Sizing and Layout Design of an Aeroelastic Wingbox through Nested Optimization Bret K. Stanford∗ NASA Langley Research Center, Hampton, VA, 23681 Christine V. Juttey Craig Technologies, Inc., Cape Canaveral, FL, 32920 Christian A. Coker z Mississippi State University, Starkville, MS, 39759 The goals of this work are to 1) develop an optimization algorithm that can simulta- neously handle a large number of sizing variables and topological layout variables for an aeroelastic wingbox optimization problem and 2) utilize this algorithm to ascertain the ben- efits of curvilinear wingbox components. The algorithm used here is a nested optimization, where the outer level optimizes the rib and skin stiffener layouts with a surrogate-based optimizer, and the inner level sizes all of the components via gradient-based optimization. Two optimizations are performed: one restricted to straight rib and stiffener components only, the other allowing curved members. A moderate 1.18% structural mass reduction is obtained through the use of curvilinear members. I. Introduction Layout optimization of a wingbox structure, a form of topology optimization, involves deciding upon the best placement of ribs, spars, and stiffeners within a semimonocoque structure. There are numerous examples of layout design in the literature, including Refs.1-6. Some of these papers adhere to conventional orthogonal layouts of ribs, spars, and stiffeners,3 others blend the roles of these components through angled and/or curved members,2,5,6 and others abandon this convention entirely with a complex network of full- and partial-depth members.1,4 Regardless of how the layout design is conducted, sizing design must be included as well, by optimizing the thickness distribution of the various shell components: ribs, spars, stiffeners, and skins (cover panels). The synergies between these two types of design variables (layout/topology and sizing) are strong, particularly in terms of skin buckling metrics. Fewer stiffeners and/or ribs increase the size of the concomitant skin panels, requiring thicker skins to preserve the required buckling and stress performance. Simultaneous optimization of sizing and topology is required to locate the best compromise to minimize weight, but this poses substantial numerical difficulties. Layout optimization typically must be conducted with non-gradient- based optimization (specifically, evolutionary optimization is used for each of the references above). This family of algorithms scales very poorly with the number of design variables, and so if sizing variables are to be included as well, they must be restricted to a small, diluted set of parameters. It is obviously preferable to work with a large set of sizing variables for detailed component sizing, driven by an algorithm that scales well with large design spaces (gradient-based optimization). In this case, sequential optimization may be considered, where the topology is optimized under a frozen set of sizing parameters, followed by sizing optimization on a frozen wingbox topology, and so on. This is considered in Ref.2, but sequential optimization cannot, in general, be expected to fully capture the rich couplings ∗Research Aerospace Engineer, Aeroelasticity Branch, [email protected], AIAA Senior Member. yResearch Engineer, Advanced Materials and Processing Branch, [email protected]. zUndergraduate Research Assistant, Department of Aerospace Engineering, [email protected], AIAA Student Member. 1 of 14 American Institute of Aeronautics and Astronautics between sizing and topology. Furthermore, layout optimization of ribs/spars/stiffeners in isolation (i.e., with a frozen set of sizing variables) will struggle to minimize structural weight while also maintaining constraints on stress, buckling, flutter, etc.7 A superior approach, though computationally expensive, is to utilize a nested optimization considered in Refs.3 and8. In this approach, the layout/topology design variables are optimized within the outer level via a global optimizer, while the sizing design variables are handled at the inner level with a gradient-based optimizer. Each set of layout design variables is evaluated by freezing the topological details, and then sizing the wingbox to minimize the structural mass of that particular topology, under a series of aeroelastic performance constraints. The final structural mass (objective function) is then returned to the topology optimizer, which then moves on to the next topology under consideration, and the process repeats. The topology optimizer is not explicitly aware that sizing optimization occurs at the nested level, and is also unaware of the existence of the performance constraints (as the sizing optimization is able to satisfy all constraints). Rather than the relatively inefficient evolutionary topology optimization considered in the previously-cited references, a Bayesian/surrogate-based optimizer is used here; the final algorithm resembles recent work conducted in Ref.9. II. Approach A. Wing Layout Generation All of the work in this paper is conducted on the conceptual Common Research Model (CRM). The 1g outer mold line for the CRM is described in Ref. 10, and a jig shape CRM wingbox, suitable for aeroelastic analysis, was subsequently developed by Kenway et al.11 This transonic transport has a wing span of 58.7 m, a mean aerodynamic chord of 7.0 m, a taper ratio of 0.25, a sweep angle of 35◦, and a cruise Mach number of 0.85. The aluminum wingbox is composed of upper and lower skins, a leading and trailing spar, ribs, and T-shaped stiffeners. The topology optimizer dictates the number and placement of the ribs and skin stiffeners, though the topology of the two spars is fixed, as is the topology of the vertical web stiffeners. The skins, spars, and ribs are modeled with shell finite elements, the skin stiffeners with beam elements, and the vertical web stiffeners are not modeled explicitly, but instead smeared into the shell stiffness properties12 of the webs. Rib stiffener pitch is fixed at 15 cm, and spar stiffener pitch at 23 cm. The baseline, \empty" wingbox, shown on the left of Fig.1, is composed of the skins, stiffened spars, a close-out rib, and a rib located at the trailing edge break: these topological features are constant. From this starting point, the layout optimizer adds ribs and skin stiffeners. Two topological parameterizations are considered here, the first utilizing only straight members (Table1) and the second allowing for curved members (Table2). For Table1, stiffener rotations are measured relative to the leading edge spar (positive angles swept forward), and stiffener spacing indicates the ratio of the stiffener pitch at the leading edge to that at the trailing edge. The topology optimizer can separately design the rib layout in the inboard and outboard portions of the wing (delineated by the fixed rib at the trailing edge break). Rib spacing in Table1 is the ratio of the rib pitch at the two ends of each rib design zone, an idea taken from the \linked shape" parameterization of Ref.2. Referencing Table2, curved stiffener paths are generated by 4 control points located from root to tip, where 2 · y=b is the relative semispan location. Stiffener rotation is dictated by the 4 design variables, and is assumed to vary in a piecewise linear manner from point to point. Spanwise integration of this piecewise linear plot yields a curvilinear path. Curved rib layouts are designed using a parameterization taken from Ref.2, where η1 and η2 dictate the normalized chordwise location of a control line at the two ends of each rib design zone, and the various ξij terms dictate the relative locations of various control points within the design zone. More details of this \linked shape" parameterization can be found in the reference. It should be noted that if η1 = η2 = 0:5, ξ12/ξ11 = ξ22/ξ21 = ξ32/ξ31, and the stiffener rotations at all 4 control points are set equal to each other, then the straight-member topological parameterization of Table1 is recovered. It is finally noted that in each parameterization, 4 of the design variables are integers, while the remainder are continuous. Two sample topologies are shown in Fig.1, one straight and one curved. In order to generate the beam/shell finite element meshes of the wingbox, each added topological component (ribs and stiffeners) is meshed independently, and then \stitched" to the outer skins and spars via multipoint constraints (MPCs). Specifically, each node of the skin stiffeners, and each node along the perimeter of a rib, is designated a slave node, whose motion is dictated by the shape functions of the skin/spar shell finite element that circumscribes 2 of 14 American Institute of Aeronautics and Astronautics Figure 1. Sample wingbox topologies with straight and curved members. Table 1. Topological parameterization with straight members only (10 total). \UB" and \LB" are the upper and lower bounds of each variable. parameter LB UB parameter LB UB # of upper skin stiffeners 8 20 # of inboard ribs 3 14 upper skin stiffener rotation -30◦ 30◦ inboard rib spacing 0.25 4.0 upper skin stiffener spacing 0.25 4.0 # of outboard ribs 10 26 # of lower skin stiffeners 8 20 outboard rib spacing 0.25 4.0 lower skin stiffener rotation -30◦ 30◦ lower skin stiffener spacing 0.25 4.0 it. This approach is found to be less computationally expensive, and (more importantly) far more robust than a situation where a monolithic mesh is computed in an automated manner (see Ref. 13 for a recent example of this). The topology optimizer will consider a wide range