Steady and eigenvalue solvers

Steady solver
The first step in any perturbative analysis (modal or resolvent analysis) is to determine a steady solution of the unsteady fluidstrucutre equation
where R_{f} denotes the fluid residual vector and R_{s} is the solid residual vector. Steady solutions, denoted , are determined by solving the steady fluidstructure equations
To that aim a classical Newton method is used to find the zero of this nonlinear equations. In this iterative algorithm, the solution is decomposed as
the sum of a (known) guess solution and an incremental (unknown) solution. As long as the guess solution does not satisfy the steady nonlinear equations (the fluid and solid residual do not vanishes), the incremental solution is obtained by solving the following linear equation
where the matrices in the left handside operator are formally defined as
The sum of the guess and incremental solution then define a new guess olsution and the above procedure is repaeted until the guess solution satisfy the steady nonlinear euqations.

Eigenvalue solver