# A Novel Optimization Framework to Replicate the Vibro-Acoustics Response of an Aircraft Fuselage
## 3 CAD/FEM Modeling  
### FEM Mesh  
The FEM model was built up by the commercial code Siemens NX Nastran 12.0 [9]. Starting from a CAD model of a fuselage barrel (Figure 1), an FEM model was generated comprising 2D/1D structural parts and 3D internal and external fluid volumes. The 2D structural part represented the key contributor to the vibration response of the model, and comprised four main surfaces made out of different materials (Figure 1(a)):  
* the external surface on which the external acoustic pressure directly impacts;
* the lining panel, realized by a sandwich suitable to reduce noise and vibration, to improve the comfort inside the cabin;
* the floor on which seats are located (seats were not modeled in this work);
* the window glasses. Figure 1: CAD model of the aircraft fuselage. The 1D structural elements (Figure 2b) were divided in: longerons and circumferential beams to stiffen the structure, transversal and axial floor support, formed by beams with different sections, windows frameworks and doors. Furthermore, the sandwich was linked to axial and circumferential beams by elastic and rigid constraints. The internal fluid cavities (Figure 2c) were: the cabin occupied by passengers, the stowage under the floor, the volume displaced between the external surface and the internal lining panels. The external air was modeled to let the sound waves propagate from the monopole sources, representative of the MDO inputs, to the structure. The external fluid modeling was also essential to measure the Acoustic Intensities (AIs), representative of the MDO outputs, in the 30 microphone locations on the external fuselage surface. Such external fluid was modeled as a cube of solid elements with a cavity in the center to accommodate the fuselage barrel. Such cube was modeled with a size large enough to comprise at least 2 wavelengths at the frequency of interest. This was required in order to prevent boundary effects since the condition of non-reflecting boundary condition was applied at the boundaries to simulate the infinite radiation [9]. The mesh comprised tetrahedral solid linear elements with variable size having at least 6 nodes per wavelength, in order to have an acceptable accuracy for fluid field calculation in the fluid-structure interface zone (close to the structure); larger elements were adopted far from the structure in order to reduce the computational burden (Figure 3b). Even if the accuracy could turn out to be not so high with 6 nodes per wavelength, it is not a problem for this work where the aim is to show that the proposed procedure is a viable approach for reducing the experimental effort, based on an acceptable agreement between the reproduced and the reference sound pressure distribution on the fuselage. More info about the level of approximation inherent the mesh density adopted could be retrieved in [10,11]. Figure 3: (**a**) External fluid mesh, (**b**) with close up to the mesh grading at the fluid-structure interface surfaces. Figure 2: Finite Element Method (FEM) model subdivision: 2D (**a**) and 1D (**b**) structural elements; (**c**) internal fluid cavities. In [10], two numerical examples, a long duct and a sedan cabin compartment have been analyzed. In the first example, the analytical solution of the corresponding one-dimensional boundary value problem has been used for comparison. Numerical investigations limited to the frequency range of \(0<\text{kl}<80\) \"Pi essentially confirmed the common rules of modeling that six constant or bilinear elements per wavelength are sufficient to obtain a solution of about 10 to 15% error. The common rule of six linear elements per wavelength, provide an error inferior than 10% in the Euclidean norm and inferior than 20% in the maximum norm. The second example confirmed most of these conclusions. In [11], [PERSON] investigated discretization requirements for sound radiation at complicated structures.