# A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems

## Summary (1 min read)

### Schematic of the control points and resulting free-form parametric deformation

- This allows the user to keep the number of FFD parameters to a desired low level (in their case roughly 5-10 parameters).
- Parametric coupling of fluid and structure.
- 1. Formulation of the coupled problem in the parameter space.
- This requires showing that the nearest point projection is continuous in the strong H 2 -norm topology.

### 5.2. Empirical interpolation method for nonaffine problems.

- Terms, and similarly for the other forms.
- In practice the EIM has been quite useful for solving nonaffinely parametrized PDEs with the reduced basis method [20, 36, 47] .
- For the free-form deformation detailed in Sect. 3.2 in fact the forms B and F are affine due to the fact that the map T FFD is polynomial.
- For generic nonpolynomial shape parametrizations the situation remains more challenging.

### 6.3. Convergence and accuracy of the coupling algorithm.

- By introducing a parametric free-form deformation of the flow geometry the fluid equations can be written as parametric partial differential equations on a fixed domain.
- The authors then applied the reduced basis method to the fluid equations to obtain an efficient reduced model with certified error bounds.
- The geometric deformation parameters were also used to couple the fluid domain to a 1-d wall equation, where the parameters acted as the coupling variables.
- The authors demonstrated that for a modest number of free-form deformation parameters an approximate coupling between fluid and structure can be achieved.
- Numerical simulations were based on the rbMIT toolkit [23] developed by the group of Anthony Patera as well as the MLife fluid mechanics solvers originally authored by Fausto Saleri.

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