Aircraft pylons are structures used to mount external equipment such as engines, fuel tanks, weapons, etc. on an aircraft. They are usually installed under the wings or the fuselage, being attached to specific locations designed to bear these extra loads that are known as hardpoints. Figure4.17shows two exampes of pylon structures attached to the wing and fuselage of an aircraft.
(a) Pylon attached to wing. Image taken from
Huber [95]
(b) Pylon attached to fuselage. Image taken from
NASA [149]
Figure 4.17: Example of pylon structures
The pylon FE model studied in this research is based on the geometry of the first Airbus pylon studied in Remouchamps et al [173] and of the engine pylon patent US 20110204179 A1, also from Airbus (Skelly and Laporte [187]). Figure 4.18 presents
images from the patent and the FE model studied, which is constructed in Altair Hypermesh (Hypermesh [97]), being the design region composed of 580285 first order tetrahedral elements (CTETRA) and the parts outside the design region (both front and rear wing and engine ties) composed of 2410 elements. The general dimensions of the pylon and material properties are obtained from Abdelkader [2].
(a) Engine pylon patent and side view of the FE model
Front engine tie
Front wing tie
Rear wing tie
Rear engine tie
Engine GC
(b) 3-D view of the pylon FE model studied
Figure 4.18: Example of pylon structures
The material is Inconel 718, which is a family of austenite nickel-chromium-based superalloys that are suited for service in extreme environments that are subjected to high pressure and heat, and is widely used in aerospace applications. The mechanical properties of Inconel 718 are presented in Table4.11, and in this example the properties used are those at T = 20oC.
Table 4.11: Mechanical properties of Inconel 718.
T [oC] 20 580
E [MPa] 140000 117320 ρ [t/mm3] 8.15 · 10−9 8.15 · 10−9
It is assumed that the pylon is connected to an engine corresponding to an aircraft of the Airbus A320 family. In this case, the engine considered in the CFM56-5A3 from
CFM International, which provides a thrust of T =118 kN and has a dry weight of W =2.27 kN. The engine is modelled as a concentrated mass situated in the location exposed in Figure 4.19, which is the gravity center (GC) of the engine. The loads considered in this study are the minor crash landing loads of the Certification Spec- ifications for Large Aeroplanes CS-25 (EASA [63]), which are specified in Table 4.12
and are applied in the concentrated mass as presented in the right side of Figure4.19. Each load is included in a separate load case.
CONM2 (2270 kg)
(a) Location of the engine (b) Set of loads acting in the pylon
Figure 4.19: Example of pylon structures
Table 4.12: Minor crash landing loads of CS-25 applied in the engine.
Loads Upward Downward Forward Rearward Side Left Side Right
Loadcase 1 2 3 4 5 6
Value [g] 3.0 -6.0 9.0 -1.5 3.0 -3.0
The strategy followed is the proposed in Section 4.3.2 (CASE II), which was also applied to the previous rear fuselage example. First the DTO is carried out aiming to minimize the volume of the structure with constraints on the displacements, in a similar way to the exposed in Remouchamps et al [173] where the displacements in the gravity center of the engine are restricted in two directions (x and y corresponding to the horizontal and transversal directions). Afterwards, the compliance of each loadcase Ci is obtained to be imposed as the probabilistic constraints in the upcoming RBTO problem. The DTO problem is defined as:
subject to:
U Xengine(d) ≤ U Xmax (4.15b)
U Yengine(d) ≤ U Ymax (4.15c)
where U Xengine and U Yengine are the displacements in x and y of the gravity center of the engine whereas U Xmax and U Ymax are the maximum values allowed, which are set to 30 mm. The optimization settings include a penalization factor of p = 3.0, a minimum member size of 40.0 mm and a simmetry imposed in the x − z plane. After 40 iterations, the objective function converged to a volume of V = 1.12 · 108 mm3 which corresponds to a volume fraction of 6.38% respect to the initial one. The dis- placement constraints are active in the loadcases 2, 5 and 6, which corresponds to the downward and both left and right sideward loads. The convergence of the objec- tive function and constraints in the constraints in the active loadcases is presented in Figure 4.20. 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 14 16 Number of Iterations Ob j. F unction V [· 10 8 mm 3 ] Objective Function
(a) Evolution of Obj. fun.
0 5 10 15 20 25 30 35 40 45 50 10 15 20 25 30 35 40 Number of Iterations Displacemen ts U X ,U Y [mm ] U XLC2 U YLC5− U YLC6 U Ymax= U Xmax
(b) Active constraints (loadcases 2,5 and 6)
Figure 4.20: Evolution of the objective function and constraints evaluation at each iteration
The compliance values obtained for each load case are presented below: • CDT O 1 = 1.52 · 106 mm/N • CDT O 2 = 6.08 · 106 mm/N • CDT O 3 = 1.58 · 106 mm/N • CDT O 4 = 4.40 · 104 mm/N • CDT O 5 = 1.00 · 106 mm/N
• CDT O
6 = 1.00 · 106 mm/N
The RBTO problem is defined in order to obtain a design as stiff as the DTO de- sign when uncertainty is considered within the material properties and load values. The definition of the random variables including the random distribution, mean and standard deviation values is shown in Table4.13.
Table 4.13: Definition, distribution and statistical moments of the random vari- ables.
Random Variable Distribution µ σ δ
Young’s modulus E [MPa] Log-Normal 140000.0 7000.0 0.05
Load values Pi Normal 1.0 0.1 0.1
The formulation of the RBTO problem is presented in Equation 4.16:
min V (d) (4.16a)
subject to:
U Xengine(d) ≤ U Xmax (4.16b)
U Yengine(d) ≤ U Ymax (4.16c)
P [Ci(d, xK) > CiDTO] ≤ Pf,i (i = 1, ..., 6) (4.16d)
where Ci is the compliance value obtained in the i loadcase of the RBTO process, CDTO
i is the compliance upper limit obtained in the i load case of the DTO, P is the probability operator and Pf,i is the probability of failure imposed in each limit-state function. Those functions are defined as Gi = 1−CDTOCi
i , which means that if Ci
> CDTO i then G < 0 and the structure is in a failure region. The optimization settings defined for the RBTO problem are the same than those of the DTO.
In this section, the values of the probabilities of failure considered in all the limit-state functions Pf,i are the same, although they could be different as it will be indicated in Section5.2. Here, three RBTO cases were performed setting as probabilities of failure Pf = 2.27 · 10−2, Pf = 1.35 · 10−3 and Pf = 3.17 · 10−5, which corresponds to reliability indexes targets of βT = 2, βT = 3 and βT = 4. The SORA method took 3 cycles to converge for all cases, being the results obtained shown in Figure 4.21. The case of βT= 2 includes the convergence of the 3 DO cycles required.
Figure4.22-Figure4.24show the structural layouts derived from the DTO and RBTOs. From these results it can be remarked that, as in the previous example, the best structural scheme changes with the level of reliability required. Not only the structural
0 1 2 3 0 2 4 6 8 10 12 14 16 Number of Iterations Ob j. F unction V [· 10 8 mm 3 ] Objective Function 0 5 10 15 20 25 30 35 0 2 4 6 8 10 12 14 16 Num. Iter. Objective Function 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 16 Num. Iter. Objective Function 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 16 Num. Iter. Objective Function
(a) Evolution of RBTO (βT= 2)
0 1 2 3 0 2 4 6 8 10 12 14 16 Number of Iterations Ob j. F unction V [· 10 8mm 3] Objective Function (b) Evolution of RBTO (βT= 3) 0 1 2 3 0 2 4 6 8 10 12 14 16 Number of Iterations Ob j. F unction V [· 10 8mm 3] Objective Function (c) Evolution of RBTO (βT= 4)
Figure 4.21: Evolution of the SORA method for the RBDO βT = 2, βT = 3 and βT = 4 cases in the aircraft pylon example.)
members that emerge from the DTO have swelled, but in the rear part of the pylons substantial differences in the distribution of material can be appreciated.
Table 4.14presents the numerical results obtained in both the DTO and RBTO, while Table4.15shows the MPP values for the set of RBTO cases. From the former it can be stated that considering the variability in the values of the loads and Young’s modulus of the material leads to an increment in the volume of the structure, which is greater when increasing the security level, but in exchange the constraints are increasingly far away their design limits in order to guarantee their accomplishment despite the variations of the random variables.
(a) DTO (b) RBTO (βT= 2)
(c) RBTO (βT= 3) (d) RBTO (βT= 4)
Figure 4.22: Three-dimensional view of DTO and different RBTO (βT = 2, βT = 3 and βT = 4) layouts for the aircraft pylon.
(a) DTO (b) RBTO (βT= 2)
(c) RBTO (βT= 3) (d) RBTO (βT= 4)
Figure 4.23: Top view of DTO and different RBTO (βT= 2, βT= 3 and βT= 4) layouts for the aircraft pylon.
The pylon example has been solved in the HPCC, taking a single FE analysis an average computational time of 180 seconds to run. The number of iterations required in the
(a) DTO (b) RBTO (βT= 2)
(c) RBTO (βT= 3) (d) RBTO (βT= 4)
Figure 4.24: Side view of DTO and different RBTO (βT= 2, βT= 3 and βT = 4) layouts for the aircraft pylon.
Table 4.14: Objective function and constraints of the DTO and different RBTO cases in the aicraft pylon.
Load case DTO RBTO(βT = 2) RBTO(βT = 3) RBTO(βT = 4) Obj. Fun. [mm3] 1.12 · 108 1.53 · 108 1.68 · 108 1.88 · 108 Vol. frac. (%) 6.38 8.90 9.83 11.08 Comp.1 [mm/N] 1.52 · 106 1.04 · 106 0.86 · 106 0.72 · 106 Comp.2 [mm/N] 6.07 · 106 4.14 · 106 3.45 · 106 2.89 · 106 Comp.3 [mm/N] 1.58 · 106 1.08 · 106 0.90 · 106 0.76 · 106 Comp.4 [mm/N] 4.40 · 104 2.99 · 104 2.49 · 104 2.13 · 104 Comp.5 [mm/N] 1.00 · 106 0.68 · 106 0.58 · 106 0.49 · 106 Comp.6 [mm/N] 1.00 · 106 0.68 · 106 0.58 · 106 0.49 · 106
Table 4.15: MPP values for the RBTO cases in the aircraft plyon example.
Load case RBTO(βT = 2) RBTO(βT = 3) RBTO(βT = 4)
Value/MPP LC 1 1.19 1.29 1.36 Value/MPP LC 2 1.19 1.29 1.39 Value/MPP LC 3 1.19 1.29 1.36 Value/MPP LC 4 1.19 1.28 1.36 Value/MPP LC 5 1.19 1.28 1.36 Value/MPP LC 6 1.19 1.28 1.36 Value/MPP E [MPa] 1.35 · 105 1.32 · 105 1.28 · 105
DTO and RA loops and total computing time measured in minutes are presented in Table4.16. All the RBTO cases required approximately the same computational effort
since the number of SORA cycles and iterations in the optimization and reliability phases is the same.
Table 4.16: Summary of the computational effort in the aicraft plyon example.
Loop DTO RBTO (βT = 2) RBTO (βT = 3) RBTO (βT= 4)
SORA cycles - 3 cycles 3 cycles 3 cycles
Optimization 40 iter. 97 iter. 97 iter. 97 iter.
Reliability - 108 iter. 108 iter. 108 iter.
DTO/RBTO 73 minutes 608 minutes 586 minutes 599 minutes
4.5
Conclusions
This chapter has tested the methodology exposed in Chapter 3 and evidences that it can perform large topology optimization problems while taking into account the uncertainty presented in both loads and material properties. This methodology has proven to be robust and efficient as it worked for different 3D structural models and the computational effort spent was acceptable regarding the time that commercial software needs to perform the classical DTO. Furthermore, the Sequential Optimiza- tion and Reliability Assessment (SORA) has proven to be successfully implementable with commercial optimization software thanks to its uncoupled formulation and the auspicious results obtained. To summarize, this research demonstrates that RBTO can be successfully applied to industry-like aircraft structural models as a further step to classical topology optimization.
In the first example it can be seen that the differences between different safety targets are mainly focused in the appearance of new members when establishing a higher level of uncertainty, as well as the fattening of the members obtained in the DTO. This concludes that, as expected, requiring a higher safety factor in the RBTO gives an increase in the amount of material needed to accomplish the constraints. Furthermore, it is noticeable that in the second and third examples, both belonging to CASE II approach, the final structural schemes obtained from the DTO and RBTO are different depending on the level of uncertainty considered. This gives an idea of how important is to include uncertain data and define the safety targets in the preliminary stages of aircraft design in order to continue with the design process from the best initial architecture.
The relevance of reliability-based
topology optimization in preliminary
phases of aerospace structural design
5.1
Introduction
Structural design is a multi-step and multidisciplinary process where several data and requirements are put in common to look for the best technical and economic solution. This process has associated uncertainties which are usually included as partial safety factors in some phases of the process. In this framework, it is important to know if it would be advantageous to include real uncertain data considering their probabilistic distributions in an initial stage of aerospace structural design, such as topology opti- mization. As exposed in Chapter 4, this discipline has become a highly valuable tool for the major aerospace manufacturers in pursuit of their goal to reduce the weight of aircraft components and structures (Grihon et al [76], Krog et al [121] or Wang et al [207]).
The topology of preliminary baseline models is critical for the following design stages. As stated in Chapter4, depending on the reliability target required by the engineers, the RBTO may provide different structural layouts that would condition the next phases of the aircraft design process. Besides, RBTO based designs might lead to lighter structures than those provided by the DTO as consequence of considering real uncertain data instead of partial safety factors throughout the design process, which ensures consistent reliability levels for specific design constraints. In this chapter the examples performed in Section4.4 are compared fairly by setting the same probability of failure for both the DTO and RBTO in order to determine which is the most efficient in terms of weight savings and establishing a coherent comparison of results in Section 5.2. The main contents of this section have been published as a research paper in López et al [137].
Moreover, the novel architectures arisen from topology optimization cannot be extrap- olated directly to subsequent phases of the design process, instead they need to be interpreted and turned into real structures through a conceptual design definition task (Harzheim and Graf [80]) and then re-tuned by applying size and shape optimization techniques. The traditional way of designing a structure through topology optimiza- tion consists of performing a DTO and extract from the topology results a structural layout that is later interpreted, defined geometrically and submitted to a size opti- mization process ([120] or Buchanan [29]). However, these studies do not mention the consideration of uncertainties throughout the design process. In practice, they are usually included in the latter phase (size optimization), either through the impo- sition of partial safety factors or more recently through RBDO. Nevertheless, other valid approach would be to consider such uncertainties from the preliminary design stages through RBTO. RBTO approaches should be interesting since the structural configurations derived from them may condition the next phases of the aircraft design process.
As exposed in Chapter4 the results obtained from the DTO and the RBTO may lead to different structural layouts, which makes difficult to decide which configuration will adapt better to the design conditions of the subsequent design phase. In this research the structures emerged from the DTO and RBTO undergo a reliability-based size optimization phase establishing the same reliability requirements than in the RBTO in order to consider uncertainties in the upcoming design phase. Comparing the results of both approaches may help to determine if it is worth to take into account such uncertainties in early design stages (topology optimization) or by contrast if it is better to include them only in the sizing phase. In Section 5.3 this strategy is detailed and applied to two application examples. The first one is a two-dimensional rectangular domain that aims to explain clearly the goal of the strategy, while the second is the rear fuselage example presented in Section 4.4.2, which represents a more industry- oriented case. In each example, two different structures were defined from the DTO and RBTO results through a conceptual design definition process similar to the carried out in Buchanan [29], which consists of transforming the topology layouts in real bar structures. Finally, the conclusions of the chapter are summarized in Section 5.4.