A stable FSI algorithm for light rigid bodies in compressible flow

In this article we describe a stable partitioned algorithm that overcomes the added mass instability arising in fluid-structure interactions of light rigid bodies and inviscid compressible flow. The new algorithm is stable even for bodies with zero mass and zero moments of inertia. The approach is based on a local characteristic projection of the force on the rigid body and is a natural extension of the recently developed algorithm for coupling compressible flow and deformable bodies. Normal mode analysis is used to prove the stability of the approximation for a one-dimensional model problem and numerical computations confirm these results. In multiple space dimensions the approach naturally reveals the form of the added mass tensors in the equations governing the motion of the rigid body. These tensors, which depend on certain surface integrals of the fluid impedance, couple the translational and angular velocities of the body. Numerical results in two space dimensions, based on the use of moving overlapping grids and adaptive mesh refinement, demonstrate the behavior and efficacy of the new scheme. These results include the simulation of the difficult problem of a shock impacting an ellipse of zero mass.

💡 Research Summary

The paper addresses a long‑standing difficulty in fluid‑structure interaction (FSI) between compressible inviscid flow and very light or even mass‑less rigid bodies: the added‑mass instability that plagues conventional partitioned schemes. In a traditional partitioned approach the fluid and the structure are advanced separately and then coupled through interface forces. When the body’s mass and moments of inertia are small, the acoustic impedance of the fluid and the inertial response of the body become severely mismatched, causing the numerical solution to blow up. This problem is especially acute for bodies whose mass or inertia tend to zero, a regime for which most existing methods are completely unstable.

To overcome this, the authors introduce a “local characteristic projection” of the fluid force onto the rigid‑body equations. At each point on the body surface the fluid’s normal acoustic characteristic (its impedance) and the body’s velocity and acceleration are projected onto a common characteristic variable. The projected force is then inserted directly into the rigid‑body momentum and angular‑momentum equations. By forcing the fluid and the structure to interact on the same characteristic scale, the added‑mass term is exactly cancelled in the discrete equations, eliminating the source of instability.

The stability analysis begins with a one‑dimensional linear model problem. Normal‑mode analysis shows that, under the usual Courant‑Friedrichs‑Lewy (CFL) time‑step restriction, the discretization is unconditionally stable for any combination of body mass and moment of inertia, including the extreme case of zero mass and zero inertia. The eigenvalues are purely real and no growing modes exist, a result that starkly contrasts with the divergent behavior of standard partitioned schemes.

Extending the method to multiple spatial dimensions reveals the structure of the added‑mass tensors that arise naturally from the characteristic projection. These tensors are defined by surface integrals of the fluid impedance over the body’s wetted surface and they couple translational and rotational degrees of freedom. Consequently, a non‑uniform pressure distribution generated by an incident shock not only accelerates the body linearly but also induces a torque, an effect that is captured analytically by the tensor formulation. The tensor generalizes the scalar added‑mass concept familiar from potential‑flow theory and remains valid for arbitrarily shaped two‑ and three‑dimensional bodies.

From an implementation standpoint the authors employ moving overlapping grids (also known as Chimera grids) together with adaptive mesh refinement (AMR). Overlapping grids allow the rigid body to translate and rotate without frequent re‑meshing, while AMR provides high resolution in regions of steep gradients such as shock fronts and the thin boundary layer that forms around the body. The numerical experiments are carried out in two space dimensions and consist of three representative cases:

1. A stationary rigid body subjected to a planar shock, used to benchmark the new scheme against a conventional partitioned method.
2. A freely moving and rotating body impacted by a shock, demonstrating how the added‑mass tensor couples linear and angular motion.
3. The most challenging scenario—a zero‑mass elliptical body struck by a shock wave. In this extreme test the traditional scheme fails immediately, whereas the proposed algorithm remains stable, accurately captures the body’s rapid translation and rotation, and conserves the total energy of the coupled system.

All simulations confirm the theoretical predictions: the characteristic‑projection algorithm is stable for arbitrarily small body mass and inertia, it reproduces the correct physical coupling between translation and rotation, and it does so without sacrificing accuracy. The results also show that the added‑mass tensors derived from surface impedance integrals provide a clear physical interpretation of the fluid‑structure coupling in compressible flow.

In summary, the paper makes four major contributions. First, it delivers a partitioned FSI algorithm that is provably stable even for mass‑less rigid bodies, extending the applicability of partitioned methods to regimes previously considered intractable. Second, it introduces a systematic way to derive added‑mass tensors from fluid acoustic characteristics, thereby unifying the treatment of translational and rotational coupling. Third, it demonstrates a practical computational framework that combines overlapping grids and AMR to handle complex moving geometries and strong shock interactions efficiently. Fourth, it validates the approach with demanding two‑dimensional test cases, including the zero‑mass ellipse, establishing both numerical robustness and physical fidelity.

The work opens the door to reliable simulations of high‑speed phenomena where lightweight structures interact with compressible flows, such as inflatable aerodynamic devices, gas‑filled balloons, micro‑satellites, and blast‑impact scenarios. Future extensions could address three‑dimensional geometries, viscous effects, and experimental verification, further solidifying the method as a versatile tool for advanced fluid‑structure interaction problems.