The Perfectly Matched Layer absorbing boundary for fluid-structure interactions using the Immersed Finite Element Method

Abstract

In this work, a non-reflective boundary condition, the Perfectly Matched Layer (PML) technique, is adapted and implemented in a fluid-structure interaction numerical framework to demonstrate that proper boundary conditions are not only necessary to capture correct wave propagations in a flow field, but also its interacted solid behavior and responses. While most research on the topics of the non-reflective boundary conditions are focused on fluids, little effort has been done in a fluid-structure interaction setting. In this study, the effectiveness of the PML is closely examined in both pure fluid and fluid-structure interaction settings upon incorporating the PML algorithm in a fully-coupled fluid-structure interaction framework, the Immersed Finite Element Method. The performance of the PML boundary condition is evaluated and compared to reference solutions with a variety of benchmark test cases including known and expected solutions of aeroacoustic wave propagation as well as vortex shedding and advection. The application of the PML in numerical simulations of fluid-structure interaction is then investigated to demonstrate the efficacy and necessity of such boundary treatment in order to capture the correct solid deformation and flow field without the requirement of a significantly large computational domain.

Figure 1

Illustration of wave attenuation on a complex domain.

Figure 2

Schematics of physical and PML domains.

Figure 3

Plane wave propagating along the length of the channel. (blue solid line): numerical solution; (red dash line): theoretical solution in absence of PML; vertical (magenta dash dot line): interface between the physical domain and PML.

Figure 4

Setup of aeroacoustic wave radiation from a duct to an open field.

Figure 5

Comparison of pressure time histories at two different locations. (cyan solid line): with 0.5 cm-thick PML; (red dash line): with 0.25 cm-thick PML; (green dash dot line): with no PML; (blue circle): with semi-circular farfield region.

Figure 6

Pressure contours of wave radiation.

Figure 7

Setup of flow past a cylinder with vortex shedding.

Figure 8

Comparison of time histories of y-velocity and vorticity history at 2.5 cm downstream of the cylinder along the centerline. (blue solid line): with PML; (green dash dot line): with large farfield region; (red dash line): with no appropriate treatment of outflow boundary.

Figure 9

Comparison of vorticity contours in cases with application of PML, shown on the same focused area near the cylinder. (a) and (b): with PML; (c) and (d): with large downstream farfield region; (e) and (f) without proper boundary treatment.

Figure 10

Setup of the fluid-structure interaction test cases.

Figure 11

Comparison of x-displacement of the upper right corner of the solid block in channel with acoustic pulses applied on the inflow boundary. (blue solid line): fluid domain with PML; (green dash dot line): 8 cm × 1 cm fluid domain; (red dash line): 2 cm × 1 cm fluid domain.

Figure 12

Comparison of x-displacement of the upper right corner of the solid block in channel with constant pressure applied on the inflow boundary. (blue solid line): fluid domain with PML; (green dash dot line): 8 cm×1 cm fluid domain; (red dash line): 2 cm × 1 cm fluid domain.

Figure 13

Vorticity contour in the case with application of PML and outflow boundary close to the block.

Figure 14

Setup of the flow past deformable leaflets with the height of 0.4 cm, 0.6 cm and 0.8 cm.

Figure 15

Comparisons of vorticity contours in the three different heights of the leaflet.

Figure 16

Traveling of vortices.

Figure 17

Comparisons for the displacement of the top right node on the leaflet. (red dashed line): Cases without PML; (blue solid line): Cases with PML.