بهینه سازی شاخ های صوتی با استفاده از عناصر محدود و الگوریتم ژنتیک

In this work the optimization of shape (geometry) of acoustic horns is analyzed. The finite element method is employed for calculating the sound pressure and optimization methods of zero order (Golden Line Search Method and Genetic Algorithm, GA) are used to obtain the optimal geometry. The shape of the horn is approximated locally using polynomials with C1 continuity with the objective to get a few design variables, to obtain smooth contours at each iteration and to eliminate the regularization of the common mesh in this type of optimization. It was also studied the influence of the size of the domain (environment) in the optimized geometry of the horn. The numerical results show the efficiency of this approach and it was also found (at least from the engineering point of view) that the solution is not unique to the geometry of the horn to single-frequency.

This paper presents the steps and procedures used for the optimization of acoustic horns, as the one shown in Fig. 1. The shape of the waves that propagate in the output of the horn can be cylindrical for problems with planar symmetry and spherical for problems with cylindrical symmetry. Finite Element Method (FEM) was used to calculate the sound pressure, and the shape optimization was performed using two different methods were used: the “Golden Line Search Method” which is employed for the problems with only one design variable, and the Genetic Algorithm method (GA), employed for the other situations.The modeled domain and the region where the variation in the horn shape is allowed are shown in Fig. 2. In this figure, Γin indicates the inflow boundary (inlet straight tube), Γout denotes the outer domain boundary (free air), which is defined by the radius RΩ. The horn walls are assumed rigid.As reported by Bängtsson et al. [1] the problem of analyzing the impedance and radiation properties of acoustic horns has been treated extensively by several authors [2], [3], [4], [5] and [6]. An excellent review of the mathematical concepts involving the problem and its numerical implementation was carried out by [1]. These authors performed the shape optimization of acoustic horns also employing the Finite Element Method (FEM) and the BFGS quasi-Newton algorithm, which uses the objective-function gradient that is numerically calculated with perturbation. In order to obtain a new geometry and a finite element mesh for each iteration, velocity fields were used (see [7]). In the present work, zeroth-order gradient-free optimization methods were used and the boundary geometry is controlled by Hermite polynomials with C1 continuity. Then, it is shown that there are two very favorable facts when this approach is used: for each new iteration, the horn boundary is smooth and with no reentrance corners, and the number of design variables for obtaining the optimum geometry is greatly reduced. Aiming to compare the results with the ones obtained by [1], we used the same numerical approach as the one employed by these authors, as summarized below.

Some conclusions can be drawn from the results presented in this work: 1. The description of the horn profile using the local formulation with Hermitian elements having a C1 continuity presents satisfactory results. The geometry of the horn at any optimization step is smooth with continuous derivative. This procedure avoids using mesh and/or geometry smoothing techniques. 2. The employment of this technique for the boundary approximation also makes it possible to carry out the shape optimization for the horn with few design variables, for these variables are associated with control points. 3. It has been demonstrated that the geometries considered optimum (at least from the engineering point of view) are not the only ones, i.e. it is possible to find more than one solution for the same frequency. This conclusion also confirms the observations of Bängtsson et al. [1] that has multiple solutions to the problem of shape optimization of acoustical horns. 4. The objective function employed for the analyses with multiple frequencies Eq. (31), originates almost equal errors in all the frequencies after convergence. 5. The GA convergence aspects follow the standards found in the literature. 6. The numerical implementation is relatively easy and it does not require the calculation for the objective function gradient; thus, not requiring from the user a great knowledge in the optimization area