## 3 The Multilevel Approach
### Second Level  
The second level optimization inherits a set of aerodynamic morphing shapes that arise from the morphing shape optimization problem described in Section 3.1. The number of aerodynamic shapes is equal to the number of multiple load conditions and defines a corresponding set of design load cases. Given these shapes and the corresponding external loads, the second level aims at designing a suitable internal compliant structure that can match, properly actuated and under external loads, the desired morphing shapes. According to the shape-adaptable design problem (3), aerodynamic shapes represent multiple objective functions at this level. Each objective function is associated with the corresponding design loads, which increases the total number of objective functions. The compliant structure must be able to satisfy the kinematic and structural requirements, for each pair of shapes and loads. This problem is a typical multiobjective design problem that can be efficiently incorporated into an EA, which can find multiple optimal solutions in one single simulation run due to its population approach. In the case of adaptive wing structures, the multiobjective design problem (4) has the shape-adaptable problems combined with the structure design problems (2). The proposed approach is based on a dedicated Multiobjective Genetic Algorithm (MOGA) coupled with a Load Path Representation. The load path representation is a design parameterization of the stiffness tensor, which makes use of load paths to represent different structural topologies, cross-sectional beam areas, and the positions of the internal points of interest for the shape problem. The load path representation is described in [23], where it is coupled with a Single-Objective Genetic Algorithm (SOGA) to solve the kinematic design, under a single load condition. This method has been further developed and extended to MOGAs, as described in this article. The same approach can be used for the synthesis of SISO compliant mechanisms [35], and for the design of SIMO compliant structures that aim at shape control. The load path representation method is coupled with a nonlinear analysis solver, and each set of load paths is transformed into a sequence of Finite Volume Beams [36]. Nonlinear analyses are performed to correctly describe the behavior of compliant structures subject to large displacements. The proposed method combines different types of techniques: sizing, geometry, and topology optimization. In the sizing optimization, the cross-sectional areas of each beam of each load path are considered to be design variables; in the geometry optimization, the design variables are the coordinates of the internal points that define the load path sequence and the intersection between the different load paths; in the topology optimization, the connectivity of the load paths is to be determined. In the second level proposed here, all of the three optimization problems are considered simultaneously, and the load path representation is essential to unifying all three optimizations. A solution represented in the GA population is a mixed vector of binary, integer and real numbers within specified ranges. The presence or absence of a load path in the compliant structure is determined by the binary variables that allows us to obtain different topologies. The approach used to solve the MOGA problem applied for our purposes is the so-called Elitist Non-Dominated Sorting Genetic Algorithm (NSGA-II) [24]. In NSGA-II, the offspring population is combined with the parent population, from which it is created, to form a population of size \(2N\), where \(N\) is the initial population size. Instead of finding the non-dominated front only, the two populations are considered together, and a non-dominated sorting is used to rank the entire population. This approach requires more computational effort, but it allows a global non-dominated check among offspring and parent solutions.