Tomi Huttunen

Solving Maxwell's equations using the ultra weak variational formulation

We investigate the ultra weak variational formulation for simulating time-harmonic Maxwell problems. This study has two main goals. First, we introduce a novel derivation of the UWVF method which shows that the UWVF is an unusual version of the standard upwind discontinuous Galerkin (DG) method with a special choice of basis functions. Second, we discuss the practical implementation of an electromagnetic UWVF solver. In particular, we propose a method to avoid the conditioning problems that are known to hamper the use of the UWVF for problems in general geometries and inhomogeneous media. In addition, we show how to implement the PML in the UWVF to accurately approximate physically unbounded problems and discuss the parallelization of the UWVF. Three-dimensional numerical simulations are used to examine the feasibility of the UWVF for simulating wave propagation in inhomogeneous media and scattering from complex structures.

Scanning path optimization for ultrasound surgery.

One of the problems in ultrasound surgery is the long treatment times when large tumour volumes are sonicated. Large tumours are usually treated by scanning the tumour volume using a sequence of individual focus points. During the scanning, it is possible that surrounding healthy tissue suffers from undesired temperature rise. The selection of the scanning path so that the tumour volume is treated as fast as possible while temperature rise in healthy tissue is minimized would increase the efficiency of ultrasound surgery. The main purpose of this paper is to develop a computationally efficient method which optimizes the scanning path. The optimization algorithm is based on the minimum time formulation of the optimal control theory. The developed algorithm uses quadratic cost criteria to obtain the desired thermal dose in the tumour region. The derived method is evaluated with numerical simulations in 3D which are applied to ultrasound surgery of the breast in simplified geometry. Results from the simulations show that the treatment time as well as the total applied energy can be decreased from 16% to 43% as compared to standard sonication. The robustness of the optimized scanning path is studied by varying the perfusion and absorption in the tumour region.

Determination of heterogeneous thermal parameters using ultrasound induced heating and MR thermal mapping

In this paper, a method for the determination of spatially varying thermal conductivity and perfusion coefficients of tissue is proposed. The temperature evolution in tissue is modelled with the Pennes bioheat equation. The main motivation here is a model-based optimal control for ultrasound surgery, in which the tissue properties are needed when the treatment is planned. The overview of the method is as follows. The same ultrasound transducers, which are eventually used for the treatment, are used to inflict small temperature changes in tissue. This temperature evolution is monitored using MR thermal imaging, and the tissue properties are then estimated on the basis of these measurements. Furthermore, an approach to choose transducer excitations for the determination procedure is also considered. The purpose of this paper is to introduce a method and therefore simulations are used to verify the method. Furthermore, computations are accomplished in a 2D spatial domain.

SIMULATION OF THE TRANSFER FUNCTION FOR A HEAD-AND-TORSO MODEL OVER THE ENTIRE AUDIBLE FREQUENCY RANGE

In this study, a method for simulating the transfer function of a head-and-torso model over the entire audible frequency range is introduced. The simulation method uses the ultra-weak variational formulation (UWVF) which is a finite element type method tailored for wave problems. In particular, the UWVF uses plane wave basis functions which better approximate the oscillatory field than a polynomial basis used in the standard finite element methods (FEM). This leads to reduction in the computational complexity at the high frequencies which, accompanied with parallel computing, extends the feasible frequency range of the UWVF method. The accuracy of the new simulation tool is investigated using a simple spherical geometry after which the method used for preliminary HRTF simulations in the geometry of a widely used head-and-torso mannequin.

COMPUTATIONAL ASPECTS OF THE DISCONTINUOUS GALERKIN METHOD FOR THE WAVE EQUATION

The Discontinuous Galerkin (DG) method is a powerful tool for numerically simulating wave propagation problems. In this paper, the time-dependent wave equation is solved using the DG method for spatial discretization; and the Crank–Nicolson and fourth-order explicit, singly diagonally implicit Runge–Kutta methods, and, for reference, the explicit Runge–Kutta method, were used for time integration. These simulation methods were studied using two-dimensional numerical experiments. The aim of the experiments was to study the effect of the polynomial degree of the basis functions, grid density, and the Courant–Friedrichs–Lewy number on the accuracy of the approximation. The sensitivity of the methods to distorted finite elements was also examined. Results from the DG method were compared with those computed using a conventional finite element method. Three different model problems were considered. In the first experiment, wave propagation in a homogeneous medium was studied. In the second experiment, the scattering and propagation of waves in an inhomogeneous medium were investigated. The third experiment evaluated wave propagation in a more complicated domain involving multiple scattering waves. The results indicated that the DG method provides more accurate solutions than the conventional finite element method with a reduced computation time and a lower number of degrees of freedom.

Estimation of aquifer dimensions from passive seismic signals with approximate wave propagation models

Recently, it has been proposed that spontaneous seismic activity could be used in the estimation of hydrological parameters of aquifers such as permeability and storage. Approximate wave propagation models such as ray tracing, which are commonly used in hydrological parameter estimation with active sources and backscattering geometry, are not feasible with passive seismological imaging. With respect to full wave propagation models, the most accurate known model for aquifers is the poroelastic model while bedrock is usually modelled as an elastic medium. Using a poroelastic model in the forward model can be a computationally impractical choice. In this paper, we carry out a feasibility study in which we attempt to estimate the aquifer depth and water table using a highly approximate elastic model also for the aquifer. We adopt the Bayesian approximation error approach in which a statistical model is constructed for the errors that are induced by using model approximations such as sparse meshing and simplified physical models. We consider the problem in a simple two-dimensional geometry and show that straightforward adoption of approximate models leads to inconsistent parameter estimates, that is, the true parameters have essentially vanishing posterior density. On the other hand, using the Bayesian approximation error approach, the parameter estimates are consistent.

A non-uniform basis order for the discontinuous Galerkin method of the 3D dissipative wave equation with perfectly matched layer

In this study a discontinuous Galerkin method (DG) for solving the three-dimensional time-dependent dissipative wave equation is investigated. In the case of unbounded problems, the perfectly matching layer (PML) is used to truncate the computational domain. The aim of this work is to investigate a simple selection method for choosing the basis order for elements in the computational mesh in order to obtain a predetermined error level. The selection method studied here relies on the error estimates provided by Ainsworth [M. Ainsworth, Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, Journal of Computational Physics 198(1) (2004) 106-130]. The performance of the non-uniform basis is examined using numerical experiments. In the simulated model problems, a feasible method choosing the basis order for arbitrary sized elements is achieved. In simulations, the effect of dissipation and the choices of the PML parameters on the performance of the DG method are also investigated.

Bayesian approximation error approach in full-wave ultrasound tomography

In ultrasound tomography, the spatial distribution of the speed of sound in a region of interest is reconstructed based on transient measurements made around the object. The computation of the forward problem (the full-wave solution) within the object is a computationally intensive task and can often be prohibitive for practical purposes. The purpose of this paper is to investigate the feasibility of using approximate forward solvers and the partial recovery from the related errors by employing the Bayesian approximation error approach. In addition to discretization error, we also investigate whether the approach can be used to reduce the reconstruction errors that are due to the adoption of approximate absorbing boundary models. We carry out two numerical studies in which the objective is to reduce the computational times to around 3% of the time that would be required by a numerically accurate forward solver. The results show that the Bayesian approximation error approach improves the reconstructions.

Optimal control in high intensity focused ultrasound surgery

When an ultrasound wave is focused in biological tissue, a part of the energy of the wave is absorbed and turned into heat. This phenomena is used as a distributed heat source in ultrasound surgery, in which the aim is to destroy cancerous tissue by causing thermal damage. The main advantages of the ultrasound surgery are that it is noninvasive, there are no harmful side effects and spatial accuracy is good. The main disadvantage is that the treatment time is long for large cancer volumes when current treatment techniques are used. This is due to the undesired temperature rise in healthy tissue during the treatment. The interest for optimization of ultrasound surgery has been increased recently. With proper mathematical models and optimization algorithms the treatment time can be shortened and temperature rise in tissues can be better localized. In this study, two alternative control procedures for thermal dose optimization during ultrasound surgery are presented. In the first method, the scanning path between individual foci is optimized in order to decrease the treatment time. This method uses the prefocused ultrasound fields and predetermined focus locations. In the second method, combined feedforward and feedback controls are used to produce desired thermal dose in tissue. In the feedforward part, the phase and amplitude of the ultrasound transducers are changed as a function of time to produce the desired thermal dose distribution in tissue. The foci locations do not need to be predetermined. In addition, inequality constraint approximations for maximum input amplitude and maximum temperature can be used with the proposed method. The feedforward control is further expanded with a feedback controller which can be used during the treatment to compensate the modeling errors. All of the proposed control methods are tested with numerical simulations in 2D or 3D.

The ultra weak variational formulation of thin clamped plate problems

We develop a new numerical scheme for a fourth order elliptic partial differential equation based on Kirchhoff@?s thin plate theory. In particular we extend the ultra weak variational formulation (UWVF) to thin plate problems with clamped plate boundary conditions. The UWVF uses a finite element mesh and non-polynomial basis functions. After deriving the new method we then prove L^2 norm convergence on the boundary. Finally we investigate numerically the feasibility of the UWVF for both homogeneous and inhomogeneous problems and show examples of p- and h-convergence.