In the context of isogeometric analysis, the subdivision surface basis function enables the representation of the identical spline surface using control meshes of different resolutions [45]. The particular division guidelines employed in this study are obtained from cubic B-splines.To achieve smooth surfaces for curves/surfaces, we employ the subdivision principle introduced by Catmull et al. [46]. This technique ensures that even unstructured meshes with irregular vertices (i.e., vertices having a varying number of adjacent edges other than 4) yield smooth surfaces. In Fig. 1, the initial mesh and convergence surface for the Catmull-Clark subdivision rule are depicted in both two and three dimensions.

Each subdivision operation generate a new vertex vx by moving the original mesh node, and new vertices ex and fx are inserted respectively at the midpoint of edges and the center of the element. All newly inserted nodes make up a new mesh level. The control vertex coordinates for a fine mesh at level k+1 are determined by the vertex coordinates of the coarse mesh at level k

(1)xk+1=Sxk,

where xk and xk+1 denote two vectors that store the vertex coordinates of meshes at the coarser level and those at the refined level, respectively. Matrix S is the global subdivision matrix with size [Nv+Ne+Nf,Nv], where Nv represents the total number of vertices. The subdivision matrix S includes standard weights for inserting junctions in regular mesh vertices. The matrix components corresponding to the vertices of the irregular region can be obtained through a particular subdivision method, which can be referred to [47]. A linear mapping can be interpreted as the continuous subdivision refinement of the control mesh:

(2)x0→Sx1→Sx2→S⋯→Sxk−1→Sxk,

The number of vertex coordinates xk increases as k increases, and the size of subdivision matrix S increases accordingly. Regardless of the arbitrary level, all control meshes converge to the same surface.

Multiresolution analysis brings editability to different levels of the subdivision surface [48–50]. In shape optimization, a well-performing geometric configuration can be obtained by alternately refining and coarsening the control mesh. Establish the relationship between the control points at levels k+1 and k through coarsening matrix R, as follows:

(3)xk=Rxk+1.

where R=(STS)−1ST is the coarsening matrix. Similar to subdivision refinement, to obtain a coarser representation of geometry, coarse matrices can be applied sequentially as follows:

(4)xk→Rxk−1→Rxk−2→R⋯→Rx1→Rx0.

The coordinates of the vertices represented by one step of refinement, followed by one step of coarsening, do not change, so the relationship between the subdivision matrix and the inverse matrix is RS=I. Under the subdivision surface, the hierarchy of control meshes is naturally used for multiresolution geometry decomposition [51,52]. For this purpose, the size of the geometric area influenced by each vertex is determined by the resolution of the control mesh, with editing coarser layers producing large-scale changes and editing thinner layers producing small-scale changes. Local subdivision refinement can be applied to the surface of the corresponding being edited.

This study considers the differential equation that governs the harmonic acoustic wave:

(5)∇2p(x)+k2p(x)=0,∀x∈Ω,

where ∇2 is the Laplace operator, p denotes the sound pressure, k=ω/c denotes the wave number, ω is the angular frequency, and c is the wave speed in the acoustic medium Ω.

For the acoustic boundary element method [53,54], the boundary integral expression and its outward normal derivative formulation of the Helmholtz equation are as follows:

(6)c(x)p(x)=∫SG(x,y)q(y)dS(y)−∫SF(x,y)p(y)dS(y)+pinc(x),

(7)c(x)q(x)=∫SG1(x,y)q(y)dS(y)−∫SF1(x,y)p(y)dS(y)+∂pinc(x)∂n(x),

where x and y are the field point and the source point, respectively. S is the boundary of the Ω. For the 3D problem the Green’s functions [55,56] are given as

(8)G(x,y)=eikr4πr,

(9)F(x,y)=−eikr4πr2(1−ikr)∂r∂n(y),

(10)G1(x,y)=−eikr4πr2(1−ikr)∂r∂n(x),

(11)F1(x,y)=eikr4πr3[(3−3ikr−k2r2)∂r∂n(x)∂r∂n(y)+(1−ikr)nl(x)nl(y)],

where r=∣x−y∣ is the distance from the source point to the field point, and nl is the Cartesian component of n(x) or n(y). The admittance boundary condition is expressed as

(12)q(y)=ikβ(y)p(y),

where β(y) represents the normalized surface admittance at point y. Note that when β(y)=0, the impedance boundary degenerates into a rigid boundary, i.e., the surface shows total reflection and does not absorb any sound waves.

After introducing the impedance boundary condition, the sound pressure and flux on the surface boundary can be discretized as follows:

(13)p(ξ,η)=∑a=1NaBa(ξ,η)pa,q(ξ,η)=ikβ∑a=1NaBa(ξ,η)pa,

where pa is the coefficients of sound pressure, Ba denotes the bicubic B-spline basis function [47] and Na represents the total number of basis functions. Substituting the above equation into Eqs. (6) and (7) after discretizing the boundary into Ne elements yields:

(14)12∑a=1NaBa(ξ,η)pa=∑e=1Ne∑a=1Naikβ[∫SeG(x,y)Ba(ξ,η)dSe]pa−∑e=1Ne∑a=1Na[∫SeF(x,y)Ba(ξ,η)dSe]pa+pinc(x),

and

(15)12ikβ∑a=1NaBa(ξ,η)pa=∑e=1Ne∑a=1Naikβ[∫SeG1(x,y)Ba(ξ,η)dSe]pa−∑e=1Ne∑a=1Na[∫SeF1(x,y)Ba(ξ,η)dSe]pa+∂pinc(x)∂n(x),

where c(x)=1/2 on the smooth surface boundary. Using the Burton-Miller method [57–59] to linearly combine Eqs. (14) and (15), then expressed in matrix form as follows:

(16)[H−G]p=pinc,

where H and G represent the matrices formed by the integrals in Eqs. (14) and (15), respectively. p denotes the vector of sound pressure, and pinc are the vectors of incident sound pressure and flux.

Once the sound pressure at the boundary is acquired, it can be represented in matrix form to express the sound pressure at various points within a given domain:

(17)pf=[Gf−Hf]p+pinf,

where the subscript f denotes the computation point is inside the acoustic domain.

In this research, topology optimization is performed for the distribution of acoustic absorbing materials with discrete values of 0 or 1 on the structure surface. AVM is adopted for the topology sensitivity analysis. The discrete values are transformed into continuous ones using the SIMP method. The design parameter for the optimization process is set to the artificial density ρe of element e. The equation of admittance by the SIMP method can be expressed as

(18)βe=β0ρeε,

where β0 represents the normalized surface acoustic admittance, ε denotes the penalization coefficient determining the intermediate density tending to zero or one. The determination of the penalty coefficient is detailed in [60]. ε is set to 3 in the present analysis.

The goal of this study is to establish a relationship between the objective function Π(pf) and the sound pressure in acoustic optimization. In accordance with Eqs. (16) and (17), the reconstructed objective function Π can be rewritten as

(19)Π=Π(pf)+ℜ{λ1T[(H−G)p−pinc]+λ2T[pf+(Hf−Gf)p−pinf]},

where Π is a real valued function, and ℜ denotes the process of extracting the real component of the complex number.

λ1T and λ2T represent the transposed adjoint vectors of the boundary of the structure and the points in the field, respectively. By differentiating the above equation with regard to the design variable ρe yields the following equation:

(20)∂Π∂ρe=∂Π(pf)∂ρe+ℜ{λ1T[(H−G)∂p∂ρe−∂G∂ρep]}+ℜ{λ2T[∂pf∂ρe+(Hf−Gf)∂p∂ρe−∂Gf∂ρep]},

The formulation of the derivative of the objective function Π(pf) can also be expressed as

(21)∂Π(pf)∂ρe=ℜ(v1+v2T∂p∂ρe+v3T∂pf∂ρe),

The values of v1, v2T, and v3T in the objective function expression are determined by the specific expression of Π(pf). Eq. (20) can be further written as

(22)∂Π∂ρe=−ℜ(λ1T∂G∂ρep+λ2T∂Gf∂ρep+v1)+ℜ[(λ2T+v3T)∂pf∂ρe]+ℜ{[λ1T(H−G)+λ2T(Hf−Gf)+v2T]∂p∂ρe},

where the adjoint vectors λ1T and λ2T can be arbitrarily selected as long as they meet the following equation:

(23){λ2T+v3T=0λ1T(H−G)+λ2T(Hf−Gf)+v2T=0

For all topology design variables, the system of adjoint Eq. (23) should be solved only once to obtain adjoint vectors λ1 and λ2. Hence, the use of AVM can greatly enhance the computational effectiveness of topology sensitivity analysis, which holds immense importance for topology optimization.

In this research, the geometry configuration of the structure is changed by modifying the location of the control points. We use AVM to perform shape sensitivity analysis and set the control points on subdivision surfaces as shape design parameters. Eqs. (6) and (7) can be derived as follows:

(24)c(x)∂p(x)∂χe=∫S[∂G(x,y)∂χeq(y)−∂F(x,y)∂χep(y)]dS(y)+∫S[G(x,y)∂q(y)∂χe−F(x,y)∂p(y)∂χe]dS(y)+∫S[G(x,y)q(y)−F(x,y)p(y)]d∂S(y)∂χe+∂pinc(x)∂χe,

and

(25)c(x)∂q(x)∂χe=∫S[∂G1(x,y)∂χeq(y)−∂F1(x,y)∂χep(y)]dS(y)+∫S[G1(x,y)∂q(y)∂χe−F1(x,y)∂p(y)∂χe(y)]dS(y)+∫S[G1(x,y)q(y)−F1(x,y)p(y)]d∂S(y)∂χe+∂2pinc(x)∂n(x)∂χe,

where χe denotes the coordinates in x, y, and z directions of control points, and ∂()∂χe represents the sensitivity values at the control points.

Afterwards, Eqs. (24) and (25) are rewritten as

(26)12∑a=1NaBa(ξ,η)∂pa∂χe=∑e=1Ne∑a=1Napa∫Se[ikβe∂G(x,y)∂χe−∂F(x,y)∂χe]Ba(ξ,η)dSe+∑e=1Ne∑a=1Na∂pa∂χe∫Se[ikβeG(x,y)−F(x,y)]Ba(ξ,η)dSe+∑e=1Ne∑a=1Napa∫Se[ikβeG(x,y)−F(x,y)]Ba(ξ,η)d∂S(y)∂χe+∂pinc(x)∂χe,

and

(27)12∑a=1NaikβiBa(ξ,η)∂pa∂χe=∑e=1Ne∑a=1Napa∫Se[ikβe∂G1(x,y)∂χe−∂F1(x,y)∂χe]Ba(ξ,η)dSe+∑e=1Ne∑a=1Na∂pa∂χe∫Se[ikβeG1(x,y)−F1(x,y)]Ba(ξ,η)dSe+∑e=1Ne∑a=1Napa∫Se[ikβeG1(x,y)−F1(x,y)]Ba(ξ,η)d∂S(y)∂χe+∂2pinc(x)∂n(x)∂χe.

Similarly, combining Eqs. (26) and (27) using the Burton-Miller method yields the system of equations for sensitivity as follows:

(28)(H−G)∂p∂χe=(∂G∂χe−∂H∂χe)p+pinc∂χe,

where matrices H and G have been computed in Eq. (16), and ∂H∂χe and ∂G∂χe are their derivative matrices. Similarly, the matrix form of the points in the domain is obtained by

(29)∂pf∂χe=(∂Gf∂χe−∂Hf∂χe)p−(Hf−Gf)∂p∂χe+∂pinf∂χe,

In shape optimization design, the boundary shape of a structure often needs to be described by functions that take multiple design variables as parameters. To improve the efficiency of acoustic sensitivity analysis of multiple design variables with increased practical significance, the discrete AVM [56] is introduced next. The acoustic pressure sensitivity calculation matrix for the boundary point can be written as

(30)∂p∂χe=(H−G)−1[(∂G∂χe−∂H∂χe)p+∂pinc∂χe],

Substituting the above expression into Eq. (29) gives:

(31)∂pf∂χe=−(Hf−Gf)(H−G)−1[(∂G∂χe−∂H∂χe)p+∂pinc∂χe]+(∂G∂χe−∂Hf∂χe)p+∂pinf∂χe,

where the inverse operation of matrix (H−G) requires considerable time, so the accompanying equation is defined as follows:

(32)(H−G)TA=(Hf−Gf)T,

where AT=(Hf−Gf)(H−G)−1, which is integrated into Eq. (31) to obtain the sound pressure sensitivity of the domain point, as shown as follows:

(33)∂pf∂χe=−AT[(∂G∂χe−∂H∂χe)p+∂pinc∂χe]+(∂Gf∂χe−∂Hf∂χe)p+∂pinf∂χe,

where adjoint matrix A is independent of shape design parameters χe. For multiple design parameter problems, its only needs to be solved once. This paper pushes further on previous research work [20] to improve the efficiency of the discrete AVM for computing shape sensitivity.

After the topology sensitivity analysis, the objective function is defined as the sound pressure level of the observer points in the acoustic domain, and the initial state is set as the structure surface fully attached to the sound absorbing material. The topology optimization model is given as follows:

(34){min:Π=20log10⁡((pfHpf)2.0×10−5),s.t.:∑e=1Neρeve−V0e∑e=1Neve≤0,Bound:0≤ρmin≤ρe≤1,e=1,…,Ne,Iter.:|Πj+1−ΠjΠj|<τ,

where Π is the objective function, ()H denotes the symbol for the conjugate transpose. The optimized constraint function is set to the volume constraint of the sound absorbing material distribution; ve is the volume of each piece of material, which corresponds to the area of the subdivision surface integral element; the volume constraint score is V0e=0.5. The corresponding material on each subdivision surface element is a topology design variable ρe. To avoid the problem of singular values, the lower limit of the topology design variable is ρmin=0.001. The number of topology design variables is equal to the number of subdivision surface integral elements for the structure surface.

According to Eq. (21), the objective function [36] here is the sound pressure level of the test points, and v1, v2T, and v3T can be set as follows:

(35){v1=0,v2T=0,v3T=10ln⁡101pfHpf⋅2pfH.

After the sound field analysis and solving of adjoint vectors λ1 and λ2, the objective function sensitivity can be obtained in the optimization process.

After the shape sensitivity analysis, using the same objective function as in topology optimization, a shape optimization model is established with the control point coordinates as design variables. The shape optimization model is given as follows:

(36){minΠ=20log10((pfHpf)2.0×10−5),s.t.V(χe)−V0⩽0,Boundχemin⩽χe⩽χemax,Iter∣Πj+1−Πj∣Πj<τ,

where V0 represents the initial volume of the structure, and V(χe) denotes the volume after the changes of control points. For each design variable, χemax and χemin are set as the top and lower bounds of its values. In this paper, a local boundary is chosen as the shape optimization region, and a single direction of control point coordinates is chosen to facilitate shape changes.

For shape optimization, the structure’s volume V can be calculated by integrating over the points x and their external normal vectors n on the subdivision surface. The volume function and its sensitivity can be expressed as follows:

(37)V=∫ΩdΩ=13∑e=1Ne∫01∫01x⋅nJe(ξ,η)dξdη.

(38)∂V∂χe=13∑e=1Ne∫01∫01[∂x∂χe⋅nJe(ξ,η)+x⋅∂n∂χeJe(ξ,η)+x⋅n∂Je(ξ,η)∂χe]dξdη,

Traditional optimization methods tend to focus on one type of optimization of the structure, such as topology or shape optimization alone. In this subsection, the combined optimization based on subdivision surface BEM is implemented to achieve efficient noise reduction. Fig. 2 displays the flowchart of the combined optimization approach for the 3D structure. The process is primarily divided into seven steps, which are as follows:

The subdivision surface model of the initial structure is input, and the appropriate optimization levels are selected according to the refinement of initial model and topology and shape design needs.

A sensitivity analysis and the combined optimization approach are calculated for a simple cylinder model. Fig. 3 shows the physical optimization model. The blue point is selected as the observation point with coordinates (0,0,3). Shape design variables include 12 points that are colored red. The cylinder’s length is 6 m, and the hemispherical’s radius is 1 m. Fig. 4 shows the multilevel subdivision meshes for the cylinder model. The number of elements and vertices of the initial mesh is 304 and 306, and the Catmull-Clark Subdivision surface of the quadrilateral element is used to obtain the subdivision meshes of different levels. The number of elements and vertices corresponding to the first subdivision level is 1216 and 1218, and the mesh of second subdivision level is selected for the objective function and sensitivity analysis.

For optimal design of a muffler pipe model [20], the combined optimization approach is taken into consideration in the following subsection. Firstly, the boundary condition is set to be a rigid boundary on the outer surface and a sound absorption boundary on the inner surface. The region for structural optimization is the inner surface of the muffler pipe, the optimized shape control points are the 38 red coordinate points shown in Fig. 17, and the optimized topology design variables are the subdivision surface elements in the inner surface adhesion of the sound absorbing material (where the red portion represents the attachment of the sound absorbing substance, while the blue portion represents the non-adhesive sound absorbing material). Under the action of the plane wave traveling along the x-axis, the observation point is set at the central position of the outlet plane of the pipeline (−3,1.5,1.5). The initial model contains 736 elements and 736 nodes, and the second level of subdivision mesh is used for sound field and optimization.

Fig. 18 shows the iterative history of the muffler pipe model at 60 and 120 Hz frequencies in different combined optimization, from which the objective function of topology optimization at 60 Hz presents signs of increase. The inner surface of the structure is initially fully covered with sound absorbing materials. The final coefficient of the topology iterative scheme setting is less than 0.5 of the initial value, which makes the final optimization result greater than the initial value. However, the objective function of the combined optimization at different frequencies is reduced. Figs. 19 and 20 show that the optimization results for different optimization schemes are in the order of topology, shape, combined A, combined B and combined C. From the final optimized distribution, local differences exist in the distribution of shape and inner surface materials.

Next, the influence of the initial value of sound absorption coefficient on the optimization result is investigated. Fig. 21 shows the optimal distributions corresponding to the initial values 0.6 and 0.8 with different optimization approaches. Different initial values correspond to different initial objective functions: Πinit is 90.695 dB with the initial value of 0.6, Πinit is 88.801 dB with the initial value of 0.8, and Πinit is 88.376 dB with the initial value of 1.0. The final objective functions of combined optimization corresponding to the three initial values are 84.467, 84.295, and 84.425 dB, respectively. The optimization results are basically consistent with only slight differences in some elements. Thus, the final optimization result is not sensitive to the initial sound absorption coefficient.

Lastly, the sound fields before and after the combined optimization are compared. Fig. 22 shows the comparison of sound fields before and after optimization corresponding to different initial values. The external sound fields of the structure are basically the same, but the internal sound fields have much differences. When the sound absorption coefficient is low, the overall internal sound pressure level is larger owing to the stronger reflection effect of the sound wave, and the optimization effect at the outlet is more significant.