# An analytically solvable benchmark problem for fluid-structure interaction with uncertain parameters

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## Abstract

In simulating fluid-structure interaction, e.g. for the biomechanical dynamics of aortic blood flow, a profound benchmarking of the numerical solver is a basic prerequisite. We consider a test scenario for a fluid-structure interaction solver including uncertain model parameters that has an analytical solution.

The solver is developed to simulate the flow and movement of the human aorta. In simulating the biomechanical dynamics of aortic blood flow, there are usually high uncertainties with respect to the model itself and with respect to the input parameters such as the stiffness of the vessel wall. If numerical simulation is considered to provide assistance in a clinical context, these uncertainties also have to be reflected in the simulation results for reliability.

To verify a solver for fluid-structure interaction with uncertain parameters, we introduce a benchmark problem for which we derive an analytical solution. The benchmark is based on the Taylor-Couette flow system with an additional solid domain surrounding the fluid domain. Whereas the analytical solution is stated in polar coordinates, the problem is non-trivial for solvers based on the Cartesian coordinate system. The stochastic space is discretised by means of a generalised Polynomial Chaos expansion. By Galerkin projection on the stochastic basis, we obtain an intrusive Uncertainty Quantification method. The benchmarking results for the implemented solver are in well accordance with the theoretically expected convergence properties.