Spatial Coupling

The analyses presented in the paper at hand concentrate on error sources in spa­tial coupling, and so the description of the projection methods shall be afforded a separate section here. To begin with, an overview of projection methods suitable for reduced structural models is given. The existing projection method based on Fi­nite Interpolation Elements (FIE) is explained. Then the newly-implemented Global Spline-Based (GSB) and Moving Least-Squares (MLS) methods are presented in detail.

In order to be valid from the physical point of view, any projection scheme has to be conservative with the following two criteria: First of all, the total force and moment vectors must be preserved during the projection. Secondly, during steady simulations the elastic strain energy of the structure must be identical to the work performed by the aerodynamic loads on the wetted surface, as implied by Eq. (2). During unsteady simulations also the instantaneous power exchanged over the coup­ling surface must be the same on both sides. From the flow solver and the volume mesh deformation code, further numerical requirements arise affecting the projec­tion of deformations from the structure back to the wetted surface: The resulting deformed surface mesh should be contiguous in particular at intersections between the surface meshes of distinct assemblies, for instance between fuselage and wing. The deformed surface mesh should be smooth in order to assure good convergence of the flow solution. One final demand is of a more practical nature: With reduced structural models, any projection scheme has to make some kind of assumption for the transfer of forces and deformations over the gap between wetted surface and structure. This assumption should not be far removed from the load paths actu­ally to be expected, i. e. some measure of locality should be preserved during the projection.

Initially, only one projection algorithm was available inside the ACM: The Fi­nite Interpolation Element (FIE) method [5, 8, 9, 25] is an uncomplicated method that uses the shape functions of the structural model to divide aerodynamic surface loads among the nodes of the closest structural element. During the first phase of MUNA, a number of alternative projection methods was reviewed for inclusion in the ACM. Many published methods are only adequate for configurations where the wetted surface and the surface of the structural model coincide up to the discret­isation error [10, 17, 22, 29]. On account of the requirements set forth for reduced

structural models such methods were excluded. Prospective methods included the Infinite-Plate Spline (IPS) method [14], the Constant-Volume-Tetrahedron (CVT) method [2, 13] and the inverse Boundary Element Method (BEM) [11]. These were extensively compared by Sadeghi et al. [27]. The GSB method [6] and the MLS method [23] constitute further alternatives suitable for reduced structural models.

With the IPS method, only the deflections normal to the wing plan form are in­terpolated from the structure to the wetted surface using splines as interpolation functions. This limits the method to (almost) planar configurations. With the CVT method, tetrahedra are spanned between the points on the wetted surface and the nodes of the closest structural element. Both the natural coordinates of the projec­tion point inside the element and the volume of the tetrahedron are kept constant for all deformation states, defining the projection. As will become evident further down, the CVT method can be regarded as an extension of the FIE method. The BEM method is the projection method which is most firmly footed on physical con­siderations instead of geometrical neighbourhood relations: The gap between wetted surface and structure is modelled as an elastic continuum, and the deformation of the structure is expressed in terms of surface deflections through the BEM. This relation then has to be inverted with the minimisation of the elastic strain energy of the continuum as an additional constraint. Of all methods presented so far this is the most demanding. Furthermore, it requires the connectivity of the wetted surface, which currently is not transmitted from the flow solver to the ACM. The GSB and MLS methods both determine a function approximation to the nodal displacement distribution and evaluate it at the surface points. The two methods differ primarily in their choice of interpolation functions. They do not interfere with the modular structure of the ACM, offer the required generality, are independent of the dimen­sionality of the structural model and involve only a moderate implementation effort. Also, there is a significant implementation overlap between them, for which reason both were selected for inclusion in the ACM.