[Paper Review] Numerical Analysis of Time-Dependent Galbrun Equation in an Infinite Duct

Kamel Berriri, Anne-Sophie Bonnet-Ben Dhia+1 more|arXiv (Cornell University)|Mar 22, 2006

Advanced Mathematical Modeling in Engineering1 references3 citations

TL;DR

This paper presents a stabilized finite element method for the time-dependent Galbrun equation in a rigid duct with uniform flow, using a regularization technique inspired by Maxwell's equations to enable stable Lagrange finite element approximation. The key contribution is a variational formulation with artificial vorticity regularization and carefully constructed absorbing boundary conditions that prevent numerical instability and allow accurate simulation of acoustic wave propagation in moving fluids.

ABSTRACT

In this paper we are interested in the mathematical and numerical analysis of the time-dependent Galbrun equa- tion in a rigid duct. This equation models the acoustic propagation in presence of flow [1]. We propose a regu- larized variational formulation of the problem, in the sub- sonic case, suitable for an approximation by Lagrange finite elements, and corresponding absorbing boundary conditions.

Motivation & Objective

• Address the numerical instability of standard nodal finite elements in solving the time-dependent Galbrun equation for acoustic propagation in a moving fluid.
• Overcome the lack of a natural functional framework for variational formulation of the Galbrun equation in the transient case.
• Develop a regularized variational formulation suitable for standard Lagrange finite elements in the subsonic, uniform flow regime.
• Construct stable, variational absorbing boundary conditions that minimize spurious reflections in infinite duct simulations.
• Enable accurate numerical simulation of transient acoustic wave propagation with convective effects due to flow.

Proposed method

• Introduce a regularization term $ s\,\text{curl}(\text{curl}\,\boldsymbol{\xi} - \psi) $ to the Galbrun equation, where $ \psi = \text{curl}\,\boldsymbol{\xi} $, to stabilize the variational formulation.
• Decouple the vorticity equation under uniform flow ($ M' = 0 $), allowing $ \psi $ to be treated as a known field.
• Formulate the regularized problem in a Hilbert space framework using $ \mathbf{H}_0(\Omega) \times L^2(\Omega)^2 $, ensuring well-posedness via Hille-Yosida theory.
• Derive a variational formulation involving bilinear forms $ a(\cdot,\cdot) $, $ b(\cdot,\cdot) $, $ c^{\Gamma^{\pm}}(\cdot,\cdot) $, and $ d^{\Gamma^{\pm}}(\cdot,\cdot) $, incorporating material derivative and boundary terms.
• Construct absorbing boundary conditions using $ c^{\Gamma^{\pm}} $ and $ d^{\Gamma^{\pm}} $ terms that are compatible with the variational structure and reduce wave reflections.
• Discretize the system using continuous Lagrange finite elements and a centered second-order finite difference scheme in time, leading to a semi-discrete ODE system.

Experimental results

Research questions

• RQ1 Can a stable and convergent finite element method be developed for the time-dependent Galbrun equation in a duct with uniform flow, despite the lack of a natural variational framework?
• RQ2 How can the numerical instability of standard nodal finite elements be overcome in the Galbrun formulation for transient acoustic problems?
• RQ3 What is the role of regularization in enabling a consistent variational formulation for the Galbrun equation using standard finite elements?
• RQ4 How can absorbing boundary conditions be designed to minimize spurious reflections while preserving the variational structure of the problem?
• RQ5 What is the impact of the regularization parameter $ s $ on the stability and accuracy of the numerical solution?

Key findings

• Without regularization ($ s = 0 $), the standard finite element method leads to numerical instability, as shown by unbounded growth in the wave amplitude in numerical simulations.
• With regularization ($ s = 1 $), the method produces stable and physically consistent results, clearly distinguishing between irrotational (outward-moving) and rotational (stationary-radius) wavefronts.
• The irrotational part of the displacement forms a circular wavefront expanding with time, while the rotational part remains localized and convected by the flow, confirming the expected physical behavior.
• The absorbing boundary conditions significantly reduce spurious reflections, as evidenced by the partial suppression of wave reflections in the second numerical experiment with a Gaussian time signal.
• The variational formulation with the regularized term ensures that the energy estimate holds, with the energy functional satisfying $ \frac{d}{dt} \left( \|\boldsymbol{\xi}_t\|^2 + \|\boldsymbol{\nabla}\boldsymbol{\xi}\|^2 - M^2 \|\partial_x \boldsymbol{\xi}\|^2 \right) \leq 0 $, indicating energy decay and stability.
• The method successfully handles the material derivative $ \frac{D}{Dt} = \partial_t + M(y)\partial_x $ in a weak form, enabling accurate simulation of convective wave propagation in a moving medium.