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 upper 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.

翻译：流固耦合模型用于研究材料在不同雷诺数下与不同流体的相互作用。针对同一模型，不仅考察不同流体作用，还考虑不同固体材料，能进一步优化建筑材料的选择。参数依赖的离散化方法或可满足这一需求。此外，降秩技术能降低计算参数依赖流固耦合离散化近似解所需的复杂度。已有研究将降秩方法应用于参数依赖的线性流固耦合离散化问题，利用算子的线性特性将方程组转化为单一矩阵方程，并通过降秩方法逼近其解。本文提出一种新方法，借助牛顿迭代将这一框架扩展到非线性参数依赖流固耦合问题。我们将参数集划分为互斥子集，在每个子集上计算与上中位数参数对应问题的牛顿近似解，并将其作为整个子集上单步牛顿迭代的初始估计。该牛顿步骤生成一个矩阵方程，其解可通过降秩方法逼近。相较于对独立问题逐一进行牛顿迭代的直接方法，本方法的牛顿迭代步数更少。实验表明，本方法计算降秩近似解的速度可达到直接方法的二十倍。