A Low-rank Method for Parameter-dependent Fluid-structure Interaction Discretizations With Hyperelasticity

In aerospace engineering and boat building, fluid-structure interaction models are considered to investigate prototypes before they are physically assembled. How a material interacts with different fluids at different Reynold numbers has to be studied before it is passed over to the manufacturing process. In addition, 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 on 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 allowed to compute a low-rank approximation within a twentieth of the time used by the direct approach.Comment: 20 pages, 6 figure

Low-rank Linear Fluid-structure Interaction Discretizations

Fluid-structure interaction models involve parameters that describe the solid and the fluid behavior. In simulations, there often is a need to vary these parameters to examine the behavior of a fluid-structure interaction model for different solids and different fluids. For instance, a shipping company wants to know how the material, a ship's hull is made of, interacts with fluids at different Reynolds and Strouhal numbers before the building process takes place. Also, the behavior of such models for solids with different properties is considered before the prototype phase. A parameter-dependent linear fluid-structure interaction discretization provides approximations for a bundle of different parameters at one step. Such a discretization with respect to different material parameters leads to a big block-diagonal system matrix that is equivalent to a matrix equation as discussed in [KressnerTobler 2011]. The unknown is then a matrix which can be approximated using a low-rank approach that represents the iterate by a tensor. This paper discusses a low-rank GMRES variant and a truncated variant of the Chebyshev iteration. Bounds for the error resulting from the truncation operations are derived. Numerical experiments show that such truncated methods applied to parameter-dependent discretizations provide approximations with relative residual norms smaller than $10^{-8}$ within a twentieth of the time used by individual standard approaches.Comment: 30 pages, 7 figure

Nonlinear dynamics and pattern formation in turbulent wake transition

Results are reported on direct numerical simulations of transition from two-dimensional to three-dimensional states due to secondary instability in the wake of a circular cylinder. These calculations quantify the nonlinear response of the system to three-dimensional perturbations near threshold for the two separate linear instabilities of the wake: mode A and mode B. The objectives are to classify the nonlinear form of the bifurcation to mode A and mode B and to identify the conditions under which the wake evolves to periodic, quasi-periodic, or chaotic states with respect to changes in spanwise dimension and Reynolds number. The onset of mode A is shown to occur through a subcritical bifurcation that causes a reduction in the primary oscillation frequency of the wake at saturation. In contrast, the onset of mode B occurs through a supercritical bifurcation with no frequency shift near threshold. Simulations of the three-dimensional wake for fixed Reynolds number and increasing spanwise dimension show that large systems evolve to a state of spatiotemporal chaos, and suggest that three-dimensionality in the wake leads to irregular states and fast transition to turbulence at Reynolds numbers just beyond the onset of the secondary instability. A key feature of these ‘turbulent’ states is the competition between self-excited, three-dimensional instability modes (global modes) in the mode A wavenumber band. These instability modes produce irregular spatiotemporal patterns and large-scale ‘spot-like’ disturbances in the wake during the breakdown of the regular mode A pattern. Simulations at higher Reynolds number show that long-wavelength interactions modulate fluctuating forces and cause variations in phase along the span of the cylinder that reduce the fluctuating amplitude of lift and drag. Results of both two-dimensional and three-dimensional simulations are presented for a range of Reynolds number from about 10 up to 1000