Erkki Heikkola

An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation

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.

Robust Formulations for Training Multilayer Perceptrons

The connection between robust statistical estimates and nonsmooth optimization is established. Based on the resulting family of optimization problems, robust learning problem formulations with regularization-based control on the model complexity of the multilayer perceptron network are described and analyzed. Numerical experiments for simulated regression problems are conducted, and new strategies for determining the regularization coefficient are proposed and evaluated.

A Parallel Fictitious Domain Method for the Three-Dimensional Helmholtz Equation

The application of a fictitious domain (domain embedding) method to the three-dimensional Helmholtz equation with absorbing boundary conditions is considered. The finite element discretization is performed by using locally fitted meshes, and an algebraic fictitious domain method with a separable preconditioner is used in the iterative solution of the resultant linear systems. Such a method is based on embedding the original domain into a larger one with a simple geometry. With this approach, it is possible to realize the GMRES iterations in a low-dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner. An efficient parallel implementation of the iterative algorithm is introduced. Results of numerical experiments demonstrate good scalability properties on distributed-memory parallel computers and the ability to solve high frequency acoustic scattering problems.

On Computation of Scattering Matrices and on Surface Waves for Diffraction Gratings

Summary. A method for the numerical computation of the so-called augmented scattering matrices (ASM) is suggested for diffraction gratings. To construct such (unitary) matrices one has to take into account not only the oscillating modes but also those which exponentially grow (attenuate) in amplitude away from the grating. The method uses an optimization procedure to identify the coefficients in the asymptotics of such modes. A justification of the approach is given and its numerical implementation is discussed. Reliable numerical results allow us to study the occurrences of surface waves by means of a general existence criterion based on the properties of ASM. To illustrate the method we give some examples of surface waves in gratings.

Assessment of the Fundamental Flexural Guided Wave in Cortical Bone by an Ultrasonic Axial-Transmission Array Transducer

The fundamental flexural guided wave (FFGW), as modeled, for example, by the A0 Lamb mode, is a clinically useful indicator of cortical bone thickness. In the work described in this article, we tested so-called multiridge-based analysis, based on the crazy climber algorithm and short-time Fourier transform, for assessment of the FFGW component recorded by a clinical array transducer featuring a limited number of elements. Methods included numerical finite-element simulations and experiments in bone phantoms and human radius specimens (n = 41). The proposed approach enabled extraction of the FFGW component and determination of its group velocity. This group velocity was in good agreement with theoretical predictions and possessed reasonable sensitivity to cortical width (r(2) = 0.51, p < 0.001) in the in vitro experiments. It is expected that the proposed approach enables related clinical application. Further work is still needed to analyze in more detail the challenges related to the impact of the overlying soft tissue.

Controllability method for the Helmholtz equation with higher-order discretizations

We consider a controllability technique for the numerical solution of the Helmholtz equation. The original time-harmonic equation is represented as an exact controllability problem for the time-dependent wave equation. This problem is then formulated as a least-squares optimization problem, which is solved by the conjugate gradient method. Such an approach was first suggested and developed in the 1990s by French researchers and we introduce some improvements to its practical realization. We use higher-order spectral elements for spatial discretization, which leads to high accuracy and lumped mass matrices. Higher-order approximation reduces the pollution effect associated with finite element approximation of time-harmonic wave equations, and mass lumping makes explicit time-stepping schemes for the wave equation very efficient. We also derive a new way to compute the gradient of the least-squares functional and use algebraic multigrid method for preconditioning the conjugate gradient algorithm. Numerical results demonstrate the significant improvements in efficiency due to the higher-order spectral elements. For a given accuracy, spectral element method requires fewer computational operations than conventional finite element method. In addition, by using higher-order polynomial basis the influence of the pollution effect is reduced.

Time-harmonic elasticity with controllability and higher-order discretization methods

The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions. The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situated at the Gauss-Lobatto quadrature points of the elements, and the Gauss-Lobatto quadrature rule is used so that the mass matrix becomes diagonal. This method is combined with the second-order central finite difference or the fourth-order Runge-Kutta time discretization. As a consequence of these choices, only matrix-vector products are needed in time dependent simulation. This makes the controllability method computationally efficient.

A Domain Embedding Method for Scattering Problems with an Absorbing Boundary or a Perfectly Matched Layer

The efficient numerical solution of the exterior Helmholtz equation modeling the acoustic scattering by an obstacle is considered. The exterior problem is approximated by truncating the domain with a rectangular boundary and using either a perfectly matched layer or a second-order absorbing boundary condition to reduce reflections from the artificial boundary. The scattering problem is solved iteratively by using a domain embedding method with an efficient separable preconditioner. With this approach, it is possible to realize the GMRES iterations in a low-dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner. The accuracy and the efficiency of the proposed solution technique are studied with numerical experiments.

Multiobjective muffler shape optimization with hybrid acoustics modeling

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.

Local control of sound in stochastic domains based on finite element models

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