In the context of MSC.FlightLoads and Dynamics, Splines provide an interpolation capability that couples the disjoint structural and aerodynamic models in order to enable the static aeroelastic analysis. The aeroelastic splines are used for two distinct purposes: as a force interpolator to compute a structurally equivalent force distribution on the structure given a force distribution on the aerodynamic mesh and as a displacement interpolator to compute a set of aerodynamic displacements given a set of structural displacements. The force interpolation is represented mathematically as:

and the displacement interpolation as:

Where G is the spline matrix, F and U refer to forces and displacements, respectively and the s and a subscripts refer to structure and aerodynamics, respectively.

The two splines given in the above relationships are used when making the force and displacement interpolations. However, virtual work principals can be applied to relate the two splines as being the transform of one another:

That is, the same set of aerodynamic and structural degrees of freedom are coupled for both interpolation functions. While this relationship is valid, this usage assumption is not necessary and can be limiting for static aeroelastic applications where the set of structural DOFs that is appropriate for load application may not be the same set that is appropriate to represent the important deflections for the aeroelastic correction. Therefore, as shown with the Type option of Aero-Structure Coupling (Ch. 5), each spline can be either General (same spline used for Force and Displacement), Displacement or Force.

Splining methods for aeroelastic analyses available in FlightLoads include the Harder‑Desmarais Infinite Plate Spline (SPLINE1 or SPLINE4 with METH=IPS), the Infinite Beam Spline (SPLINE2),the Thin Plate Spline (SPLINE1 or SPLINE4 with METH=TPS) and the Finite Plate Spline (SPLINE1 or SPLINE4 with METH=FPS). A fifth method that employs an MPC-like interpolator (SPLINE3) is available in MSC.Nastran, but is not supported in FlightLoads. The SPLINE3 allows the user to build an interconnection between select aerodynamic DOFs and select structural DOFs and is not discussed further here.

The IPS, FPS and the linear spline assume that the aerodynamic and structural points for a single interpolation matrix lie on or can be projected to the same plane. They relate structural displacements normal to that plane to aerodynamic displacements normal to the plane and to an aerodynamic slope (rotation about a single axis lying in the plane). The TPS is a three dimensional extension of the existing IPS spline. The FPS uses a virtual planar finite element mesh to interpolate between the two meshes.

As stated in (B‑1) and (B‑2), the two basic relationships that must be developed are the displacement transformation and the force transformation. In general, the structural displacements are the usual six global displacement degrees of freedom and the forces are the usual three forces and three moments. The aerodynamic degrees of freedom depend on the aerodynamic method, but must include displacements normal to a local surface and rotations about an axis lying in the osculatory plane since these are the degrees of freedom used in the aerodynamic methods of Panel Aerodynamics (App. A). The corresponding aerodynamic forces are a normal force and a local pitching moment.

Each set of structural points and aerodynamic points may be related via a pair of unique spline transformations of the form of (B‑1) and (B‑2). The total transformation matrices for all the aerodynamic and structural DOFs are then assembled from the individual spline matrices. In FlightLoads, the structural points are taken as the independent degrees of freedom in the spline relationships, so the same structural point may appear in more than one spline relation. However, each aerodynamic point may appear in only one.

The force transformation must be computed such that the resultant structural loads are statically equivalent to the aerodynamic loads:

where is the transformation from global to basic coordinates. For moments, the following condition must be satisfied:

where the are, respectively, the vectors between the (arbitrary) moment center and the structural and aerodynamic mesh points in the basic coordinate system. These two requirements are imposed on the individual spline matrices on a component-by-component basis, thus ensuring that the relationship will hold for the assembled spline transformation.

Each of the spline methods yields a relationship:

where are the weighted coefficients of the interpolant (usually determined by boundary conditions on the function, e.g., equilibrium) and are the coefficient matrix and the applied load respectively. The coefficient matrix is a function only of geometry and the form of the interpolant. The evaluation uses the structural geometry alone in (B‑6) to evaluate the coefficients:

and then uses (B‑6) again, with both geometries to evaluate the displacement function at the aerodynamic points given the solution of (B‑7) (which are loads at the structural grids) for point displacements at the structural grids.

In other words, to create the spline transformation matrix, (B‑6) is evaluated for point loads at the structural points to form basis vectors at the aerodynamic points that are the columns of the displacement transformation of (B‑1).