This paper proposes a novel optimization framework in passive control techniques to reduce noise pollution. The geometries of the structures are represented by Catmull-Clark subdivision surfaces, which are able to build gap-free Computer-Aided Design models and meanwhile tackle the extraordinary points that are commonly encountered in geometric modelling. The acoustic fields are simulated using the isogeometric boundary element method, and a density-based topology optimization is conducted to optimize distribution of sound-absorbing materials adhered to structural surfaces. The approach enables one to perform acoustic optimization from Computer-Aided Design models directly without needing meshing and volume parameterization, thereby avoiding the geometric errors and time-consuming preprocessing steps in conventional simulation and optimization methods. The effectiveness of the present method is demonstrated by three dimensional numerical examples.

Installing sound-absorbing materials on structures is considered as an effective passive control technique to reduce noise level [

As a versatile numerical technique, the finite element method (FEM) is widely used in topology optimization. For acoustic problems, the boundary element method (BEM) is also commonly used for its advantages in dealing with unbounded domain problems [

Subdivision surface modeling is an important CAD technique to represent the complex surface geometries of structures [

This study aims to fill this research gap. We propose a topology optimization procedure to design the distribution of sound-absorbing materials, in which the Catmull-Clark subdivision surface is adopted to construct geometric models and its basis functions are used to simulate acoustic fields. The objective is to minimize the noise level subject to the volume constraints of absorption materials. As an extension of our previous work [

The remainder of this paper is organized as follows. In

The first step for a designer to build geometries with Catmull-Clark subdivision surfaces is to construct a control grid, which is a polygon mesh with quadrilateral elements whose vertices are called control points. Subsequently, the control grid is subdivided, new control points are introduced, and the positions of the existing control points are updated in

E-vertex: Let the two vertices of the inner edge be ^{k+1} generated by this inner edge is

F-vertex: If the vertices on each face are ^{k+1} generated by the face is

V-vertex: For an internal vertex ^{k}, we denote its one-neighbor vertices by ^{k}, and the ones with even subscripts are on the diagonal line of the quadrilateral element. Correspondingly, the updated position of vertex ^{k+1} is

where

After the control points of the subdivided control grid are determined, a smooth surface can be constructed through a linear combination of control points and basis functions:

where _{i}

where the symbols

Numerical simulation using the IGABEM can be used to compute the objective function and evaluate the design performance at each iterative step in the topology optimization process. The acoustic field is governed by the Helmholtz equation in domain

where

where

where

where

The conventional boundary condition may yield fictitious eigen-frequencies. This problem can be resolved by Burton-Miller method [

where

For a structural surface that is modeled using Catmull-Clark subdivision surfaces, whose basis functions _{i}

where _{e}_{e}

To transform the boundary integral equation to linear algebraic equations, we placed the source points at a set of discrete collocation points on the boundary and enforce the governing equations to be satisfied on them. In conventional BEM, the nodes of the mesh are normally chosen as collocation points. In the present work, the collocation points are selected as the points on the smooth surface that are mapped from the nodes of the control grid. With the collocation scheme, the system equation can be rewritten as

where _{e}

where

Because the kernel functions are singular at the source points, the singular integral arises in

We consider an optimization problem for the absorbing-material distribution to minimize sound pressures subject to material volume constraints. The topology optimization problem can be formulated as follows:

where the objective function _{e}_{v}

where matrices

In the present work, the density-based method is used for topology optimization. As a gradient-based optimization algorithm, it is more mathematically sound and converges to the optimized solution more rapidly than the gradient-less optimization methods like genetic algorithms [

where

After rearrangements,

where

As mentioned above, the adjoint vectors

According to the Delany-Bazley-Miki model [

where

To make the material distribution close to a 0–1 design, a material interpolation model, the so-called solid isotropic material with penalization (SIMP) method, is used to interpolate the element admittance:

where

After the sensitivities are evaluated, a gradient-based optimizer can be used to update the design variables until it converges. In the present work, the method of moving asymptotes is used as the optimizer and the convergence criterion is set as

where

A numerical example is presented in this section to investigate the effectiveness of the topology optimization approach using the IGABEM with Catmull-Clark subdivision surfaces. The iterative convergence criterion is set to

Now, we consider the model mentioned in

Similarly, when

The effect of different parameters on the objective function and material volume constraints is investigated. First, the effect of penalty factor on optimization results is considered. Theoretically, the larger the value of

We consider the effects of the initial values of design variables on the optimization results.

In this example, a car body shell model is constructed using Catmull-Clark subdivision surfaces with an initial control grid with 2288 elements and 2290 vertices in

Herein, the volume ratio constraint is 0.5, that is, at most

In the iteration process of the optimization, the objective and material volume fraction functions vary with the number of iterative steps, as shown in

As the number of iterative steps increases, the optimized distribution of sound-absorbing materials is shown in

In this study, we propose a method to optimize the distribution of sound-absorbing materials in noise pass control techniques based on isogeometric boundary element methods. The complex geometries of components or structures are modeled using Catmull-Clark subdivision surfaces, whose basis functions are also employed for acoustic analysis. The proposed method eliminates geometric errors and cumbersome preprocessing steps in traditional topology optimization procedures. The numerical results indicate that sound pressure levels can be significantly reduced subject to the given constraint of the volume of absorbing materials. However, the present paper only considers topology optimization. Combining shape and topology optimization with subdivision surfaces will deliver a more flexible design [