A preconditioned Krylov subspace approach to a tightly coupled aeromechanical system

A tightly coupled approach is attempted to compute a modest fluid-structure interaction for high subsonic flow through a converging nozzle with deformable walls. A globally convergent Newton statement and a matrix-free GMRES linear equation solver are used to linearize and solve the coupled system of equations without explicitly forming the left hand side jacobian matrix associated with the Newton method. A variable forcing function term is successfully incorporated into the Newton statement to balance inner (linear) and outer (nonlinear) iterations. The fluid-structure system is solved for comparison purposes using a loosely coupled approach. Residual convergence stagnated in the tightly coupled system approach but converged successfully in the loosely coupled approach using the same coding for domain calculations.

A novel approach using time derivative preconditioning is incorporated to speed convergence of the GMRES linear equation solver. No algebraic preconditioning is used. The fluid flow equations showed significant improvements using the time derivative preconditioning method but the error term generated in the structural equations overwhelmed the physical solution increment.

The Taylor Weak Statement derivation of the finite element form of the fluid flow equations with time derivative preconditioning shows a strong connection to the Streamwise Upwind Petrov Galerkin (SUPG) method. This connection is exploited to develop a theoretical basis for the damping term and the time scale parameter common to the SUPG method.