Local control of sound in stochastic domains based on finite

A numerical method for optimizing the local control of sound in a stochastic domain is developed. A three-dimensional enclosed acoustic space, for example, a cabin with acoustic actuators in given locations is modeled using the (cid:28)nite element method in the frequency domain. The optimal local noise control signals minimizing the least square of the pressure (cid:28)eld in the silent region are given by the solution of a quadratic optimization problem. The developed method computes a robust local noise control in the presence of randomly varying parameters such as variations in the acoustic space. Numerical examples consider the noise experienced by a vehicle driver with a varying posture. In a model problem, a signi(cid:28)-cant noise reduction is demonstrated at lower frequencies.


Introduction
Machine generated noise is an increasing problem in modern working environments. Rotating and constantly moving parts such as wheels, engines and cooler fans are typical noise sources. Noise control applications are found especially in factory environment, engineering vehicles and passenger cars. It is possible to reduce noise signicantly by dierent methods, which are often classied as either active or passive techniques [4,13]. Probably the best situation would be to remove or reduce important noise source mechanisms by suitable design choices such that noise control measures would become unnecessary. In many cases, 1 however, this is not possible or the design is limited by other more important factors than noise.
Passive noise reduction by absorbing and insulating acoustic elements is effective for high frequency noise but typically less eective for low frequency noise, as long waves require large elements. On the other hand, active noise control (ANC) is most eective for low frequency noise. Active attenuation is based on generating antisound with actuators, such that original noise is canceled. The antisound must have the same amplitude as the noise to be canceled, but the opposite phase so that destructive interference occurs. If the noise contains both high and low frequency components, the best noise attenuation is obtained by combining both active and passive methods together. It is challenging to estimate the eectiveness of local sound control in a complicated three-dimensional domain like in passenger cars and other vehicles. In the passenger car, low frequency noise sources are mainly due to structural vibration from engine and tires [18]. Especially structure-borne noises are low frequency, whereas airborne noises often have higher frequencies. Tires cause high frequency noise due to aerodynamic phenomena. The mechanical vibratory noise from tires is mainly below 1 kHz. The most important noise components originating in the passenger car engine are below 500 Hz. The resonance of car cabin is also an important low frequency noise source. Accurate mathematical modeling of acoustics in such cabins is a formidable task [10].
As there are low-frequency noise sources, the local sound control can provide a signicant noise reduction to the car cabin environment. More advanced methods designing and assessing such systems employ numerical simulation and optimization. The mathematical side of these problems have been considered in [7,11,12], for example. Approaches using nite element modeling, are presented in articles [19,5,17]. In [19], resonance modes for mining vehicle are studied by modal coupling analysis and antinoise is optimized by using FEM model to obtain global noise control in the cabin. In [5], a local active noise control method based on the nite element method is described which minimizes noise locally in microphone locations. A method to determine the optimal locations for antinoise actuators is also presented. In [17], the locations of control sensors and actuators for global sound control are optimized based on nite element models for a complicated geometry. In [22], an optimal active noise control implementation based on quadratic programming and boundary element method (BEM) is presented.
Often it is necessary to control local sound in time varying domains. For example, parts of machinery move or like in this article, the driver of a car moves. It is convenient to use stochastic domains to model such changes in geometry instead of deterministic models. A vast amount of research has been performed on partial dierential equations (PDEs) with stochastic coecient, but PDEs in stochastic domains have had much less attention. In [21], a mapping from a random domain to a xed domain is used to transform the problem to be one with stochastic coecients. Fictitious domain approach is used in [6,16] and an extended nite element method is employed in [15] to treat stochastic domains. For the local sound control problem considered here, a noise measure needs to be computed for a small subdomain. The expected noise can be conveniently computed in a stochastic domain by integrating numerically the product of the noise and the probability distribution function over possible domains. Thus, a functional of the solution of a stochastic PDE is computed directly without approximating the stochastic solution. This is a non-intrusive approach, that is, a solution method for non-stochastic problems like the one in [1,2] can be employed without any modications.
Here, a novel modeling method for the local control of sound by antinoise actuators is introduced in stochastic domains. It can be used to assess the possibilities of active noise control in enclosed acoustic spaces such as vehicle cabins. The method is based on using acoustic nite element modeling. The antinoise is optimized by minimizing the expected value of the noise computed using the nite element method. By including the stochasticity of the cavity domain in the model the optimal performance of a local sound control can be determined more accurately and reliably than with earlier methods. The numerical example, optimization of local sound control in a car cabin model, shows the eciency of the presented method.
This article is organized as follows. In Section 2, a mathematical model of sound propagation, the Helmholtz partial dierential equation, and a numerical method to solve it are briey presented. In Section 3, the local noise control in a stochastic domain is formulated as a quadratic optimization problem. In Section 4, an example of local noise control in car driver's ears is described. In Section 5, the numerical results of ANC performance in three-dimensional car cabin problem are studied and analyzed. In Section 6, conclusions are given.

Acoustic model
The time harmonic sound propagation can be modeled by the Helmholtz equation where ρ (x) is the density of the material at location x, and c (x) is the speed of sound in the material. The complex pressurep (x) denes the amplitude and phase of the pressure. The sound pressure at time t is obtained as e −iωt p, where ω is the angular frequency of sound and i = √ −1.
A sound source f acting on a part S of the boundary ∂Ω is modeled via a boundary condition. A partially absorbing wall material is described by the impedance boundary conditions where η (x) is the absorption coecient depending on the properties of the surface material. The value η = 1 approximates a perfectly absorbing material and the value η = 0 approximates a sound-hard material (the Neumann boundary condition). An approximate solution for the partial dierential equation (PDE) Eq. (1) can be obtained using a nite element method [20]. The nite element discretization transforms Eq. (1) into a system of linear equations Ax = b, where the matrix A is generally symmetric, large, and sparse. Due to the large size and structure of A, direct solution methods are computationally too expensive. Instead an iterative solution methods like GMRES needs to be used. Solving the system with a reasonable number of iterations is, however, challenging as the matrix A is badly conditioned and especially so when the calculation domain is large and the frequency is high. In the numerical example in Section 5, the solutions are computed after the systems are preconditioned by a damped Helmholtz preconditioner described in [1,2]. 3 The noise control problem An acoustic model in an enclosed stochastic domain Ω (r) is considered, where r is a random vector that conforms to a known probability distribution F (r). The pressure p (x, r, γ) is the sum of the sound pressures caused by noise and n antinoise sources where the pressure amplitude p 0 is due to the noise source, p j is due to the jth antinoise source, and γ j is a complex coecient dening the amplitude and phase of the jth antinoise source. The noise and antinoise sources are located on the boundaries of Ω. The antinoise dened by the coecients γ j is optimized so that the noise is minimized in a subdomain denoted by Ξ (r) ⊂ Ω (r). For this, a noise measure is dened as where g (x) is a weighting function andp is the complex conjugate of p.
As the domain Ω is stochastic, the expected value of the noise measure is given by where F (r) is the probability distribution of r. The objective function J for optimization is chosen to be an approximation of the integral (5) and it is given by the numerical quadrature where the pairs (r j , w j ) give the quadrature points and weights. The optimization problem is dened as where Γ is the set of feasible controls. For example, in practical applications it is necessary to bound the amplitude of the antinoise sources.
In order to give the objective function in a compact form, the following notations are introduced: wherep is the elementwise complex conjugate of the vector p, and the superscripts T and H denote the transpose and the Hermitian conjugate, respectively. By expanding terms and by using the notations in Eq. (8), the objective function in Eq. (6) can be expressed in a compact form In the case that there are no constraints, that is, Γ = C n in (7), the optimal complex coecients γ i that give phases and amplitudes for antinoise actuators, are now given by the optimality condition ∇ γ J = 0. This leads to a system of linear equations Aγ = −b, which has the solution The case with constrains can be transformed to a real-valued optimization problem for a vector of length 2n consisting the real and imaginary parts of γ j s. The objective function has a quadratic form with a 2n × 2n symmetric and positive denite matrix. There are several ecient methods available for such optimization problems (see [14]), and they could be applied for this problem. For technical simplicity, however, we restrict this work to the case with no constraints, which leads to the solution of linear systems of the form (10).

Sound control in a car interior
As an example of application of the numerical method, noise control in BMW 330i car interior is studied, see Fig. 1. The interior of the car excluding the driver is the domain Ω (r). The objective of the noise control is to minimize noise in driver's ears. Thus, Ξ is dened as a set where e l (r) and e r (r) are the co-ordinates of the left and right ear, respectively. The noise measure in Eq. (4) has now the expression N (r, γ) = |p (e l , r, γ)| 2 + |p (e r , r, γ)| 2 .
It is assumed that there is only the driver and no other passengers or signicant objects in the car that would inuence the sound propagation. Driver's variable properties like shape and posture have an impact on reections and propagation of sound, so they must be taken into account. Especially the posture and position of head aect the sound heard by ears. As the posture varies to some extent, it is better to minimize the expected value of the sound level in ears. This leads to a stochastic domain in the computation. The driver is modeled by using the freely available Animorph library, that is based on ideas and algorithms presented in [3]. With Animorph, it is possible to model driver's geometry with a rich set of parameters changing the posture and shape. Three parameters are considered here: r 1 is driver's sideways bending angle, r 2 is forward bending angle, and r 3 is head rotation angle to left/right. These parameters are illustrated in Fig. 2. Now the random variable r = (r 1 , r 2, r 3 ) T determines the posture of the driver, where the value of each parameter is limited by condition L i < r i < H i , The probability distribution function F is given by a piecewise trilinear function dened by the nodal values on the lattice {L 1 , C 1 , H 1 }×{L 2 , C 2 , H 2 }× {L 3 , C 3 , H 3 } and elsewhere by trilinear interpolation. The integral in Eq. (12) is estimated by the three-dimensional generalization of the trapezoidal quadrature rule. The numerical integration of expression in Eq. (12) gives the objective function where w i is a weight coecient from the trapezoidal rule for the integral of the probability distribution function F and r i is the co-ordinate triplet of the ith quadrature point.
To evaluate the objective function in Eq. (13), the pressure amplitude caused by each noise and antinoise source is needed in ears for each driver sample r i . The acoustic reciprocity principle allows here a signicant computational saving. The principle says that the observation stays the same when the locations of sound source and observer are exchanged. For more details about the principle and its applications, see [9,8]. This is employed in the following way. First, a nite element model which has a point noise source at the ear e i is set up. Then, the pressure amplitude is studied in a noise or antinoise surface S. By the 7 reciprocity the following holds: the sound emitted by the source S measured at the ear e i has the same pressure amplitude as the sound emitted from the ear e i measured over the surface S. Thus, the sound pressure amplitude caused by many dierent sound sources can be resolved by just performing one simulation for each combination of sampled driver's posture, sampled frequency, and ear. The pressure amplitude heard at the ear e i is given by the integral where p S (e i ) is the sound pressure propagated from the surface S that is heard at the ear e i , f (x) is the force term for the sound source on S, and p ei (x) is the sound pressure propagated from the ear e i at the point x on the surface S. With a point antinoise source S, the integral in Eq. (14) is replaced by the point value at x = S.  To solve the Helmholtz equation in Eq. (1) with the nite element method, a collection of meshes consisting of linear tetrahedra and triangles were generated with Ansys ICEM CFD. Each mesh corresponds to dierent driver posture and they were generated so that there are at least 10 nodes per wavelength at f = 1000 Hz. The total number of meshes is 5 3 = 125 which is the number of parameter combinations (r 1 , r 2 , r 3 ).
The noise and antinoise sources are presented in Fig. 3. The noise source is modeled by a uniformly vibrating surface behind the leg room, which is a simplication of the real noise source. The antinoise sources are labeled as follows. Antinoise panels on the roof are labeled as Axx, where xx is 00, 01, 10, and 11. Point actuators are labeled as AxDOOR, AxBACK, APx2, APx5, APx7, APx9, APx12, APx14, where x is here L for the left side sources and R is for the right side sources. Actuators AxDOOR are located on front doors and actuators APxxx are located on front side window frames, see Fig. 3. On inner surfaces, the absorbing boundary condition in Eq. (2) is posed with the absorbency coecient η = 0.2.
The study was done in the frequency range 501000 Hz with 25 Hz steps. This means that 39 frequencies are sampled. By employing the reciprocity principle a sound source is placed in an ear. The acoustic model is solved for all 125 sampled driver's postures for both ears. Thus, discrete Helmholtz equations are solved 125 × 39 × 2 = 9750 times for the optimal antinoise control.

Actuator quality evaluation
It is possible to enhance the noise control by choosing good locations for antinoise actuators and by increasing their number. However, increasing the number of actuators also increases the costs and complexity of the noise control system. Thus, it is worthwhile to remove the actuators that have only minor contribution to the desired sound control. The graphs in Figs 46 study the quality of noise control and evaluate how each actuator contribute to the noise control quality. In Fig. 4, the expected value of the noise attenuation and its weighted standard deviation have been plotted at each driver's ear with dierent actuator combinations. By using two door loudspeakers (AxDOOR) as antinoise actuators, a satisfactory noise control is obtained within the engine noise frequency range, below 500 Hz, as Fig. 4 d shows. By this choice, however, the noise reduction result is not good at higher frequencies, although the expected value of the attenuation stays negative, i.e. noise is reduced. In Fig. 4 e, 12 additional point antinoise actuators have been placed on side window frames. By these additional actuators, a good attenuation of ca. 10 dB is obtained over the whole studied frequency range.
In Figs 4 a and c, it can be seen that by using only planar actuators on the roof (Axx), the attenuation at low frequencies is not good, but at higher frequencies (700900 Hz) it is reasonable. When comparing Figs 4 c and d, it is clear that the planar roof actuator (Axx) together with the side door actuators (Ax-DOOR) is signicantly better than the side door actuators (AxDOOR) alone. The attenuation prole is more at, and even at high frequencies (6001000 Hz) more than 5 dB expected attenuation is obtained.
If the antinoise itself is very loud, it may cause high sound pressure levels in some parts of the car cabin. By good placement, the amplitude of each actuator can be kept comfortable. In Fig. 5, the amplitude of each antinoise actuator is plotted to evaluate the actuator selections. Fig. 5 a shows that the amplitude of planar roof loudspeakers (Axx) is over 10 dB louder than the amplitude of front door loudspeakers (AxDOOR), especially at high frequencies. In Fig. 5 b, it is seen that the amplitude of back window loudspeakers (AxBACK) is signicantly lower than the amplitude of front door loudspeakers (AxDOOR). However, when comparing Figs 4 d and f, it is seen that the contribution of back window loudspeakers (AxBACK) is insignicant for the noise control. In Fig. 6, the contribution of each actuator to the noise control is plotted in the following way. The noise levels are compared in both ears when the chosen actuator is enabled and when it is disabled, i.e. γ i = 0. For each examined frequency, the worst attenuation result is selected from left or right ear. From Fig. 6, the benet of each actuator can be evaluated. As already has been suggested, it is seen from Fig. 6 Figure 6: The contribution of each actuator to the noise control system. The noise level is compared between the case when the inspected actuator is enabled and when it is disabled, i.e. γ i = 0. For each examined frequency, the worst attenuation is selected from left or right ear and it is plotted in the graph in dB. The antinoise actuators are used as follows: (a) Axx, (b) AxDOOR, AxBACK, Axx, (c) AxDOOR, Axx, (d) AxDOOR, AxBACK.

Attenuation plots
In Fig. 7, there are example plots of the attenuation when the driver is at different postures. The two front door loudspeakers (AxDOOR), the back window (AxBACK) and the planar roof (Axx) loudspeakers are used as the antinoise actuators. When the frequency is less than 400 Hz, there is more than 10 dB attenuation in almost every posture. At frequencies higher than 400 Hz, there is mostly signicant, over 5 dB attenuation, but there are also occasional postures that lead to noise amplication, i.e. additive interference of sounds. However, strong noise peaks are unlikely and on average the noise is reduced signicantly.   8 demonstrates the eect of active noise control with two actuators (AxDOOR) at single frequency f = 300 Hz. It is seen that the method reduces noise eectively near the ears and also in a wider region around the ears. At higher frequencies, the silent area is smaller and the noise is increased in other parts of the car. Figure 8: The noise control at frequency f = 300 Hz for the basic driver's parameters r 1 = r 2 = r 3 = 0. The modulus of pressure amplitude |p| is plotted on the logarithmic color scale. On left plots, the acoustic eld is depicted without noise control. On right plots, the noise control is enabled. The attenuation at both ears in this case is ca. −30 dB. The front door (AxDOOR) loudspeakers are used in the noise control.

Conclusions
A novel method is introduced to assess the eectiveness of the optimal antinoise for local sound control in a stochastic domain. The acoustic modeling is performed in the frequency domain using a sequence of nite element discretizations of Helmholtz equations. The optimization of antinoise is performed by minimizing the expected value of the noise at each frequency. This leads to a robust and accurate noise control in varying domains.
The sound control in a car interior with a driver in varying postures is considered as an example and numerical results are presented. A good attenuation noise is obtained for probable postures at lower frequencies, say, below 300 Hz. At higher frequencies, the noise reduction can be improved by increasing the number of actuators. 14