Acoustic Topology Optimization Research Articles

Topology optimization of reactive acoustic mufflers using a bi-directional evolutionary optimization method

This article proposes an acoustic muffler design procedure based on finite element models and a Bi-directional Evolutionary Acoustic Topology Optimization. The main goal is to find the best configuration of barriers inside acoustic mufflers used in the automotive industry that reduces sound pressure level in the outlet of the muffler. The acoustic medium is governed by Helmholtz equation and rigid wall boundary conditions are introduced to represent acoustic barriers. The continuum problem is written in the frequency domain and it is discretized using the finite element method. The adopted objective function is Transmission Loss (TL). Increasing TL guarantees that the sound pressure level ratio between outlet and inlet of the muffler is reduced. To find the configuration of acoustic barriers that increases the Transmission Loss function of the muffler an adaptation of the Bi-directional Evolutionary Structural Optimization (BESO) method is used. Applying the proposed design procedure topologies in 2D models are reached, which raises the Transmission Loss function for one or multiple frequencies. Three examples are presented to show the efficiency of the proposed procedure.

An isogeometric approach of two dimensional acoustic design sensitivity analysis and topology optimization analysis for absorbing material distribution

A study of structural shape optimization and absorbing material distribution optimization of noise barrier structures based on the recently proposed isogeometric analysis method with exact geometric definitions is presented. The acoustic scattering is approximated using the basis functions that represent the geometry. A fast multipole method is applied to accelerate the solution of the boundary element method (BEM). The Burton–Miller formulation is used to overcome the fictitious frequency problem when a single Helmholtz boundary integral equation is used for the exterior boundary–value problem. The strongly singular integrals in the Burton–Miller formulation using the isogeometric BEM are evaluated explicitly and directly, particularly for the sensitivity formulation with a hyper-singular integral. The optimality criteria method is used for two types of optimization analyses, shape optimization and material distribution topology optimization. For the shape optimization, the design variables can be set to the locations of the control points because the control points determine the shape of structure. For the second optimization, a new material interpolation scheme for acoustic problems based on the solid isotropic material with penalization (SIMP) method is given, where the interpolation variable is not the real structural density used in a conventional SIMP, but a fictitious material density that determines the normalized surface admittance. Several examples are presented to demonstrate the validity and efficiency of the proposed algorithm.

Study on the Optimization of the Distribution of Absorbing Material on a Noise Barrier

Noise barriers have been widely used to decrease the transportation noise. How to utilize limited structure and material to achieve the environmental requirements for noise control is an important research topic. This research combines the use of boundary element method and SIMP to generate an acoustic topology optimization method and applies this optimization method to optimization distribution of absorbing material installed on the noise barrier’s edges. Optimality criteria method is used to update the design variables and look for the final optimal solution. A new material interpolation scheme for acoustic problems based on SIMP is given, where the interpolation variable is not real structural density used in conventional SIMP, but fictitious material density deciding the normalized surface admittance. It is noteworthy that gray elements exist in the acoustic optimization analysis based on SIMP. However, a modified method based on a smoothed Heaviside function is applied to eliminate the gray elements. In order to demonstrate the validity and efficiency of the proposed algorithm in this paper, vertical barrier and T-shaped barrier with two wells are used for the numerical analysis, respectively.

A Comparative Study on Acoustic Optimization and Analysis of CLD/Plate in a Cavity Using ESO and GA

An acoustic radiation model of a cavity with a flexible plate treated with constrained layer damping (CLD) is developed by a combination of finite element method (FEM) and boundary element method (BEM). An acoustic topology optimization model is established with the objective of minimizing sound radiation power at specific modal frequency and design variables defined as locations of CLD treatments. The evolutionary structural optimization (ESO) method and genetic algorithm (GA) are employed to search optimal CLD configurations. Sound power sensitivity for CLD/plate is derived to determine search direction in ESO optimization procedure. The optimal CLD layouts for the flexible plate with two different boundary conditions are obtained and analyzed. Computational time, optimal layouts, and minimum sound power obtained using ESO and GA are compared. The results demonstrate effectiveness of the two methods, and ESO is more efficient to obtain deterministic and more practical optimal CLD material layouts for minimizing sound radiation power. The influences of CLD materials thickness and exciting force locations on optimal results obtained using ESO are discussed in detail. It is shown that the optimal rejection ratio varies with thicknesses of CLD materials and distribution of normal velocity of the flexible plate. Variation trend of the optimal rejection ratio is opposite for the two boundary conditions.

トポロジー最適化による防音エンクロージャーの防音性能の向上に関する検討

トポロジー最適化による防音エンクロージャーの防音性能の向上に関する検討

Acoustic topology optimization of sound power using mapped acoustic radiation modes

An efficient approach for acoustic topology optimization to minimize the radiated sound power from a vibrating structure is described. The topology optimization is implemented by modifying the local stiffness at discrete locations on the surface of the structure. The radiated sound power level from the structure is directly chosen as the objective function to be minimized. A sensitivity analysis is then implemented to further optimize the layout of the locations of the modified local stiffness. To speed up the computational process, the radiated sound power is computed based on mapped acoustic radiation modes. To demonstrate the acoustic topology optimization using mapped acoustic radiation modes, the radiated sound power of a compressor housing is examined. Based on results from the numerical model, the local stiffness of a compressor housing was experimentally modified. Good agreement in sound power reduction obtained both numerically and experimentally was observed for the overall trend for the sound power levels as a function of one-third octave frequency bands.

덕트 밖에서 계산된 삽입 손실을 고려한 머플러 최적 설계

Abstract : In this study, we formulate an acoustical topology optimization problem to optimally design a partition layout inside the expansion chamber of a muffler. The lower-limit insertion loss value at a target frequency is constrained, and the partition volume is selected as an object function. In this study, we calculate the insertion loss outside the duct, while to determine the noise-attenuation performance, we use the insertion loss value calculated inside the duct or transmission loss value obtained in a previous study. We employ the finite-element model for acoustical analysis, and we determine the transmission of an incident acoustic wave through each finite element using the functions of design variables that change continuously between “0” and “1.” The rigid body elements, which totally reflect incident waves, build up partitions. Finally, we compare optimal topologies that depend on the target frequency and the allowed lower-limit value of insertion loss. § 이 논문은 대한기계학회 창립 70주년 기념 학술대회(2015. 11. 10-14., ICC제주) 발표논문임. † Corresponding Author, jinwoolee@ajou.ac.kr Ⓒ 2016 The Korean Society of Mechanical Engineers

Optimal partition layout of expansion chamber muffler with offset inlet/outlet

An optimal partition layout inside an expansion chamber muffler with an offset inlet/outlet is systematically designed by using topology optimization to achieve the desired characteristics in terms of acoustics and fluid mechanics. To that end, a partition volume minimization problem is formulated by applying acoustical and flow topology optimization methods. The partition volume is set as an objective function with constraints imposed on the target values of the transmission loss and pressure drop. The finite element method is employed for the acoustical and flow analyses. A design variable is assigned to each finite element such that it changes continuously between 0 and 1 to determine the state of the associated finite element. The design variables are updated during the optimization process and parameterized to converge to 0 or 1 at the end of the process. Finite elements with design variables of 1 build up rigid partitions which are optimally placed to achieve the target values of transmission loss and pressure drop. Different optimal partition layouts are obtained depending on the target frequency, the target values of transmission loss and pressure drop, and the initial values of the design variables. An experiment-based validation strongly supports the validity of the proposed muffler design method.

Two-step design process for optimal suction muffler in reciprocating compressor

A systematic design process for an optimal suction muffler is proposed to reduce the noise of a reciprocating compressor. Because the outer shape of a suction muffler is complicated, the well-known internal configuration for simple expansion chamber mufflers is not easily applicable to the suction muffler design problem. To achieve an optimal design of a suction muffler, two sequential optimization problems are formulated to maximize the transmission loss value at a target frequency: acoustical topology and shape optimization problems. The key idea in the suggested method is to use an optimal topology obtained by solving the topology optimization problem as an initial shape for the shape optimization problem. The formulated optimization problems are solved at several target frequencies, and the acoustical characteristics of the optimal shapes are closely investigated. The experimental results for two optimal suction mufflers support the validity of our suggested two-step design process for optimal suction mufflers.

Optimal rigid and porous material distributions for noise barrier by acoustic topology optimization

This research applies acoustic topology optimization (ATO) for noise barrier design with rigid and porous materials. Many researchers have investigated the pressure attenuation phenomena of noise barriers under various geometric, material, and boundary conditions. To improve the pressure attenuation performance of noise barriers, size and shape optimization have been applied, and ATO methods have been proposed that allow concurrent size, shape, and topological changes of rigid walls and cavities. Nevertheless, it is unusual to optimize the topologies of noise barriers by considering the pressure attenuation effect of a porous material. The present research develops a new ATO considering both porous and rigid materials and applies it to the discovery of optimal topologies of noise barriers composed of both materials. In the present approach, the noise absorption characteristics of porous materials are numerically modeled using the Delany–Bazley empirical material model, and we also investigate the effects of some interpolation functions on optimal material distributions. Applying the present ATO approach, we found some novel noise barriers optimized for various geometric and environmental conditions.

Optimal topology of reactive muffler achieving target transmission loss values: Design and experiment

A topology-optimization-based muffler design method for a reactive muffler is proposed and experimentally validated. In a reactive muffler design problem, rigid partitions should be located optimally inside the muffler to improve its acoustical attenuation performance in the target frequency range. In an optimal-performance muffler, the partition volume should be made as small as possible, and the transmission loss value in the target frequency range should be high enough for flow noise reduction in a duct. To this end, a partition-volume-minimization problem achieving target transmission loss values is formulated by using acoustical topology optimization. The formulated muffler design problem is solved for several target frequencies, and the effect of the initial values of the design variables on the optimal topology is investigated. Numerical simulation results show that the proposed formulation requires a smaller volume of partition than the previous topology-optimization-based formulation. The calculated transmission loss curves of the optimal mufflers agree well with the measured transmission loss curves of mufflers made of acrylic.

Acoustical topology optimization of Zwicker’s loudness with Padé approximation

Zwicker’s loudness is a conventional standard index for measuring human hearing annoyance and has been widely considered in many industrial fields for noise evaluations. The calculation of Zwicker’s loudness, which is needed for a wide range of frequency responses with a fine frequency resolution, using the finite element procedure usually requires significant computation time, since a numerical solution must be obtained for each considered frequency. Furthermore, if the analysis is the basis for an iterative optimization procedure such as a gradient-based acoustical topology optimization, this approach imposes prohibitively high computational costs. In this research, we propose a computationally-efficient approach to resolve the computational issue in the computation and optimization of Zwicker’s loudness. We present an efficient approach which combines the finite element method (FEM) with the Padé approximation (PA) procedure for obtaining Zwicker’s loudness and for applying it in a gradient-based acoustical topology optimization procedure applied to the design of acoustic devices to minimize Zwicker’s loudness. In this respect, the calculation of Zwicker’s loudness is represented by the PA. The main specific loudness considered as an objective function is evaluated using the PA procedure with a sufficient number of subintervals and expansion terms. An adjoint variable formulation for the design sensitivities that uses the advantages of the PA is used. We compare the performance of the proposed algorithm with the standard FEM in terms of accuracy and the CPU-time required for the calculation of Zwicker’s loudness. In addition, we also compare the optimized designs obtained by the proposed method to optimized designs obtained by the standard method in terms of objective values, optimized topology, and iterations or CPU-times needed for the optimization. Through several examples including 2-D and 3-D acoustics, the efficiency and reliability of using PA for computation and acoustical topology optimization of Zwicker’s loudness are compared and validated.

Acoustic topology optimization of fibrous material with Delany–Bazley empirical material formulation

This research details a new acoustic topology optimization (ATO) framework with an empirical material formulation for fibrous material. Despite the importance of considering pressure attenuation not only by internal solid structures but also by fibrous (porous) structures in acoustic design, a systematic ATO approach with an empirical material formulation has not yet been proposed. Thus, in this paper, an empirical material formulation called the Delany–Bazley model is implemented for the development of an ATO framework for fibrous material with porosity close to 1. By means of the SIMP (solid isotropic material with penalization) interpolation functions developed for multiple structural materials, ATO processes for fibrous structures as well as internal solid structures are carried out. In addition, a heuristic filter method that allows fibrous material to emerge only at the boundaries or rims of an internal solid structure is presented. Finally, the effect of the pressure attenuation on the topological layout for fibrous materials is investigated by solving several illustrative topology optimization examples.

Microstructural topology optimization with respect to sound power radiation

The paper deals with the problem of topological design of microstructure with respect to minimization of the sound power radiation from a vibrating macrostructure. The macrostructure is excited at a single or a band of excitation frequencies by a time-harmonic mechanical loading with prescribed amplitude and spatial distribution. The structural damping is considered to be proportional damping. The sound power is calculated using a high frequency approximation formulation and thus the sensitivity analysis may be performed in a very efficient manner. The microstructure composed of two different solid isotropic materials is assumed to be identical from point to point at the macro-level which implies that the interface between the structure and the acoustic medium is unchanged during the design process. The equivalent material properties of the macrostructure are calculated using homogenization method and the bi-material SIMP model is employed to achieve zero-one design at the micro-scale. Numerical examples are given to validate the model developed. Some interesting features of acoustic microstructure topology optimization are revealed and discussed.

Acoustical topology optimization for Zwicker’s loudness model – Application to noise barriers

Traditionally, the objective of design optimization of an acoustic system is to reduce physical acoustic properties, i.e., sound pressure and power. However, since these parameters are not sufficient to present the relation of physical sound stimulus with human perceptual judgment, physical acoustic properties may not represent adequate parameters for optimizing acoustic devices. In this paper, we first present a design method for acoustical topology optimization by considering human’s subjective conception of sound. To consider human hearing characteristics, Zwicker’s loudness is calculated according to DIN45631 (ISO 532B). The main objective of this work is to minimize the main specific loudness of a target critical band rate by optimizing the distribution of the reflecting material in a design domain. The Helmholtz equation is used to model acoustic wave propagation and, it is solved using the finite element method. The sensitivity of the main specific loudness is calculated using the adjoint variable method and the chain rule. To demonstrate the effectiveness of the proposed method, various examples of noise barriers are presented with different source and receiver locations. The results obtained, using the optimized noise barriers that consider Zwicker’s loudness, are compared with the results for straight and T-shaped barriers. The results are also compared with topology optimization using 1/3-octave band level as an objective function. The optimized noise barrier using the proposed method shows the best result with respect to a human’s hearing sensation.

Systematic design of reversal flow mufflers by topology optimization

A new muffler design method is suggested for systematic design of reversal flow mufflers. In the new method, a muffler design problem is reformulated as an acoustical topology optimization problem, where the transmission loss at the frequency of interest is maximized. A finite element model is employed for acoustical analysis, and one design variable is assigned to each finite element and changes continuously from zero to one. When the design variable becomes one, the associated finite element is filled with rigid body and an incident acoustic wave is fully reflected. The rigid bodies build up partitions, which improve the acoustical characteristics of flow-reversing chambers. When the design variable becomes zero, an incident acoustic wave is freely transmitted to the other side. Since the optimal location and length of the partitions are determined automatically by the suggested muffler design method, the internal configuration of the reversal flow mufflers does not depend on the designers' intuition and experiences. Several numerical results prove the feasibility of the suggested muffler design method.

Topology optimization of muffler internal partitions for improving acoustical attenuation performance

AbstractThe internal partition configuration of an expansion chamber muffler affects significantly its acoustical transmission characteristics, but the use of systematic optimization methods to muffler design problems is rare. The main objective of this research is to maximize the transmission loss at target frequencies by optimizing partition layouts inside a muffler chamber by formulating an acoustical topology optimization problem. The selected target frequencies include the deep frequencies of a nominal muffler in order to see the critical effects of partition configurations on the acoustical transmission characteristics. The effects of partition volume constraint ratios are also investigated and physics behind the optimized layouts is investigated. Numerical results show that mufflers with optimized partition layouts outperform nominal mufflers considerably, but the shapes and locations of the optimized partitions should be much different from those of conventional partitions. Copyright © 2009 John Wiley & Sons, Ltd.

Optimal distribution of holes in a partition interfacing two cavities for controlling the eigenfrequencies by acoustical topology optimization

When two acoustic cavities are connected through an opening, the acoustical characteristics of the resulting double cavity can be considerably affected by the opening. Typical examples include a passenger vehicle where a passenger compartment cavity is acoustically coupled with the trunk compartment cavity through small holes in the package tray. Although the locations and cross-sectional areas of these holes are known to significantly affect the acoustical characteristics, there is no systematic design method available to determine an optimal multi-hole distribution. Since the hole distribution problem can be formulated as a topology optimization problem, the eigenfrequencies of the double cavity system having a partition with holes can be controlled by an optimal multi-hole distribution solved in the topology optimization setting. After demonstrating that it is difficult to find satisfactory results by using only a single initial guess, significant efforts are made to develop a systematic procedure to generate a suitable set of initial guesses. The proposed method uses the so-called cross-modes that appear due to the presence of a hole. The obtained results are compared with those obtained by using randomly-generated initial guesses. In addition, the physics behind the optimized results, obtained by using the developed method, is explained by the concept of the added length associated with the cross-modes of evanescent waves. The effectiveness of the developed approach is tested with a three-dimensional simplified half-scaled car cavity connected to the car trunk cavity by a package tray with holes.

Rigid body modeling issue in acoustical topology optimization

For gradient-based acoustical topology optimization, the acoustical properties of an acoustic medium are interpolated. Although the lower limit values of the density and sound speed simulating a rigid body affect the accuracy of the final results significantly, an in-depth study on the selection of these limit values is rare. In the first part of this work, the importance of the proper choice of the limit values was demonstrated with an example to show that improperly-selected limit values can result in erroneous eigenvalues and eigenmodes of an optimized acoustic cavity. Then, by using two theoretical models representing typical air-rigid body configurations, the effects of the density and sound speed values of a simulated rigid body were investigated by the power transmission coefficient. From the findings of the theoretical investigation, a guideline for properly selecting the low limit values of the density and sound speed of a simulated rigid body is suggested. The validity of the guideline will be checked in numerical case studies.

이중 공동의 고유 주파수 최대/최소화를 위한 위상 최적화 기반 격벽 설계

The position and size of holes in the partition of a double cavity are known to strongly affect the eigenfrequency of the longitudinal eigenmodes of the double cavity. To maximize or minimize the eigenfrequency of the hole-partitioned double cavity, two acoustical topology optimization problems are formulated and solved. While two sub-cavities are filled with air, a partition between them is assumed to consist of sub-partitions of variable acoustical properties. One design variable is assigned to each sub-partition, whose material properties are interpolated as those of an intermediate material between air and a rigid body. The penalty parameter of the used interpolation function is adjusted to obtain a distinct air and rigid body distribution at the converged stage in each acoustical topology optimization problem. A special attention is paid to the selection of initial values of design variables to obtain solutions as close to global optimum and symmetric as possible. To show numerical characteristics of these optimization problems, the formulated problems are first solved for the one-dimensional partition design domain and then for the two-dimensional partition design domain.