Accuracy and Stability of Finite Element Schemes for the Duct Transmission Problem

Abstract

A one-dimensional numerical model is used to investigate the characteristics of finite element computational schemes for linearized acoustical transmission in ducts with flow. Primitive variables and coupled first-order equations are used. The relative performances of Lagrangian and Hermitian elements with Galerkin and residual least squares formulations are assessed. Results of the numerical study are shown to correlate with the characteristics of analytic solutions for the equivalent regular grid difference equations. Galerkin solutions are shown to introduce spurious nonphysical modes which must be eliminated by careful attention to the local resolution requirements of the finite-element mesh. Residual least squares formulations do not introduce spurious numerical modes but result in significant numerical damping. This is particularly severe if Lagrangian elements are used.

Stability and accuracy of finite element methods for flow acoustics. I: general theory and application to one‐dimensional propagation

“…The sawtooth nature of the pressure field corresponding to one of these spurious modes is given in Figure 8. Similar spurious modes have already been reported for the one-dimensional linearized Euler equations solved with standard finite elements [9]. The spurious modes in the mixed Galbrun scheme are present for all values of the parameters.…”

Frequency-Domain Linearized Euler Model for Turbomachinery Noise Radiation Through Engine Exhaust

“…Astley et al [15] have found that solving the LEE with a standard finite element formulation leads to unstable numerical models with the presence of spurious oscillations in the solutions. A possible stabilization strategy, called the algebraic subgrid-scale stabilized (ASGS) method [16], consists of adding to the left-hand side of Eq.…”

Finite Element-Integral Acoustic Simulation of JT15D Turbofan Engine

“…Obviously, the bandwidth of the solution matrix increases with the Hermitian formulation; however, fewer elements are needed to resolve the pressure field. Also, use of the Hermitian formulation tends to reduce spurious oscillation excited by inaccurate matching at boundaries [12]. Application of the Galerkin finite element method yields the following relationship.…”

“…Along the axis of symmetry, and the acoustically hard walls (centerbody, part of the duct wall and the nacelle exterior wall) the normal gradient of the acoustic velocity potential is </>"=0 (12) The boundary condition at the surface of a locally reacting sound absorbent soft-wall duct can be expressed in terms of specific acoustic impedance. Z. Majjigi [9] has shown that a proper formulation of the requirement of continuity of particle displacement yields…”