THE APPLICATION OF HIGHER-ORDER ADAPTIVE FINITE ELEMENTS IN VlBRO-ACOUSTIC ANALYSES

THE APPLICATION OF HIGHER-ORDER ADAPTIVE FINITE
ELEMENTS IN VlBRO-ACOUSTIC ANALYSES
Gaurav Kumar Siemens Industry Software Limited, UK
Koen Vansant Siemens Industry Software Limited. Belgium
1 INTRODUCTION
Vibro-acoustic analysis has become a mainstream engineering tool in the quest for quieter cars,
railway vehicles. and aircrafts. Moreover, for acoustics applications, the engineers are often
interested in modelling the sound eld over the entire audible frequency range (20 to 20,000 Hz),
which means solving the Helmholtz equation over a large frequency range. Following the nature of
the solution, the requirements in terms of mesh resolution vary drastically over this frequency range,
In conventional nite element (FE) formulations, smaller wavelengths translate into smaller element
sizes, therefore, leading to extremely large system of matrices and lengthy computations.
Furthermore. in conventional FE formulations♥with 1☁L or 2"☁-degree polynomial shape functions♥
typically 6 to 8 elements per wavelength are required for the accurate representation of the sound
wave. Therefore. ideally, a separate mesh should be generated for each frequency of interest: i.e.
an optimum mesh (h renement in hp-FEM) for each frequency step. On the contrary, it is common
practice to use a single FE mesh, with element density suited to represent the sound waves at the
highest frequency of interest. Such practical considerations are compounded by more fundamental
limitations of the conventional nite element methods at mid to high frequencies. In particular, the
pollution effect (i.e. the cumulative build-up of dispersion error over the computational domain)
leads to a rapid increase in numerical error at high frequencies [1]. The dispersion error is the
difference between the theoretical wavenumber k and the wavenumber k actually observed in the
numerical model. The presence of this pollution error enhances even further the differences in mesh
resolutions required for low and high frequencies.
This paper presents an efcient implementation of the higher-order nite elements, which keeps
nite element mesh density same throughout the frequency range of interest, and yet maintains the
accuracy of the solution, by increasing the polynomial order (p renement in hp-FEM) of the nite
elements A key feature of the proposed method is the ability to select automatically the order of
interpolation in each element so as to obtain a target accuracy for each individual frequency. while
minimizing the cost. This is achieved using a simple local a priori error indicator. For simulations
involving several frequencies, the. use of hierarchical shape functions leads to an efficient strategy
to accelerate the assembly of the nite element model. ☂
This p-adaptive higher-order nite element method has been implemented in the FEM A0 (Adaptive
Order) solver of the LMS VirtualLab'M. The performance of the FEM A0, compared against
conventional FEM, is illustrated using three different examples: a. prediction of acoustic transfer
function for pass-by-noise simulation; b. prediction of fan noise for an aero♥engine; c. prediction of
transmission loss of an industrial mufer. The examples show that FEM AO delivers equally
accurate results 2 to 20 times faster compared to a conventional FEM and with a more efcient use
of in-core memory.