MSC.FlightLoads and Dynamics supports three aerodynamic theories:

Each of these methods is described in this section, but they all share a common matrix structure.

Three matrix equations summarize the relationships required to define a set of aerodynamic influence coefficients [see Rodden and Revell (1962)]. These are the basic relationships between the lifting pressure and the dimensionless vertical or normal velocity induced by the inclination of the surface to the airstream; i.e., the downwash (or normalwash),

the substantial differentiation matrix of the deflections to obtain downwash,

and the integration of the pressure to obtain forces and moments,

where:

The three matrices of (A‑1), (A‑2), and (A‑3) can be combined to give an aerodynamic influence coefficient matrix:

which relates the force at an aerodynamic grid point to the deflection at that grid point and a rigid load matrix:

which provides the force at an aerodynamic grid point due to the motion of an aerodynamic extra point.

All methods compute the , , and matrices as a function of Mach number. The matrix is only a function of the model geometry and is therefore calculated only once per configuration.

The Doublet‑Lattice method (DLM) can be used for interfering lifting surfaces in subsonic flow. The theory is presented by Albano and Rodden (1969), Giesing, Kalman, and Rodden (1971), and Rodden, Giesing, and Kalman (1972) and is not reproduced here. The following general remarks summarize the essential features of the method.

The theoretical basis of the DLM is linearized aerodynamic potential theory. The undisturbed flow is uniform and is either steady or varying (gusting) harmonically. All lifting surfaces are assumed to lie nearly parallel to the flow. The DLM is an extension of the steady Vortex‑Lattice method to unsteady flow.

Each of the interfering surfaces (or panels) is divided into small trapezoidal lifting elements (“boxes”) such that the boxes are arranged in strips parallel to the free stream with surface edges, fold lines, and hinge lines lying on box boundaries. The unknown lifting pressures are assumed to be concentrated uniformly across the one-quarter chord line of each box. There is one control point per box, centered spanwise on the three-quarter chord line of the box, and the surface normalwash boundary condition is satisfied at each of these points.

The code for computing the aerodynamic influence coefficients was taken from Giesing, Kalman, and Rodden (1972b). Any number of arbitrarily shaped interfering surfaces can be analyzed, provided that each is idealized as one or more trapezoidal planes. Aerodynamic symmetry options are available for motions which are symmetric or antisymmetric with respect to one or two orthogonal planes. The user may supply one-half (or one-fourth) of the model and impose the appropriate structural boundary conditions. The full aircraft can also be modeled when the aircraft or its prescribed maneuvers lack symmetry.

ZONA51 is a supersonic lifting surface theory that accounts for the interference among multiple lifting surfaces. It is an optional feature in MSC.Nastran (available as the Aero II option). It is similar to the Doublet‑Lattice method (DLM) in that both are acceleration potential methods that need not account for flow characteristics in any wake. An outline of the development of the acceleration-potential approach for ZONA51 is presented by Liu, James, Chen, and Pototsky (1991), and its outgrowth from the harmonic gradient method (HGM) of Chen and Liu (1985) is described. ZONA51 is a linearized aerodynamic small disturbance theory that assumes all interfering lifting surfaces lie nearly parallel to the flow, which is uniform and either steady or gusting harmonically. As in the DLM, the linearized supersonic theory does not account for any thickness effects of the lifting surfaces.

Also, as in the DLM, each of the interfering surfaces (or panels) is divided into small trapezoidal lifting elements (“boxes”) such that the boxes are arranged in strips parallel to the free stream with surface edges, fold lines, and hinge lines lying on box boundaries. The unknown lifting pressures are assumed to be uniform on each box. There is one control point per box, centered spanwise on the 95 percent chord line of the box, and the surface normalwash boundary condition is satisfied at each of these points.

The code for computing the aerodynamic influence coefficients, , was integrated into MSC.Nastran by Zona Technology, Inc., taking full advantage of the extensive similarities with the DLM. Any number of arbitrarily shaped interfering surfaces can be analyzed, provided that each is idealized as one or more trapezoidal planes. Aerodynamic symmetry options are available for motions that are symmetric or antisymmetric with respect to the vehicle centerline. Unlike the DLM, symmetry about the XY-plane is not supported. The user may supply one half of the vehicle model and impose the appropriate structural boundary conditions.

The method of images, along with Slender Body Theory, has been added to the Doublet‑Lattice method (DLM) in Giesing, Kalman, and Rodden (1972a, 1972b, and 1972c). The DLM is used to represent the configuration of interfering lifting surfaces, while Slender Body Theory is used to represent the lifting characteristics of each body (i.e., fuselage, nacelle, or external store). The primary wing‑body interference is approximated by a system of images of the DLM trailing vortices and doublets within a cylindrical interference body that circumscribes each slender body. The secondary wing‑body interference that results from the DLM bound vortices and doublets is accounted for by a line of doublets located on the longitudinal axis of each slender body. The boundary conditions of no flow through the lifting surfaces or through the body (on the average about the periphery) lead to the equations for the lifting pressures on the surfaces and for the longitudinal (and/or lateral) loading on the bodies in terms of the normalwashes on the wing‑body combination.

The code for computing the aerodynamic matrices was adapted for MSC.Nastran from Giesing, Kalman, and Rodden (1972b). The adaptation required a matrix formulation of all of the body interference and body loading calculations. These equations are written using the symbols adopted for MSC.Nastran and showing the equivalences to names used in the documentation of Giesing, Kalman, and Rodden (1972b).

The program of Giesing, Kalman, and Rodden (1972b) finds the forces on the lifting boxes and bodies of an idealized airplane in terms of the motions of these elements. The lifting surfaces are divided into boxes. The bodies are divided into elements. There are two types of body elements: slender elements, which are used to simulate a body’s own motion, and interference elements, which are used to simulate the interaction with other bodies and boxes. The body elements may have Z (vertical), Y (lateral), or both (ZY) degrees of freedom.

The basic method is the superposition of singularities and their images. There are two basic singularity types: “forces” and modified acceleration potential “doublets.” Each “force” singularity is equivalent to a line of doublets in the wake. As discussed, the wing boxes use the “force” type of singularity concentrated along the box quarter chord. The interference elements use the “doublet” type of singularity. The slender body elements use both types.

An extensive set of matrix equations dealing with slender body theory as adapted to the Doublet Lattice Method are described in the MSC.Nastran Aeroelastic Guide, and are not reproduced here.