Deriving correction functions to model the efficiency of noise barriers with complex shapes using boundary element simulations

Abstract

The boundary element method (BEM) is a commonly used method to compute the insertion loss of noise barriers having arbitrary cross-sections. For large scale three-dimensional problems, however, the BEM is not feasible. On the other hand, standardized calculation methods for noise mapping are efficient, but shapes other than the straight barrier cannot be properly calculated. Attempts to merge these two approaches by using BEM to derive correction functions based on geometrical quantities such as source and target angle as well as the path length elongation between source and receiver caused by the barrier were usually focused on a small set of barrier types, dimensions, absorptive configurations, source or receiver positions. The main objective of this study is to investigate which functions based on the most common geometrical parameters are well suited for approximating the efficiency of different types of barriers, dimensions and absorptive configurations. To achieve this, numerous combinations of 7 different barrier types, different heights and widths as well as 3 different absorptive configurations were simulated using the 2D BEM for 8 different source positions. The octave-band-wise efficiency, i.e. the frequency-dependent gain in insertion loss compared to an equally high, fully reflective straight barrier was used as a basis for the correction functions. Linear as well as polynomial models were compared yielding a polynomial of third degree in the source and fourth degree in the target angle as the best model. Effects on the error using uniform sampling in the target angle instead of a uniform receiver grid as a basis for the correction functions are also investigated. Furthermore, wide-band efficiencies based on standardized traffic emission spectra are calculated showing small errors compared to single-band errors, in particular in the high-frequency range. A linear interpolation scheme is suggested to deal with barriers having dimensions not simulated in this work.

Introduction

In the last two to three decades there has been an increasing interest in the calculation of the insertion loss of noise barriers with various cross-sectional designs different from the simple straight barrier using the boundary element method (BEM) [e.g. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]]. The BEM is particularly well suited for this type of problem as only the barrier outline and possibly the traffic lanes or railway tracks need to be discretized. Despite numerous studies there is still a gap between numerical simulations using BEM and standardized algorithms implemented in noise mapping software, which are mostly based on ray-tracing methods. These algorithms are, in general, very fast and efficient, but their use is problematic for complex geometries. The BEM, on the other hand, can be used for complex geometries but the computational demand is high even for relatively simple problems. Thus, large scale calculation as they usually occur in noise mapping problems cannot be efficiently calculated using the BEM.

There has been some effort to derive approximate expressions for the insertion loss of the complex barrier or correction functions for the efficiency of complex structures compared to straight barriers partly based on the BEM calculations [1], [5], [10], [13]. Studies applying BE methods usually attempt to derive correction functions for the efficiency that are based on geometrical quantities such as the angle between the barrier top and the source and/or target position as well as the increased path length between source and target due to the barrier [5], [10]. Usually, however, these studies focus only on a single type of barrier. An overview focusing on the T-profile is provided in Oldham and Egan [10] where a particularly easy scheme to approximate the cap efficiency of T-barriers as a function of the additional path due to a straight barrier of the same height was derived. Another approach is to use some kind of equivalent straight barrier [1], [14] having a different height and position depending on the type and overhang of the complex barrier (in the ASJ RTN-Model 2008, for example, source and target angle are the same for the equivalent barrier [14]). In the same manner certain profiles (e.g. T- or L-profile) can be approximated by a thick barrier [14] combined with the use of Maekawa’s diffraction charts [15] for different diffraction paths depending on the receiver and source position. This approach has further been corroborated using BEM calculations showing the efficiency of T-profile with a T-cap depth of up to one meter to be mainly an effect of the width [13]. Finally, such correction terms for the efficiency can of course also be derived from measurements as shown e.g. in [16] where different modified tops were investigated for different source and target angles.

One important drawback of many of these studies is that they only focus on the area that lies below the height of the barrier, often by choosing receiver points in the symmetry plane. Points lying in the shaded region above the height of the barriers (e.g. upper floors in high-rise buildings) are either neglected [e.g. [1], [5], [10], [13]] or, as in Okubo et al. [16], interpolated between the unshaded region, where the barrier is assumed to have zero effect, and the region below the barrier edge. Another interesting point is that some of these studies compare straight noise barriers with barriers of complex shapes using single wide-band values [1], [10] even though, as also pointed out in Oldham and Egan [10] such values are clearly dependent on the emission spectrum used and thus would need to be calculated for different noise sources separately.

Since no study has been made to derive such band-wise correction functions for a larger set of different geometries including absorption properties of the barriers, the main objective of this study is to investigate which types of functions based on the most common geometrical parameters are well suited for approximating the efficiency of different types of barriers, dimensions and absorptive configurations. In an initial analysis [17] a single-variable linear model using the additional path length on the octave frequency bands was introduced. These first results indicated that there is a need for more elaborate models. Here, this work is extended by investigating the gain in accuracy using multi-variable linear as well as non-linear, i.e. polynomial, regression. Further, the set of source distances originally ranging from 4 to 10 m is increased to cover a range up to 40 m from the noise barrier, a distance that can occur in large multi-lane highways.

To this end, a large number of 2D BEM simulations were performed to calculate the insertion loss for a set of complex structures comprising 4 different basic shapes (T-profile, L-profile, arched top and 45 degree slanted top). The latter three barriers were simulated directed towards and away from the source resulting in 7 types in total. 5 different heights were modeled ranging from 3 to 8 m and different widths up to 4.5 m were used. Three different configurations concerning the absorption material were used in this study, resulting in a total of 393 barriers simulated for 8 different source positions ranging from 4 to 40 m. The insertion losses of a set of straight, fully reflective, barriers of the same heights were calculated to provide a basis for the estimation of the efficiency of the complex structures compared to the straight barriers. A set of different linear and non-linear correction functions was then derived for each complex barrier and frequency band. A further point in the investigation was to find an interpolation scheme for the correction functions to be able to also predict the effect for barriers not included in the simulations described above (same shapes but different heights and/or widths). Model errors for different methods were compared on a global scale but also position-dependent. Furthermore, the effect of unequal sampling of the geometrical quantities was investigated in detail. The interpolation scheme was analyzed for the best overall model. Additionally, the difference between single-band error and the wide-band error using standardized emission spectra was analyzed.