A combined shape and topology optimization algorithm based on isogeometric boundary element method for 3D acoustics is developed in this study. The key treatment involves using adjoint variable method in shape sensitivity analysis with respect to non-uniform rational basis splines control points, and in topology sensitivity analysis with respect to the artificial densities of sound absorption material. OpenMP tool in Fortran code is adopted to improve the efficiency of analysis. To consider the features and efficiencies of the two types of optimization methods, this study adopts a combined iteration scheme for the optimization process to investigate the simultaneous change of geometry shape and distribution of material to achieve better noise control. Numerical examples, such as sound barrier, simple tank, and BeTSSi submarine, are performed to validate the advantage of combined optimization in noise reduction, and to demonstrate the potential of the proposed method for engineering problems.

Isogeometric analysis (IGA) [

The problem of sound emission of structures has attracted much attention in engineering practice. Structures such as sound barriers have also been investigated for their acoustic performance [

Shape optimization plays a great role in engineering problems to obtain the optimal shape under certain constraints with objectives. In this rational and automatic process, the geometry shape needs to be regenerated in each step, which causes limitation in conventional numerical analysis. In IGA, the structure surface is constructed by NURBS control points with weights and knot vectors; thus, the surface is easy to change by defining the control points as design parameters. Although IGA FEM has been firstly combined with shape optimization in fluid mechanics [

The previous works mainly conduct one type of optimization method to improve acoustic performance of structures. The present study focuses on combining the shape and topology optimization together efficiently to ensure better noise control for 3D acoustic problems. For combined optimization, researchers have done many works with the level set method (LSM) to conduct shape and topology update simultaneously by a uniform manner [

To implement combined optimization, the gradient-based optimizer is generally used due to its high efficiency, for example, the method of moving asymptotes (MMA) [

The present work is devoted to extending the combined optimization algorithm to 3D acoustics. The rest of this paper is organized as follows. In Section 2, the IGA BEM for 3D acoustics with the impedance boundary conditions are reviewed. Section 3 discusses the sensitivity analysis via AVM with respect to the NURBS control points and the artificial densities of SAM, respectively. Section 4 describes the three optimization procedure: shape optimization, topology optimization, and combined optimization. Section 5 introduces several numerical examples to validate the proposed approach. Finally, Section 6 provides the conclusions of this work.

In IGA, we can define the knot vector _{g}

For _{g}

Then, a B-spline curve’s formulation can be described by

where

For B-spline surface, we need two knot vectors

By introducing a weight

Similarly, the NURBS surface formulation can be described as follows:

where the number of control points is represented by

Considering a domain

where

For boundary element method, by applying Green’s second theorem and letting point

where

The Green’s function for 3D acoustics is given as follows [

where

In this study, we consider the impedance boundary condition

where

In IGA BEM, the NURBS interpolations are applied to both the geometry and physical fields. In this study, we adopt different NURBS interpolation formulations to suit physical analysis, which means that the physical field is separated from the geometry. The knot vectors of the physical space can be represented as

where _{l}_{l}_{d}

The location of collocation points in parametric space can be obtained by the Greville abscissa as follows:

where _{d1} and _{d2} denote the orders of basis function of each dimension, respectively; _{d}_{d}

Similarly, after discretizing the boundary into

where

When the parameter

By adopting the Burton–Miller formulation [

where

After

where the computations of matrices

Sensitivity analysis can obtain the derivatives of objection function with respect to different kinds of design variables. Thus, this type of analysis plays a critical role in the optimization process. The shape sensitivity and topology sensitivity analyses are presented in this section.

Control points usually control the configuration of structure surface in IGA, which means that they can be naturally set as the design parameters in shape optimization. In this study, we set certain control points of the NURBS surface as design parameters to change the shape, and the AVM proposed by Zheng et al. [

In DDM,

where the sensitivities of the kernel function are presented as

The sensitivity interpolation by NURBS basis functions are presented as

where

Similarly, using the same discretization as in

Subsequently, still based on the Burton–Miller formulation,

where matrices

For sensitivity analysis via DDM, after

where the computation of

However, some engineering problems may require more design parameters to guarantee the complexity of the structure, and the objective function in the optimization process may be merely related to physical values at field points. Thus, the AVM is adopted for ordinary optimization design. In accordance with

where the calculation of

and thus, the value of adjoint matrix

Finally, by substituting

Evidently, the adjoint matrix

In this study, we investigated the topology optimization of the distribution of SAM on structural surface, which is a problem with discrete values of 0 or 1, so that the mathematical computation in sensitivity analysis can be conducted. The SIMP method [

where

As the artificial density of the material of each element is set as a design variable, the AVM is also implemented for higher efficiency in topology sensitivity analysis. Usually, the objective function is related to the sound pressure of the field points as

where

The derivatives of objective function

where the term _{3} does not contain any term about

Combining

As both

and by solving

Evidently, both the adjoint operators _{3} are determined by objective function. The objective function

In this work, we present the combined shape and topology optimization to change the geometric shape and distribution of SAM on structural surface simultaneously. IGA BEM is applied to gradient-based optimization algorithm to construct a bridge between these two types of optimization process. After obtaining the two types of sensitivities in Section 3, we select the method of moving asymptotes (MMA) developed by Svanberg [

The optimization model of shape optimization can be described as follows:

where the objective function _{f}_{s}

_{0}, where we called the maximum volume constraint. The formulations of volume constraint and its sensitivity are expressed as

where

In each iteration step, the design parameters are updated until the objective function

As mentioned in Section 3.2, we change the distribution of SAM to minimize the average SPL of observed points. The optimization model is expressed as

where ^{s}_{e}^{s}_{0} denotes the volume constraint of the total materials, and we usually set it as half of the surface volume of the structure, which means that the volume fraction should be less than 0.5.

In the topology optimization, we select the same objective function and its sensitivity as those in shape optimization, where the values of _{3} in

Then, we can compute

According to Chen et al. [

In this study, the target is to combine the geometry shape change and distribution of SAM simultaneously to achieve better acoustic performance than the single type of optimization. Thus, the key is to select a suitable iteration scheme to combine these two types of optimization process. Considering the computational efficiency and their features in the optimization process, the presented scheme is shown in

Several numerical examples are presented to validate the applicability of the proposed approach and show its potential in engineering problems. Here, all the examples are exterior acoustic problems, and the parameter ^{−4}. The code is parallelized by using the OpenMP tool in Fortran to improve the computational efficiency.

A scattering sphere model is considered to verify the shape sensitivity analysis and topology sensitivity analysis of the present approach. The sphere center is located at point (0, 0, 0) with a radius

For the computation of sound sensitivities with respect to shape parameters, the

The verification of topology sensitivity analysis is also shown in

Although sound barriers have been simplified as 2D problems for optimization design by using the IGA BEM in the work of Liu et al. [

where

As shown in

Design parameter | Initial value | Final value |
Final value |
---|---|---|---|

4.25769 | 4.15769 | 4.35769 | |

4.25769 | 4.15769 | 4.35769 | |

4.38141 | 4.28141 | 4.48141 | |

4.38141 | 4.46237 | 4.48141 | |

4.50513 | 4.40513 | 4.60513 | |

4.50513 | 4.60513 | 4.40513 | |

4.62885 | 4.52885 | 4.63175 | |

4.62885 | 4.72885 | 4.62144 | |

4.75256 | 4.84276 | 4.85256 | |

4.75256 | 4.75652 | 4.85256 | |

4.87628 | 4.97628 | 4.97628 | |

4.87628 | 4.92489 | 4.97628 |

For the topology optimization at a higher frequency of _{min}_{min}

The example of a underwater simple tank presented by Chen et al. [

Design parameter | Initial value | Lower bound | Upper bound | Final value |
Final value |
---|---|---|---|---|---|

3.75000 | 0.75000 | 7.00000 | 7.00000 | 7.00000 | |

3.75000 | 0.75000 | 7.00000 | 0.75000 | 0.84480 | |

3.75000 | 0.75000 | 7.00000 | 7.00000 | 4.46602 | |

3.75000 | 0.75000 | 7.00000 | 1.63348 | 4.39776 | |

3.75000 | 0.75000 | 7.00000 | 7.00000 | 5.46867 | |

3.75000 | 0.75000 | 7.00000 | 5.82688 | 1.25608 |

An obvious checker-board phenomenon is also indicated by _{min}_{min}

To exhibit the capability of the proposed algorithm in optimization design of complicated geometry, the BeTSSi submarine model described by Venås et al. [

Design parameter | Initial value | Lower bound | Upper bound | Final value |
Final value |
---|---|---|---|---|---|

2.00000 | 1.00000 | 3.00000 | 1.00000 | 1.00000 | |

2.00000 | 1.00000 | 3.00000 | 1.00000 | 1.25217 | |

1.73738 | 1.00000 | 3.00000 | 2.64019 | 3.00000 | |

1.46476 | 1.00000 | 3.00000 | 3.00000 | 1.00000 | |

2.00000 | 1.00000 | 3.00000 | 1.00000 | 1.00000 | |

2.00000 | 1.00000 | 3.00000 | 1.00000 | 1.41316 | |

1.73738 | 1.00000 | 3.00000 | 3.00000 | 3.00000 | |

1.46476 | 1.00000 | 3.00000 | 3.00000 | 1.00000 |

The iteration histories of objective function through the three optimization methods at different frequencies are shown in

In this study, we developed a combined shape and topology optimization algorithm for 3D structures based on IGA BEM. Different from the past shape optimization method on rigid structures, the impedance boundary condition is applied to the optimization process, where the structure surfaces are covered with SAM with artificial densities. The NURBS model constructs a bridge between geometry and physics, when surface shapes and the distribution of material change in each optimization iteration step. The control points are selected as the design parameters for shape optimization on account of their convenience and flexibility in shape control. With the application of SIMP method, the artificial densities of SAM located in integral elements are set as the design variables for topology optimization. The AVM is applied to the sensitivity analysis with respect to shape design parameters and topology design variables. Considering the computational efficiency and features of the two types of optimization, an iteration scheme for combined optimization, including a convergent topology optimization and a one-step shape optimization, is investigated in this study. Several numerical examples, including a complex submarine scattering model, are performed to demonstrate the potential of the proposed combined optimization in achieving improved noise reduction, compared with the single type of shape optimization or topology optimization. All the optimization processes obtain frequency-dependent results, where the optimized shape and distribution of material at higher frequency tend to show a better noise reduction.

In the future, we aim to apply the fast multipole method to the acoustic analysis and sensitivity analysis to expand the developed method to larger-scale engineering problems. The level set method is also considered replacing the SIMP method in topology optimization to eliminate the medium densities in the distribution of SAM.

The reconstructed BeTSSi submarine model consists of 7 NURBS patches, which are shown in