Optimization of thin noise barrier designs using Evolutionary Algorithms and a Dual BEM Formulation

Optimization of thin noise barrier designs using Evolutionary Algorithms and a Dual BEM Formulation∗ R. Toledo, J. J. Azn´arez, O. Maeso, D. Greiner Instituto Universitario de Sistemas Inteligentes y Aplicaciones Num´ericas en Ingenier´ıa (SIANI) Universidad de Las Palmas de Gran Canaria Edificio Central del Parque Cient´ıfico y Tecnol´ogico Campus Universitario de Tafira, 35017, Las Palmas de Gran Canaria, Spain {rtoledo, jjaznarez, omaeso, dgreiner}@siani.es, web: http://www.siani.es April 2014 Abstract This work aims at assessing the acoustic efficiency of different thin noise barrier models. These designs frequently feature complex profiles and their implementation in shape optimization processes may not always be easy in terms of determining their topological feasibility. A methodology to conduct both overall shape and top edge optimisations of thin cross section acoustic barriers by idealizing them as profiles with null boundary thickness is proposed. This procedure is based on the maximization of the insertion loss of candidate profiles proposed by an evolutionary algorithm. The special nature of these sort of barriers makes necessary the implementation of a complementary formulation to the classical Boundary Element Method (BEM). Numerical simulations of the barriers performance are conducted by use of a 2D Dual BEM code in eight different barrier configurations (covering overall shaped and top edge configurations; spline curved and polynomial shaped based designs; rigid and noise absorbing boundaries materials). Results obtained show the usefulness of representing complex thin barrier configurations as null boundary thickness-like models. Keywords: Thin noise barriers, Shape optimisation, Genetic Algorithms, Dual Boundary Element Formulation 1 Introduction The inclusion of sound barriers for abating the negative effects of road traffic noise near residential areas is a broadly used strategy. Considerable research work and studies focused on sound diffraction around barriers have been carried out in the past two decades, specifically in the prediction of the performance and the development of more efficient designs. Among all of the different numerical methods available concerning the issue, the Boundary Element Method (BEM hereinafter) is one of ∗This is the pre-peer reviewed version of the following article: Toledo R, Aznarez JJ, Maeso O, Greiner D. Optimization of thin noise barrier designs using evolutionary algorithms and a Dual BEM formulation. J Sound Vib 2015;334:219-38, which has been published in final form at http://dx.doi.org/10.1016/j.jsv.2014.08.032 1 the broadly used. Remarkable work using the singular BEM for assessing the acoustic efficiency of sound noise barriers have been carried out to date. In 1980, Seznec [1] implements this methodology to assess the diffracted sound field behind a barrier. Hothersall et al. [2, 3] make use of this technique to study the performance of a vertical screen and compare it with different types of barriers with diffusive elements on their top. Watts and Morgan [4] predict the acoustic behaviour of a sound-interferencetype device added on the top of an existing straight barrier, yielding a significant improvement in the screening performance. Crombie et al. [5] study the performance of multiple-edge barriers, concluding that the addition of side-panels leads to a significant increase in acoustic efficiency over a simple vertical screen. Monazzam and Lam [6] carry out a comparison study between noise barriers with quadratic residue diffuser (QRD) tops and different top-edge profiles barriers, for both rigid and with absorptive coverage. In the same line, Ishizuka and Fujiwara [7] conclude that providing the top of noise barriers with soft edges significantly improves their efficiency. Configuration modifications provide only a slight improvement, though. Okubo and Fujiwara [8] assess the acoustic efficiency of the so called waterwheel cylinder installed on the top-edge of noise barriers to produce an approximate soft surface (a surface with a null sound pressure level), concluding that these designs are strongly frequency dependent. Jean et al. [9] study the influence of both source and ground type in the assessment of the efficiency of a straight, a T-shaped and a cylindrical top barrier. To supplement this compendium, other notable work for assessing the acoustic efficiency of noise barriers conducted by Maeso and Azn´arez can be consulted in [10]. From a broader point of view, to the authors’ knowledge, there are some noteworthy work involving the coupled use of BEM in outdoor acoustics in the literature. In this line, Tadeu et al. [11] propose a coupled BEM-TBEM formulation to model the propagation of sound in the presence of very thin elements. de Lacerda et al. [12] propose a 2D Dual BEM formulation for the treatment of non-thickness configurations and applied it to the assessment of a vertical and a T-shaped noise barrier modelled as thin bodies over an absorbing ground. Chen et al. [13] make use of a Dual BEM formulation to suppress the fictitious frequencies that arise when handling with non-thin elements. In particular to the concerning issue here presented, the combined used of BEM and evolutionary algorithms has been used for shape design optimization in outdoor acoustics problems. Duhamel [14] starts off with a rectangular volumetric structure built of equally-sized bricks to lead to the final optimised shapes with non-inner holes and fillings. Baulac et al. [15] assess the performance of Tshaped barriers with different series of wells covered with a reactive surface on the top. Greiner et al. [16, 17] conduct the study of a singleand a multi-objective design optimization of a Y-shaped noise barrier; the consideration of uncertainties in the optimum design have also been handled in [18]. Grubeˇsa et al. [19] carry out a 3D optimization of both acoustic performance and economical feasibility of a noise barrier built from different modules with varying cross-sections. A more recent research, also covers the inclusion of an innovization procedure for multiobjective noise barrier optimum design in Deb et al. [20]. In this line, a procedure for the shape design optimisation of noise barriers by coupling BEM with an evolutionary algorithm is conducted in this paper. Two-dimensional sound propagation problems concerning an infinite, coherent mono-frequency source of sound, placed parallel to an infinite noise barrier that stands on a flat plane (ground) of uniform admittance are studied. The sound propagation analysis is performed in the frequency domain. Expression of the fitness function to be maximized throughout the shape optimisation process is written in terms of this response. 2 The principal novelty of this work lies in the fact that the proposed Dual BEM formulation is applied in the study of noise barriers featured with very thin boundaries, idealized as null boundary thickness-like models. This simplification of reality greatly facilitates the geometric definition of barrier profiles, having no major influence on the acoustic performance [12]. The special nature of these type of barriers makes every node of the discretization hold both the pressure and the flux value at each side of it, i.e., 2nunknowns per nnodes. The inclusion of an additional BEM formulation (hyper-singular) combined with the classical one (singular) provides a compatible system of equations that allows the problem to be solved. The coupling of an evolutionary algorithm with the Dual BEM code allows to obtain interesting acoustic solutions avoiding the complexity associated with the geometric generation of volumetric structures. To the authors’ knowledge, the procedure described in this paper is the first joint implementation of evolutionary algorithms and a Dual BEM formulation concerning this issue. Fig. 1 shows the usefulness of representing complex volumetric structures as null boundary thickness-like models. The procedure for the geometric definition of the studied noise barriers, the fundamental aspects of the Dual BEM formulation implemented and the main features of the evolutionary algorithm software used are thoroughly and clearly explained in the next sections. As application, eight different barrier configurations of practical interest in the topic here presented, proposing more efficient designs in each case, are assessed. These profiles cover a wide range of designs, from complex straight boundary configurations to curve-shaped profiles. The paper is structured as follows: in section 2, the modelling and discretization by implementation of a Dual BEM formulation is explained. Section 3 deals with the noise barrier design problem definition and section 4 relates the shape optimisation formulation. Section 5 follows with the application of the proposed methodology to the assessment of the acoustic efficiency of different barrier designs. Finally, section 6 shows results and discussion, and section 7 covers the conclusions of the paper. 2 Modelling and discretization by implementing a Dual BEM formulation The next lines are focused on the description of the implemented Dual BEM formulation to make possible the numerical treatment of thin noise barriers idealized as null boundary thickness profiles (see Fig. 1). The special nature of these type of barriers makes necessary the addition of a complementary formulation (hyper-singular) that coupled with the conventional BEM formulation yields a compatible system of equations. 2.1 Singular BEM formulation The integral equation for the iboundary point to be solved by the singular BEM formulation can be expressed as follows: cipi+− ZΓb p∂p∗ ∂nj dΓ = p∗ 0+ZΓb ∂p ∂nj p∗dΓ (1) This integral equality just involves the boundary of the barrier under investigation. The − Rsymbol represents the integral along the boundary to be understood in the Cauchy principal value sense, once 3 Figure 1: (a) Generic thin barrier modelled as a volumetric structure. (b) Idealization of the former barrier as a null boundary thickness profile. the singularity around the collocation point ihas been extracted (ci). In (1), pis the acoustic pressure field over the barrier surface and p∗is the half-space fundamental solution (the acoustic pressure field when the source is placed at the collocation point iover a plane with admittance βg(ground admittance)) and ciis the free term. As a general rule: ci=θ/2π, where θrepresents the inner angle to the boundary measured in radians. It is easily shown that ci= 0.5 for smooth boundaries. The expressions of the fundamental solution and its derivative for a perfectly reflecting ground (βg= 0) for bi-dimensional, harmonious problems are: p∗(k, r) = 1 2π[K0(ikr) + K0(ikr)] ∂p∗ ∂nj =−ik 2πK1(ikr)∂r ∂nj +K1(ikr)∂r ∂nj(2) being ithe imaginary unit, kthe wave number, and r,rthe distances to the observation point from the collocation point and its symmetric point with respect to the ground plane, respectively. K0and K1are the Bessel modified functions of order 0 and 1, respectively. The application of (1) on each inode of the boundary discretization leads to the following system of equations: (Cs+H)·P=G·Q+P∗ 0(3) where Csis a diagonal matrix whose entries involve the free term values at the nodes of the discretization, P,Qare the pressure and the flux (the derivative of the pressure with respect to the normal at 4 each boundary node) vectors, P∗ 0vector stores the values of the fundamental solution concerning the external noise source, and H,Gare matrices whose entries are associated with the integration cores of the singular BEM formulation involving just the variables of the problem along the barrier boundary: hij k=ZΓj ∂p∗ ∂nj φkdΓj;gij k=ZΓj p∗φkdΓj(4) with ibeing the collocation point, jthe observation point and φkthe shape functions with quadratic approximation of the local variable ξalong the element under integration. 2.2 Hyper-singular BEM formulation The integral equation for the iboundary point to be solved by the hyper-singular BEM formulation can be written as follows: ci∂pi ∂ni+= ZΓb p∂2p∗ ∂ni∂nj dΓb=− ZΓb ∂p∗ ∂ni ∂p ∂nj dΓb+∂p∗ 0 ∂ni (5) where the = Rand − Rsymbols represent the integral along the boundary to be understood in the Hadamard finite part integral and in the Cauchy principal value sense, respectively. As the H¨older condition must be satisfied at the collocation point i, hyper-singular formulation of the method demands the source placement to be inside the element (non-nodal collocation). Therefore, ci= 0.5 in (5) in all situations. Expression (6) shows the values of the derivatives of the fundamental solution implied in the hyper-singular integral equation (5): ∂p∗ ∂ni =−ik 2πK1(ikr)∂r ∂ni +K1(ikr)∂r ∂nI ∂2p∗ ∂ni∂nj =(ik)2 2πK2(ikr)∂r ∂ni ∂r ∂nj +1 rK1(ikr)ni·nj+ K2(ikr)∂r ∂nI ∂r ∂nj +1 rK1(ikr)nI·nj (6) As in (1), iis the imaginary unit, kthe wave number and r,rthe distances to the observation point from the collocation point and its symmetric point with respect to the ground plane, respectively. It is worth making a distinction here regarding the normal vectors involved in the expressions above. njis the normal to the boundary at the integration point and ni(ni x, ni y), nI(ni x,−ni y) represent the normal vectors to the real boundary at the collocation point (i) and at its symmetric point (I) placed on a fictitious, symmetric boundary with respect to the ground plane, respectively. K1and K2 represent the Bessel modified functions of order 1 and 2, respectively. The application of (5) on each inode of the boundary discretization leads to the following system of equations: M·P=L−Ch·Q+Q∗ 0(7) 5 where Chis a is a diagonal matrix with entry values of 0.5, P,Qare the pressure and the flux (the derivative of the pressure with respect to normal at each boundary node) vectors, Q∗ 0array stores the values of the derivative of the fundamental solution concerning the external noise source, and M,Lare matrices whose entries are associated with the integration cores of the hyper-singular BEM formulation involving just the variables of the problem along the barrier boundary: mij k=ZΓj ∂2p∗ ∂ninj φkdΓj;lij k=ZΓj ∂p∗ ∂ni φkdΓj(8) The numerical strategies employed in the evaluation of both the singular and the hyper-singular BEM intregrals have been developed and implemented in a computer code by following the patterns proposed by S´aez et al. [21]. 2.3 Dual BEM formulation Fig. 2(a) represents a generic thin-cross section noise barrier to be solved by the Dual BEM formulation. After a discretization process, each node holds the values of pressure and flux with respect to the boundary normal (p+,q+,p−,q−hereinafter). Figure 2: (a) Idealization of a generic thin-cross section noise barrier profile as null thickness boundaries. (b) Strategy used to avoid the singularity around the collocation point in BEM formulation. Fig. 2(b) represents the strategy used to isolate the singularity of the method in this type of domains. Thus, the matrix equality of the singular BEM formulation for thin-cross section noise 6 barriers can be expressed as follows: cip+ i+p− i+ N X j=1 H+ jp+ j+H− jp− j= N X j=1 G+ jq+ j+G− jq− j+p∗ 0(9) being Nthe overall nodes number of the discretization over the boundary. Taking into account that n+=−n−, it is easily shown that: H+ j=−H− j;G+ j=G− j(10) Following the mathematical notation used by de Lacerda et al. [22], the final expression can be written for the sake of clarity as follows: ciΣpi+ N X j=1 H+ j∆pj= N X j=1 G+ jΣqj+p∗ 0(11) where: Σpi=p+ i+p− i; ∆pj=p+ j−p− j; Σqj=q+ j+q− j(12) For smooth boundaries ci= 0.5 in (11). Furthermore, considering that these type of profiles demand a non-nodal collocation at unbound extremes of boundaries, the free term is equally assumed as 0.5 in such cases. According to Fig. 2, the hyper-singular expression concerning these type of geometries can be written as follows: 1 2∂p+ i ∂n+ i +∂p− i ∂n+ i+ N X j=1 M+ jp+ j+M− jp− j= N X j=1 L+ jq+ j+L− jq− j(13) where: ∂p− i ∂n+ i =−qi;M+ j=−M− j;L+ j=L− j(14) The hyper-singular formulation of the method requires the collocation point jto be inside the element. In this way, the final expression can be expressed as follows: 1 2∆qi+ N X j=1 M+ j∆pj= N X j=1 L+ jΣqj+q∗ 0(15) The absorptive capacity of the barrier boundary is usually determined by means of the Robin boundary condition, so the pressure value and its derivative at each node are related: q+ j=−i k β+ Γp+ j;q− j=−i k β− Γp− j(16) 7 with βΓbeing a complex value (based on the empiric relation proposed by Delany and Bazley [23]) that represents the admittance of the boundary Γ for a particular frequency. In this way, the following can be written: ∆qj=A−Σpj+ A+∆pj Σqj=A+Σpj+ A−∆pj (17) being: A+=−ik 2β+ Γ+β− Γ; A−=−ik 2β+ Γ−β− Γ(18) Substituting (17) into (11) and (15) the following system of equations is obtained:      I 2-G+A+H+-G+A− A 2 − I-L+ A+A 2 + I+M+-L+A−        ΣP ∆P  =  P∗ 0 Q∗ 0  (19) with Ibeing the identity matrix. The matrix system above represents the final Dual BEM expressions for thin cross section barriers. For cases in which the boundaries are perfectly rigid (β+ Γ=β− Γ= 0) the variables of the problem uncouple and ∆pjis then directly obtained, resulting in faster computational times with respect to other cases. Once the variables of the problems are known, their corresponding values at any point of the domain can be easily obtained by applying (20). pi=p∗ 0+  N X j=1 G+ jΣqj− N X j=1 H+ j∆pj (20) 2.4 Discretization The Dual BEM code in this paper uses quadratic elements with three nodal points both to get the acoustic pressure level along the boundary (21) and to fit the barrier profile. This discretization process is frequency-dependent (with four elements per wavelength, at least). pi=φ1pi 1+φ2pi 2+φ3pi 3(21) being: φ1=ξ 2(ξ−1) ; φ2= 1 −ξ2;φ3=ξ 2(ξ+ 1) (22) where ξrepresents the local coordinate within the element with side limits (-1,1) (see Fig. 3). Fig. 3 represents the strategy used in the code for the hyper-singular BEM formulation. As previously mentioned, a non-nodal collocation is required in the extreme nodes of the elements. Extensive 8 Figure 3: Non nodal collocation points at the bound limits of the element when dealing with the hypersingular BEM formulation [21]: (1) collocation point PC1; (2) collocation point PC2; (3) collocation point PC3. references concerning the choice of δvalue can be found in scientific literature. For the cases here presented, the reallocation of such nodes has been carried out with a well-proven distance of δ= 5% of the element length for the point displacement towards inside. Differences in results associated with the election of this strategy are negligible, particularly given the fact that sound pressure levels of interest here are those neither at the barrier boundaries nor at the barrier corners but at the receiver points. As for the singular BEM formulation, the code here presented makes use of nodal collocation with the exception of the nodes placed at non-connected extremes of boundaries, where a non-nodal collocation strategy is employed. Some special procedures to tackle both sharp-angled boundaries and sharp angles between boundaries are considered. In order to assure the convergence of the numerical integrations of nearly singular integrals, the computing code used in this work implements two strategies. One of them is based on the procedure proposed by Telles [24], consisting in the reallocation and concentration of the Gauss points around the point with the minimum rdistance within the element under integration. The other strategy consists in the subdivision of the associated element from the barrier discretization into multiple subintervals, depending on the minimum rdistance to the cuasi-singular point. The final result is the overall sum of the numerical integration applied to each subinterval of the element. 9 Figure 6: Barrier models studied. 16 Figure 7: Convenience of the choice of a parametric representation to generate a multiple splinesbased curve. 17 0.5 1.0 1.5 2.0 2.5 3.0 3.5 a) 3-SIDED POLYGONAL-SHAPED BARRIER y [m] FF=19.27 [dBA] (SD=2.58 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 IL [dBA] AVERAGE FREQUENTIAL IL OF ALL RECEIVERS Straight barrier (SB) - FF=14.54 [dBA] 3-sided polygonal (3P) - FF=19.27 [dBA] AVERAGE IL SPECTRUM ALONG RECEIVERS’ HEIGHT (3P) y=0 [m] (3P) y=1 [m] (3P) y=2 [m] (3P) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 b) 5-SIDED POLYGONAL-SHAPED BARRIER y [m] FF=20.54 [dBA] (SD=3.20 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 IL [dBA] Straight barrier (SB) - FF=14.54 [dBA] 5-sided polygonal (5P) - FF=20.54 [dBA] (5P) y=0 [m] (5P) y=1 [m] (5P) y=2 [m] (5P) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −0.5 0 0.5 e) Y-SHAPED BARRIER y [m] x [m] FF=19.29 [dBA] (SD=2.47 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 IL [dBA] f [Hz] Straight barrier (SB) - FF=14.54 [dBA] Y-shaped barrier (Y) - FF=19.29 [dBA] 1 2 3 4 5 6 7 8 9 10 Distance from median axis of feasible region [m] (Y) y=0 [m] (Y) y=1 [m] (Y) y=2 [m] (Y) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] Figure 8: Overall shape design optimisation for polygonaland Y-shaped models (model a), b) and e), respectively). ’Case 1’ (rigid boundaries). 18 0.5 1.0 1.5 2.0 2.5 3.0 3.5 c) 3 CUBIC SPLINES-SHAPED BARRIER y [m] FF=19.03 [dBA] (SD=2.50 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 IL [dBA] AVERAGE FREQUENTIAL IL OF ALL RECEIVERS Straight barrier (SB) - FF=14.54 [dBA] 3-cubic splines (3S)- FF=19.03 [dBA] AVERAGE IL SPECTRUM ALONG RECEIVERS’ HEIGHT (3S) y=0 [m] (3S) y=1 [m] (3S) y=2 [m] (3S) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −0.5 0 0.5 d) 5 CUBIC SPLINES-SHAPED BARRIER y [m] x [m] FF=19.32 [dBA] (SD=2.90 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 IL [dBA] f [Hz] Straight barrier (SB) - FF=14.54 [dBA] 5-cubic splines (5S) - FF=19.32 [dBA] 1 2 3 4 5 6 7 8 9 10 Distance from median axis of feasible region [m] (5S) y=0 [m] (5S) y=1 [m] (5S) y=2 [m] (5S) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] Figure 9: Overall shape design optimisation for multiple cubic splines-based models (model c) and d)). ’Case 1’ (rigid boundaries). 19 0.5 1.0 1.5 2.0 2.5 3.0 3.5 f) TREE-SHAPED BARRIER y [m] FF=20.52 [dBA] (SD=2.94 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 IL [dBA] AVERAGE FREQUENTIAL IL OF ALL RECEIVERS Straight barrier (SB) - FF=14.54 [dBA] Tree-shaped barrier (T) - FF=20.52 [dBA] AVERAGE IL SPECTRUM ALONG RECEIVERS’ HEIGHT (T) y=0 [m] (T) y=1 [m] (T) y=2 [m] (T) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 g) Y-VARIANT-SHAPED BARRIER y [m] FF=21.29 [dBA] (SD=3.08[dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 IL [dBA] Straight barrier (SB) - FF=14.54 [dBA] Y-variant barrier (YV) - FF=21.29 [dBA] (YV) y=0 [m] (YV) y=1 [m] (YV) y=2 [m] (YV) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −0.5 0 0.5 h) FORK-SHAPED BARRIER y [m] x [m] FF=21.20 [dBA] (SD=3.14 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 IL [dBA] f [Hz] Straight barrier (SB) - FF=14.54 [dBA] Fork-shaped barrier (F) - FF=21.20 [dBA] 1 2 3 4 5 6 7 8 9 10 Distance from median axis of feasible region [m] (F) y=0 [m] (F) y=1 [m] (F) y=2 [m] (F) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] Figure 10: Top edge optimisation for f) tree-, g) Y-variantand h) fork-shaped model. ’Case 1’ (rigid boundaries). 20 0.5 1.0 1.5 2.0 2.5 3.0 3.5 f) TREE-SHAPED BARRIER y [m] FF=21.41 [dBA] (SD=2.85 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 IL [dBA] AVERAGE FREQUENTIAL IL OF ALL RECEIVERS Straight barrier (SB) - FF=14.54 [dBA] Tree-shaped barrier (T) - FF=21.41 [dBA] AVERAGE IL SPECTRUM ALONG RECEIVERS’ HEIGHT (T) y=0 [m] (T) y=1 [m] (T) y=2 [m] (T) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 g) Y-VARIANT-SHAPED BARRIER y [m] FF=22.00 [dBA] (SD=3.28[dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 IL [dBA] Straight barrier (SB) - FF=14.54 [dBA] Y-variant barrier (Y) - FF=22.00[dBA] (YV) y=0 [m] (YV) y=1 [m] (YV) y=2 [m] (YV) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 −0.5 0 0.5 h) FORK-SHAPED BARRIER y [m] x [m] FF=21.78 [dBA] (SD=3.75 [dBA]) −40 −35 −30 −25 −20 −15 −10 −5 0 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 IL [dBA] f [Hz] Straight barrier (SB) - FF=14.54 [dBA] Fork-like barrier (F) - FF=21.78 [dBA] 1 2 3 4 5 6 7 8 9 10 Distance from median axis of feasible region [m] FF=21.78 [dBA] (SD=3.75 [dBA]) (F) y=0 [m] (F) y=1 [m] (F) y=2 [m] (F) y=3 [m] (SB) y=0 [m] (SB) y=1 [m] (SB) y=2 [m] (SB) y=3 [m] Figure 11: Top edge optimisation for f) tree-, g) Y-variantand h) fork-shaped model. ’Case 2’ (absorbing boundaries). 21 Table 2: Design variables in the transformed domain of the best individual found along all optimisation runs for each barrier model. Case Model ξ1η1ξ2η2ξ3η3ξ4η4ξ5η5ξ6η6η7 1 a) 3-sided polygonal 0.17843 - 0.50000 0.94510 -0.46078 0.06275 -0.33137 - - - - - - b) 5-sided polygonal -0.50000 - -0.34314 0.69804 0.46471 0.95294 0.12353 0.89020 -0.39412 1.00000 -0.06078 - - c) 3-cubic splines 0.10392 - 0.46471 0.96471 -0.17059 0.56471 -0.49216 - - - - - - d) 5-cubic splines 0.15882 - 0.14706 0.63137 0.47647 0.78431 0.30784 0.89804 -0.20980 0.23922 -0.45686 - - e) Y-shaped -0.12353 0.05882 -0.30784 1.00000 0.50000 0.96471 - - - - - - - f) Tree-shaped -0.11177 0.87843 -0.47255 1.00000 0.50000 0.72941 0.21765 0.88235 - - - - g) Y-variant-shaped - 0.04314 -0.45324 1.00000 -0.22500 0.76863 0.10814 1.00000 0.33147 0.92549 - 0.80392 - h) Fork-shaped - 0.98824 - 0.48628 - 1.00000 - 0.25490 - 0.83529 - 0.84706 0.72549 2 f) Tree-shaped 0.30784 1.00000 -0.50000 1.00000 0.50000 0.83529 0.06078 1.00000 g) Y-variant-shaped - 1.00000 -0.30235 1.00000 -0.08775 1.00000 0.15520 0.96863 0.38265 0.86667 0.74902 - h) Fork-shaped - 1.00000 - 1.00000 - 0.92549 - 0.83529 - 0.74118 - 0.56863 0.24314 22 From the analysis of the results obtained some conclusions on the response of the models studied and, most importantly, on the procedure described in this work may be drawn: •In line with other authors ([4], [6], [7], [8], [28]), acting on the top of the barrier is found to be an appropriate strategy to minimize the acoustic impact. This is illustrated by the fact that, on the whole, models based on their top edge optimisation feature a better acoustic performance than those whose overall shape has been optimised. Furthermore, the latter models equally display a tendency towards the modification of their top edge in the search for the best acoustic performance. •The barrier performance when applying sound-absorbing materials to some boundaries of its optimised reflecting profile (Case 1 absorbing) may lead to unexpected results, as the configuration of the top of sound reflecting barriers plays an important role by producing reflected waves that help partly offset the incident ones. The incidence of this effect largely depends on the crowning configuration of the models studied here, ranging from a gain of 0.4 dBA for the tree-shaped barrier (model f)) to a loss of the same value for the fork-shaped barrier (model h)) when compared to their respective performance for rigid boundaries condition (Case 1). •Considering absorbing boundaries condition within the optimisation process (Case 2) is necessary to give assurance that the search leads to the best affordable profiles in terms of acoustic efficiency. This is supported by the fact that the performance of the best individuals from the top edge optimisation of f) tree-, g) Y-variantand h) fork-shaped barrier under this consideration, clearly outperforms the acoustic efficiency of such models from Case 1 absorbing (between 0.5 and 1.0 dBA). •The average IL spectrum values tend to remain roughly regular with the receiver distance to the barrier for the range studied. The fork-shaped barrier (model h)) shows a far better acoustic behaviour for close receiver points (between 5 and 10 dBA), though. 7 Conclusions A methodology to successfully optimize thin noise barriers by idealizing their profiles as null crosssection boundaries has been presented. With the purpose of highlighting the robustness, flexibility and the wide range of possibilities of the method, some template configurations have been analysed in this work, ranging from complex straight boundary configurations to curve-shaped profiles, from overall shape designs to top edge configurations. Nevertheless, this methodology may be applied to any real geometric thin design with immediate practical application for its performance to be improved. The versatility of the algorithm responsible for the geometry generation of the barrier makes the building of the profile to be easily accomplished. The Dual Boundary Element formulation here presented allows a simple treatment of the geometric shape of thin complex barriers. This is a significant advantage over the case when dealing with geometries of real barrier profiles, as the evaluation process for the feasibility of the design proposed by an evolutionary algorithm is often cumbersome and difficult to establish. To the auhtors’ knowledge, the procedure described in this work is the first joint implementation of evolutionary algorithms and a Dual BEM formulation concerning this issue. 23 The procedure presented is a useful method to assess the acoustic behaviour of thin complex noise barriers configurations and yields conclusions that might have been hardly drawn without its implementation. Acknowledgments This work was supported by the Subdirecci´on General de Proyectos de Investigaci´on of the Ministerio de Econom´ıa y Competitividad (MINECO) of Spain and FEDER through research project BIA2010-21399-C02-01 and also by the Agencia Canaria de Investigaci´on, Innovaci´on y Sociedad de la Informaci´on (ACIISI) of the Government of the Canary Islands and FEDER through research project ProID20100224. R. Toledo is a recipient of a fellowship from the Subprogram of Predoctoral Fellowships of Research Personnel in Trainning (FPI-MICINN), granted by the Ministerio de Ciencia e Innovaci´on of Spain. The authors are grateful for this support. References References [1] S. Seznec. Diffraction of sound barriers: use of the Boundary Element technique. Journal of Sound and Vibration, Vol. 73, 195–209, 1980. [2] D. Hothersall, S. Chandler-Wilde and M. Hajmirzae. Efficiency of single noise barriers, Journal of Sound and Vibration, Vol. 146, 303–322, 1991. [3] D. Hothersall, D. Crombie and S. Chandler-Wilde. The performance of T-profile and associated noise barriers. Applied Acoustics, Vol. 32, 269–287, 1991. [4] G. Watts and P. Morgan. Acoustic performance of an interference type noisebarrier profile. Applied Acoustics, Vol. 49(1), 1–16, 1996. [5] D. Crombie, D. Hothersall and S. Chandler-Wilde. Multiple-edge noise barriers. Applied Acoustics, Vol. 44, 353–367, 1995. [6] M. Monazzam and Y. Lam. Performance of profiled single noise barriers covered with quadratic residue diffusers. Applied Acoustics, Vol. 66, 709–730, 2005. [7] Ishizuka and K. Fujiwara. Performance of noise barriers with various edge shapes and acoustical conditions. Applied Acoustics, Vol. 65, 125–141, 2004. [8] Okubo and K. Fujiwara. Efficiency of a noise barrier on the ground with an acoustically soft cylindrical edge. Journal of Sound and Vibration, Vol. 216(5), 771-790, 1998. [9] P. Jean, J. Defrance and Y. Gabillet. The importance of source type on the assessment of noise barriers. Journal of Sound and Vibration, Vol. 226(2), 201–206, 1999. [10] O. Maeso and J. J. Azn´arez. Estrategias para la reducci´on del impacto ac´ustico en el entorno de carreteras. Una aplicaci´on del M´etodo de los Elementos de Contorno. Universidad de Las Palmas de Gran Canaria. ISBN: 84-6890340-X, http://hdl.handle.net/10553/1500, doi:846890340X, 2005. 24 [11] A. Tadeu, J. Ant´onio, L. Godinho and P. A. Mendes. Simulation of sound absorption in 2D thin elements using a coupled BEM/TBEM formulation in the presence of fixed and moving 3D source. Journal of Sound and Vibration, Vol. 331, 2386–2403, 2012. [12] L. de Lacerda, L. Wrobel and W. Mansur. A dual boundary element formulation for sound propagation around barriers over an impedance plane. Journal of Sound and Vibration, Vol. 202(2), 235–247, 1997. [13] I. Chen, J. Lee, Y. Hsiao and J. Chen. On physical and numerical resonances for water wave problems using the dual boundary element method. Engineering Analysis with Boundary Elements, Vol. 36, 1571–1580, 2012. [14] D. Duhamel. Shape optimization of noise barriers using genetic algorithms. Journal of Sound and Vibration, Vol. 297, 432–443, 2006. [15] M. Baulac, J. Defrance and P. Jean. Optimisation with genetic algorithm of the acoustic performance of T-shaped noise barriers with a reactive top surface. Applied Acoustics, Vol. 69, 332–342, 2006. [16] D. Greiner, J. J. Azn´arez, O. Maeso and G. Winter. Shape design of noise barriers using Evolutionary optimisation and Boundary Elements. The Fifth International Conference on Engineering Computacional Technology, Civil-Comp-Press, Stirlingshire, UK: Civil-Comp Press, Vol. 43, 2006. [17] D. Greiner, J. J. Azn´arez, O. Maeso and G. Winter. Singleand multi-objective shape design of Ynoise barriers using Evolutionary computation and Boundary Elements. Advances in Engineering Software, Elsevier, Vol. 41, 368–378, 2010. [18] D. Greiner, B. Galv´an, J. J. Azn´arez, O. Maeso and G. Winter. Robust design of noise attenuation barriers with Evolutionary multiobjective algorithms and the Boundary Element Method. NCS, Evolutionary Multi-Criterion optimisation, Eds: M. Ehrgott et al., Springer, Vol. 5467, 261–274, 2009. [19] S. Grubeˇsa, K. Jambroˇsi´c and H. Domitrovi´c. Noise barriers with varying cross-section optimized by genetic algorithms. Applied Acoustics, Vol. 73, 1129–1137, 2012. [20] K. Deb, S. Bandaru, D. Greiner, A. Gaspar-Cunha and C. Celal Tutum. An integrated approach to automated innovization for discovering useful design principles: Case studies from engineering. Applied Soft Computing, Vol. 15, 42–56, 2014. [21] A. S´aez, R. Gallego and J. Dom´ınguez. Hypersingular quarter-point boundary elements for crack problems. International Journal for Numerical Methods in Engineering, Vol. 38, 1681–1701, 1995. [22] O. von Estorff (Ed.). Boundary Elements in Acoustics. Advances & Applications, 2000. [23] M. Delany and E. Bazley. Acoustical properties of fibrous absorbent materials. Applied Acoustics, Vol. 3, 105–116, 1970. [24] J. Telles. A self-adaptative co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical Methods in Engineering, Vol. 24, 959–973, 1987. 25