Shape Optimization of Acoustic Barriers Based on Subdivision Surfaces BEM

As an effective and economical noise reduction tool, the sound barrier has been widely used in traffic noise control. The noise reduction effect of a sound barrier is related to the shape, size and material properties of the barrier [1, 2]. The optimization design of sound barrier can effectively improve its noise reduction effect. The isogeometric BEM based on NURBS has been applied to the structural optimization [3–5]. Subdivision surface can construct discon- tinuous discrete meshes of any topology into an overall smooth surface. In essence, a certain subdivision rule is used to repeat the subdivision processing of the polygonal mesh until the subdivision convergence limit is reached. At present, the optimization by means of subdivi- sion surfaces is mainly applied to the distribution of sound absorbing materials [6], electrostatics [7] and elasticity problems [8]. This work aims to extend the shape optimiza- tion approach based on the subdivision surfaces BEM for sound barrier problems. We use fine control meshes for BEM discretisation and coarser control meshes for geometric modifications. The optimization is started from the coarsest initial control mesh, and the finer control meshes are gradually obtained. A key treatment of the present optimization approach is that a reverse processing is applied to the original optimized structure for reducing its oscillation.

Constructing a subdivision curve requires a control polygon and a subdivision algorithm. A recursive refinement scheme is used to generate new polygons from the initial control poly- gon. The result of the limiting process is a smooth subdivision curve. The subdivision algorithm is a corner cutting scheme which can successively cut the corner with a fixed ratio.

Givenan initialcontrol polygon(P00 ,P10 ,P20 ,...,Pn0 ), forChaikinalgorithmshown inFig. 1a, ith level ofrefinement onjth segment adoptsthefollowingequations:

For Catmull–Clark algorithm shown in Fig. 1b, ith level of refinement on jth segment adopts the following equations:

It has been shown that the limiting of a Chaikin curve is identical to a quadratic B-spline curve, while the Catmull–Clark curve converges to a cubic B-spline one. Figure 2 shows different levels of the subdivision curves from a concave polygon, obtained by the Chaikin and Catmull–Clark algorithms.

Here, we write the subdivision process as a linear mapping that maps a coarser control mesh at level l to a finer control mesh at level l + 1

where x ^l and x^{l +1} are two vectors containing the coordinates of all the vertices of the coarser and its refined polygons. A is the subdivision matrix and it contains the weights given by the subdivision stencils. The subdivision process can be interpreted as the chain of linear map- pings for obtaining increasingly finer control meshes.

In this work, we introduce the coarsening of control meshes that are also obtained with subdivision. The linear coarsening matrix B maps the given control points at level l + 1 to the control points at its coarser level l

The coarsening matrix B is formed by a least-squares fitting

which leads to A^T Ax^l = AT x^{l +1}. By comparing with eqn (4), A and B have the following relationship:

In words, one step of subdivision refinement of a given control mesh followed by coarsen- ing does not change the nodal coordinates. The refinement and coarsening processes enable us to use different control meshes of the same geometry for optimization and analysis.

For acoustic scattering problem, the total acoustic pressure is the sum of acoustic pressures caused by scattered acoustic wave and incident wave, i.e. p(x) = p _inc (x) + p_scat (x). The Bur- ton–Miller method that combines the conventional boundary integral equation and its normal derivative formulation overcomes the so-called fictitious eigenfrequency problem [3]. The two equations are as follows:

where $x$ is the source point, $y$ is the field point, $q(y)=\partial p(y) / \partial n(y)$ and $n(y)$ denotes the outward normal vector at point $y$. The coefficient $c(x)=1 / 2$ if the boundary $\Gamma$ is smooth around the source point $x$. The kernel functions $G(x, y), F(x, y), G^1(x, y)$ and $F^1(x, y)$ can be found in [3]. In this study, the geometry can be expressed by B-spline curves as follows:

where $m_i$ is the number of B-spline basis functions and $P_i$ is the control point for the geometry. The quadratic and cubic B-spline basis functions can be expressed as

Similarly, the sound pressure and flux at the boundary are interpolated using B-spline basis functions as follows:

where $p_i$ and $q_i$ represent the nodal parameters associated with the sound pressure and normal flux, respectively. The Burton-Miller formulation can be discretized in matrix form as:

Acoustic sensitivity analysis is a critical step in the shape optimization of acoustic struc- tures. Using the direct differentiation method, the equations are given as

where the upper dot (^.) denotes the differentiation with respect to the design variable. The shape sensitivities of sound pressure and its flux are interpolated from their respective nodal values

Here, the matrix form of the combined Burton–Miller formulation of eqns (14) and (15) can be expressed as

where H. and G. are the corresponding sensitivity coefficient matrices and p. and q. are the sensitivities of p and q.

The acoustic shape optimization problem can be formulated as

where $\mathbf{p}_f$ is the sound pressure vector at the reference point $x_f$ and $\overline{\mathbf{p}}_f$ denotes its complex conjugate. In this work, the coordinates of control points are set as design variables. $\Pi$ is the objective function. The optimization constraint is the designed area $A$ smaller than the initial area $A_0$, and $x_i^l$ and $x_i^u$ represent the lower and upper limits of the design variables, respectively.

The sensitivity of the objective function to the design variable can be expressed as

where $\Pi$ is the real part of the complex number. The expression for the sound pressure vector $\mathbf{p}_f$ in the acoustic domain can be calculated by the discretized formulation:

By differentiating $\mathbf{p}_f$ with respect to design variable $x_i$, the following formulation can be obtained:

Here, we employ the method of moving asymptotes (MMA) as the optimizer to update the design variables as shown below:

where $t$ denotes the iteration step and $\Pi^t$ represents the value of objective function at $t$ th iteration. $\tau$ is the iterative convergence criterion, set as $1.0 \times 10^{-4}$.

To demonstrate the validity and applicability of the developed optimization algorithm, we consider the analysis and design domain of a vertical noise barrier shown in Fig. 3. The thick- ness of the barrier is set as 0.25 m and a monopole source is located at (0, 1)m. As the figure shows, the left vertical boundary of the noise barrier is selected as the line to be optimized. The design variables are the horizontal coordinates of the N control points of the designing vertical line, and the optimization objective is to minimize the mean sound pressure on the reference plane.

The acoustic scattering of the sound barrier is a half-space problem, and the kernel func- tion can be written as

where r ′ =| x ′ − y | and x ′ is the mirror point of the source point x. Make the same changes to other kernel functions in the above formulations. After obtaining the basic sound pressure and sensitivity information, use the MMA solver to iterate the design variables until the iter- ation convergence accuracy is satisfied.

Figure 4 shows the objective function varying with the iteration step for different numbers of control points. Obviously, with an increase in design variables, the number of iteration steps also tends to increase. A larger N produces a smaller optimal value of the objective function, where the convergent value is 0.00122 when N = 30 and 0.00117 when N = 60.

Figures 5 and 6 show the shape configurations of the sound barrier at different control points N with frequency 300 Hz. Adding geometric control points can improve the flexibility in shape optimization, and we can find that the oscillation of the optimization line increases obviously. In order to eliminate this feature, the reverse matrix processing method is used to reduce the local oscillation of the optimized boundary. Figure 7 shows the detailed optimiza- tion process in Fig. 6. After the directly optimized Chaikin curve 1 is obtained, the boundary consisting of control points is coarsened once through the coarsening matrix, so as to obtain the boundary of the third subgraph shown in Fig. 7, and finally it is subdivided into a Chaikin curve 2 to meet the design requirements of the structure.

The sensitivity distributions of the control points at different frequencies are different, and the corresponding optimization results are also different. Now the number of geometric control points is fixed at N = 60. Figures 8 and 6 show the final shapes of the sound barrier after optimization at 150 and 300 Hz, respectively. For the optimized boundary, more peaks and valleys are found in higher frequencies. Table 1 shows the comparison of the noise reduc- tion effect before and after optimization. The optimization value 1 represents the result of the direct optimization of the initial sound barrier, and the optimization value 2 represents the optimization result obtained through the inverse matrix processing. It can be found that the two optimized results are basically the same, indicating that the method is feasible and effective.

A shape optimization approach based on subdivision surfaces is proposed for the sound bar- rier in this work. Numerical tests show the validity of the proposed approach and the necessity of an optimal design. The optimal noise reduction structure is obtained through the refine- ment of the optimization results and the reverse operation processing, which provides a set of reasonable solutions for the production and design structure.