### Sound Scattering Optimization Using the Topological Derivative and the Boundary Element Method

#### Abstract

Today, reduction of sound emission plays a vital role while designing objects of any kind.

Desirable aspects might include decreased radiation in certain directions of such an object. This work shows an approach to iteratively compute the shape of an obstacle which fulﬁls best to prescribed design variables using the framework provided by the topological derivative and the boundary element method (BEM).

At the beginning of the process a design space is deﬁned in which in iterative steps the shape will be developed. A regular array of points is set over the entire design space. The objective function is given by a set of prescribed pressure values for the scatter pattern on a circle around this design space.

The object, which acts as a scatterer, is considered acoustically rigid. The shape of the object builds up cumulatively, adding in each iterative step a rigid inclusion at the position that the topological derivative identiﬁes as the most effective to achieve the prescribed design values. The procedure is repeated until a given stopping criteria is satisﬁed. The proposed method requires the computation of a forward problem and an adjoint problem for each step. The ﬁrst is solved using a standard BEM for 2D acoustics, while the latter is solved backwards using the prescribed pressure values. The insertion of the rigid inclusions in each step is done by removing points from the design space. The BEM model geometry is updated automatically using a weighted Delaunay triangularization algorithm capable of detecting ‘holes’ at those positions where the points have been eliminated.

The capabilities of the proposed strategy are demonstrated by solving some examples.

Desirable aspects might include decreased radiation in certain directions of such an object. This work shows an approach to iteratively compute the shape of an obstacle which fulﬁls best to prescribed design variables using the framework provided by the topological derivative and the boundary element method (BEM).

At the beginning of the process a design space is deﬁned in which in iterative steps the shape will be developed. A regular array of points is set over the entire design space. The objective function is given by a set of prescribed pressure values for the scatter pattern on a circle around this design space.

The object, which acts as a scatterer, is considered acoustically rigid. The shape of the object builds up cumulatively, adding in each iterative step a rigid inclusion at the position that the topological derivative identiﬁes as the most effective to achieve the prescribed design values. The procedure is repeated until a given stopping criteria is satisﬁed. The proposed method requires the computation of a forward problem and an adjoint problem for each step. The ﬁrst is solved using a standard BEM for 2D acoustics, while the latter is solved backwards using the prescribed pressure values. The insertion of the rigid inclusions in each step is done by removing points from the design space. The BEM model geometry is updated automatically using a weighted Delaunay triangularization algorithm capable of detecting ‘holes’ at those positions where the points have been eliminated.

The capabilities of the proposed strategy are demonstrated by solving some examples.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**