A preconditioner defined by an algebraic multigrid cycle for a damped Helmholtz operator is proposed for the Helmholtz equation. This approach is well-suited for acoustic scattering problems in complicated computational domains and with varying material properties. The spectral properties of the preconditioned systems and the convergence of the GMRES method are studied with linear, quadratic, and cubic finite element discretizations. Numerical experiments are performed with two-dimensional problems describing acoustic scattering in a cross section of a car cabin and in a layered medium. Asymptotically the number of iterations grows linearly with respect to the frequency while for lower frequencies the growth is milder. The proposed preconditioner is particularly effective for low-frequency and mid-frequency problems.

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This paper considers the combined use of a hybrid numerical method for the modeling of acoustic mufflers and a genetic algorithm for multiobjective optimization. The hybrid numerical method provides accurate modeling of sound propagation in uniform waveguides with non-uniform obstructions. It is based on coupling a wave based modal solution in the uniform sections of the waveguide to a finite element solution in the non-uniform component. Finite element method provides flexible modeling of complicated geometries, varying material parameters, and boundary conditions, while the wave based solution leads to accurate treatment of non-reflecting boundaries and straightforward computation of the transmission loss (TL) of the muffler. The goal of optimization is to maximize TL at multiple frequency ranges simultaneously by adjusting chosen shape parameters of the muffler. This task is formulated as a multiobjective optimization problem with the objectives depending on the solution of the simulation model. NSGA-II genetic algorithm is used for solving the multiobjective optimization problem. Genetic algorithms can be easily combined with different simulation methods, and they are not sensitive to the smoothness properties of the objective functions. Numerical experiments demonstrate the accuracy and feasibility of the model-based optimization method in muffler design.

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 nite element method in the frequency domain. The optimal local noise control signals minimizing the least square of the pressure 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 signicant noise reduction is demonstrated at lower frequencies.

An optimal local active noise control method based on stochastic finite element models Airaksinen, T.; Toivanen, J.Airaksinen, T. & Toivanen, J. (2013). An optimal local active noise control method based on stochastic finite element models. Journal of Sound and Vibration, 332 (2013) pp.6924-6933. doi:10.1016/j.jsv.2013 AbstractA new method is presented to obtain a local active noise control that is optimal in stochastic environment. The method uses numerical acoustical modeling that is performed in the frequency domain by using a sequence of finite element discretizations of the Helmholtz equation. The stochasticity of domain geometry and primary noise source is considered. Reference signals from an array of microphones are mapped to secondary loudspeakers, by an off-line optimized linear mapping. The frequency dependent linear mapping is optimized to minimize the expected value of error in a quiet zone, which is approximated by the numerical model and can be interpreted as a stochastic virtual microphone. A least squares formulation leads to a quadratic optimization problem. The presented active noise control method gives robust and efficient noise attenuation, which is demonstrated by a numerical study in a passenger car cabin. The numerical results demonstrate that a significant, stable local noise attenuation of 20-32 dB can be obtained at lower frequencies (< 500 Hz) by two microphones, and 8-36 dB attenuation at frequencies up to 1000 Hz, when 8 microphones are used.

Processes that can be modeled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmoltz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Laplacian operator; i.e. the damped Helmholtz operator as a preconditioner. The second approach returns to a transient wave equation, and finds the time-periodic solution using a controllability technique. We concentrate on acoustic problems, but our methods can be used for other types of Helmholtz problems as well. Numerical experiments show that the control method takes more CPU time, whereas the shifted-Laplacian method has larger memory requirement.

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