
0
10
Penalized objective function
Table 4
First form.
MDF Formulation 1 Formulation 2 Formulation 3
6
10
5
0
2
4
6
Time (h)
Fig. 9
8
10
10
0
2
4
6
8
Time (h)
Comparison of MDF, first, second, and third stagewise decomposition methods.
10
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Table 5
Characteristics of the different methods
First feasible design
MDF F1 F2 F3
Best design after 10 h
Computation time
GLOW, t
Gap with respect to ref. optimum
3h49 5h28 1h32 0h14
152:97 158.75 156.83 162.63
11:92% 16.15% 14.74% 18.99%
section. In this section, we will compare the results found in the phase I with a same random initialization and very large variation domains on the design variables. B. Comparison of MDF and Stagewise Decomposition Formulations 1. Methodology of Comparison
We have chosen to perform the comparison between the different formulations and the MDF method in the phase I to be consistent with the study cases addressed in recent literature concerning the LVD [3,19,20]. These papers use GA to solve analog MDO problems as the one presented in the previous section but do not involve a decomposition into the different stages. The objective of the proposed comparison is to quantify the ability to find feasible designs and to optimize them in the case of large search space bounds on optimization variables (100 km on altitude, 4 km=s on velocity, 50 deg on angles, 2 m on diameters, etc.), and without a priori knowledge on the feasible solutions domain. In that way, we have chosen to perform the optimization using the same GA formulation as [3,19]. For this purpose, the formulations have been modified to address all the equality constraints as a penalization on the objective function [27]. Thus, the global objective function is a weighted sum of the real objective (GLOW) and relative differences on the satisfaction of the equality constraints. The generic problem reformulation to use a GA as the system-level optimizer is the following: Minimize fx; y
Gap reduction during the optimization
GLOW, t Gap with respect to ref. optimum 152.54 149.67 145.85 141:27
2:64% 41:14% 54:49% 82:31%
11.60% 9.50% 6.71% 3:36%
different methods. The second column indicates the elapsed computation time to find a feasible design. The third column shows the GLOW of the first feasible design and the column 4 indicates the gap between the first feasible design and the value of the optimum found after phase II by the third method (reference design, taking approximately one more hour using a gradient-based method at the system level). In the same manner, the columns 5 and 6 present the obtained GLOW at the stopping time of the optimization process (10 h) and the gap with respect to the reference optimum. Finally, the column 7 shows the reduction of this gap during the optimization process. From Fig. 9 and Table 5, we can see that the third formulation leads to the best results in terms of computation time to find a feasible design and quality of the design found at the stopping time of the algorithm. Indeed, this method allows to find a feasible design after 14 min whereas MDF requires 3 h 49 min to obtain a feasible design. Moreover, the design found by the third formulation is 10 tons better than the design found by MDF. Concerning the computation time to find a feasible design, the second and third methods find faster a feasible design than MDF. The first method is the slowest one to find a feasible design. This is principally due to the mass coupling constraints [Eqs. (28) and (29)], which are difficult to satisfy. Handling these constraints by a collaborative formulation (second formulation) allows to greatly improve the behavior of the optimization process.
Minimize fx; y
P jhi x;yj i
Hi
P jcj x;yj j
Cj
With respect to x With respect to x gx; y 0 ! Subject to hx; y 0 Subject to gx; y 0 cx; y 0
where Hi and Cj are normalization coefficients on the equality and coupling constraints. To compare the efficiency of the proposed methods, we use the same random initialization, the same optimization algorithm and the same tunings at the system level. Depending on the different methods used, the objective function can vary according to the number of equality constraints present in the optimization problem.
Figure 10 shows the evolution of the real objective function (GLOW) with respect to the computation time for only the feasible designs. At the stopping time of the optimization process, all the proposed methods allow to obtain a better design than MDF, although they find a worse first feasible point than MDF. Concerning Evolution of GLOW during the optimization 165
Algorithms Employed
At the system level (including the MDF optimizer), we use a genetic algorithm with 100 individuals per generation. At the subsystem level (only for stagewise decomposition formulations), we use a SQP algorithm. 3.
Results
Figure 9 presents the results obtained in this study. In this figure are shown the evolution of the penalized system-level objective and the penalization with respect to the computation time. We consider that the design is feasible when the penalization is lower than 104 which corresponds to a satisfaction of the equality constraints with an accuracy of 100 m on altitude, 10 ms1 on velocity, 0.1 deg on angles, and 100 kg on masses. Table 5 summarizes the behavior of the
MDF Form. 1 Form. 2 Form. 3 Ref. opt
160 155 GLOW (t)
2.
150 145 140 135
0
2
4
6
8
Time (h)
Fig. 10 Evolution of GLOW for the feasible designs.
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Fig. 11 Best found designs.
the improvement of the gap with respect to the optimum, the third method allows to reduce the gap of 82% while the first (resp. second) method reduces this gap of 41% (resp. 54%). MDF presents some difficulties in improving the objective function (gap reduction of only 2.6%). Finally, Fig. 11 shows the schematic representations of the obtained launch vehicles and the corresponding trajectories. From this figure, we can see that the launch vehicle geometries found by the first and second formulation are the closest to the reference optimal one. Even if the geometry of the design found by the third formulation is relatively far from the optimal one, the found trajectory is very close to the reference optimal one, that explains the good value of the found GLOW. Concerning the MDF method, the design found at the stopping time of the optimization process is far from the reference optimum with respect to the geometry as much as to the trajectory. C.
Advantages and Drawbacks of the Proposed Methods
The best advantage of the proposed methods is to reduce the search domain dimension at the system level. Indeed, the analytical analysis of the problem dimension has shown that the number of variables at the system level can be reduced threefold with using the third formulation. Consequently, the proposed formulations allow to move the design complexity from the system level to the subsystem level. In the same manner, the number of constraints at the system level is also reduced. In particular, the third formulation does not involve any constraint at the system level, that allows the use of no-constrained optimization algorithms. Since the search domain dimension is reduced, the different proposed methods are more adapted to the exploratory search (heuristic) than MDF. Indeed, for the considered study case, with the same computation time, all the proposed methods find a better design than MDF. Moreover, by handling the couplings by a collaborative form (second formulation) or directly at the subsystem level (third formulation), the proposed methods allow to find a feasible design in less computation time than MDF. The main drawback of the method is the need of fast and robust optimizers at the subsystem level. Indeed, to be competitive, the proposed formulations require that the stages optimizers produce results efficiently and can be correctly initialized with respect to the system-level instructions.
VI. Conclusions In this paper, three new MDO methods adapted to the LVD problem have been described. These methods use the stagewise decomposition to split up the global optimization problem into smaller ones, and to reduce the search domain dimension and the number of equality constraints at the system level. Thus, the initial optimization problem is reduced to the coordination of elementary design problems and the problem complexity is shared between the system and the subsystem levels. The proposed formulations have been compared with the MDF method in a global search study case by using a genetic algorithm. Results show that the proposed methods work well from random initializations whereas the MDF method presents difficulties to find a feasible design and optimize it. Thus, with using an appropriate problem formulation, the stagewise decomposition makes the use of the global search algorithms more efficient to solve the LVD problem, from random initialization and without requiring specific knowledge of the optimization variables variation domains. The proposed method and comparison only apply to “sequential staging,” where one stage can be ignited after the previous stage has been jettisoned. The consideration of parallel staged launch vehicles with overlapping thrusting would require additional research.
Appendix To evaluate the robustness of the MDF method and its ability to find the reference optimum in case of large search space using SQP algorithm, we have generated 250 randomized initializations. The obtained results are summarized in Table A1. This table shows that the coupled use of MDF and a gradient-based algorithm poses important problems in terms of design process behavior and robustness to the initialization (only 10% of success). Table A1
Results of the MDF robustness study
Number of initializations Number of achieved optimizations Success rate Average optimum Average relative difference with respect to the reference optimum
250 25 10% 149 t 9%
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Indeed, the average found optimum is relatively far from the reference optimum (9%). The results of this study motivate the use of a global search algorithm when the MDF is employed, and the need of a dedicated method (formulation and optimization algorithm) to efficiently solve the LVD problem.
[13]
Acknowledgment
[14]
The work presented in this paper is part of a Ph.D. Thesis cofunded by Centre National d’Etudes Spatiales and Onera (funding research contract No.2008/82208).
[15]
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O. de Weck Associate Editor
