A low‐rank method for parameter‐dependent fluid‐structure interaction discretizations with hyperelasticity

Fluid‐structure interaction models are used to study how a material interacts with different fluids at different Reynolds numbers. Examining the same model not only for different fluids but also for different solids allows to optimize the choice of materials for construction even better. A possible answer to this demand is parameter‐dependent discretization. Furthermore, low‐rank techniques can reduce the complexity needed to compute approximations to parameter‐dependent fluid‐structure interaction discretizations. Low‐rank methods have been applied to parameter‐dependent linear fluid‐structure interaction discretizations. The linearity of the operators involved allows to translate the resulting equations to a single matrix equation. The solution is approximated by a low‐rank method. In this paper, we propose a new method that extends this framework to nonlinear parameter‐dependent fluid‐structure interaction problems by means of the Newton iteration. The parameter set is split into disjoint subsets. On each subset, the Newton approximation of the problem related to the median parameter is computed and serves as initial guess for one Newton step on the whole subset. This Newton step yields a matrix equation whose solution can be approximated by a low‐rank method. The resulting method requires a smaller number of Newton steps if compared with a direct approach that applies the Newton iteration to the separate problems consecutively. In the experiments considered, the proposed method allows to compute a low‐rank approximation up to twenty times faster than by the direct approach.