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 fluid-strucutre 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 fluid-structure equations

To that aim a classical Newton method is used to find the zero of this non-linear 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 non-linear 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 hand-side 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 non-linear euqations.

• Eigenvalue solver