4. Structural Solver

The Newmark method, [FAR 95], is used to integrate the structure's equation of motion:

(19)

with acceleration, , velocity and and displacement q at time t ^{n +1}, starting from a discrete initial condition, with airloads { F( t)} at time step t ^{n +1}. It can be considered as a generalisation of the constant average acceleration method.

The following approximations are used:

(20)

The parameters ( ? = 1/2, ? = 1/4) give the constant average acceleration method and ( ? = 1/2, ? = 1/6) the linear acceleration method. The limit of unconditional stability of the Newmark method is given by ? ? 1/4( ? + 1/2) ², see Bathe [BAT 82]. Hence, the constant average acceleration method is unconditionally stable. Moreover, this method has no numerical dissipation which is very suitable for flutter calculations.

The complete system [9] now has to be solved combining the individual solvers fluid and structure in a fully coupled way. The goal is to devise techniques that integrate the fluid and the structure at the same time level and hence should preserve energy.

The equation of motion [19] is then solved for the acceleration with the airloads F( t) at t ^{n +1}.

A time integration procedure then uses predicted displacements and corrected aerodynamic steps as discussed in the section 5.3.