The computers of today make it possible to do direct simulation of aeroacoustics, which is very computational demanding since a very high resolution is needed. In the present thesis we study issues of relevance for aeroacoustic simulations. Paper A considers standard high order difference methods. We study two different ways to apply boundary conditions in a stable way. Numerical experiments are done for the 1D linearized Euler equations. In paper B we develop difference methods which give smaller dispersion errors than standard central difference methods. The new methods are applied to the 1D wave equation. Finally in Paper C we apply the new difference methods to aeroacoustic simulations based on the 2D linearized Euler equations. Taken together, the methods presented here lead to better approximation of the wave number, which in turn results in a smaller L2-error than obtained by previous methods found in the literature. The results are valid when the problem is not fully resolved, which usually is the case for large scale applications.

We study robust, preconditioned, iterative solution methods for large-scale linear systems of equations, arising from different applications in geophysics and geotechnics. The first type of linear systems studied here, which are dense, arise from a boundary element type of discretization of crack propagation in brittle material. Numerical experiment show that simple algebraic preconditioning strategies results in iterative schemes that are highly competitive with a direct solution method. The second type of algebraic systems are nonsymmetric and indefinite and arise from finite element discretization of the partial differential equations describing the elastic part of glacial rebound processes. An equal order finite element discretization is analyzed and an optimal stabilization parameter is derived. The indefinite algebraic systems are of 2-by-2-block form, and therefore block preconditioners of block-factorized or block-triangular form are used when solving the indefinite algebraic system. There, the required Schur complement is approximated in various ways and the quality of these approximations is compared numerically. When the block preconditioners are constructed from incomplete factorizations of the diagonal blocks, the iterative scheme show a growth in iteration count with increasing problem size. This growth is stabilized by replacing the incomplete factors with an inner iterative scheme with a (nearly) optimal order multilevel preconditioner.

The aim of this study is to develop numerical techniques for the analysis and optimization of acoustic horns for time harmonic wave propagation. An acoustic horn may be viewed as an impedance transformer, designed to give an impedance matching between the feeding waveguide and the surrounding air. When modifying the shape of the horn, the quality of this impedance matching changes, as well as the angular distribution of the radiated wave in the far field (the directivity). The dimensions of the horns considered are in the order of the wavelength. In this wavelength region the wave physics is complicated, and it is hard to apply elementary physical reasoning to enhance the performance of the horn. Here, topology optimization is applied to improve the efficiency and to gain control over the directivity of the acoustic horn.

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In this master thesis, a thermal design optimization has been performed. By solvingthe two dimensional steady state conduction convection equation using the finiteelement method in a unit square domain with a source term corresponding tocomponents heated by resistive heating, the objective functional was formulated as aminimization of the combination of a low temperature and small temperaturedifferences inside the domain. The design parameters are based on geometricproperties such as length, width, angle or position of the heat sources. The heatsources were parameterized by combining two smooth exponential functions thatexplicitly depended on the position and size of the heat source. The problem wasthen solved as a PDE constrained optimization problem using MATLAB's built infunction fmincon. Three different 1D test cases were implemented to investigate how the solverbehaved and that the parameterization was correctly implemented. Then the solverwas extended to 2D and three heat sources were placed in the domain. The optimalangle of rotation of the sources where the heat transfer was governed by conductionand convection were found. This was followed by an optimal placement of two heatsources in the domain. Three cases with a different convective field in each case wereinvestigated. In the last examples, four heat sources were placed inside the domain.One geometric property of each heat source was allowed to change. The fourdifferent parameters were length, width, angle of rotation and position. Themotivation was to test the functionality of the solver using different design parameterswith different sensitivities. The results showed that the derived objective functional fullfilled the purpose tominimize the temperature and temperature deviation from the mean temperature,respectively. In the 1D cases it was concluded that there exist several local minimawhen adding a heat source and a heat sink of unequal magnitude. Optimal angles ofthree heat sources in 2D showed a trivial solution and fast convergence. The optimalplacement of two heat sources converged rapidly when the forced convection was setto zero. When adding convection the number of iterations increased and the optimalplacement was highly dependent of the type of convective field and boundaryconditions. When constructing a non symmetric problem the optimization loopedover several random initial positions in order to find the best optimal solution. Forthe last examples, narrow bounds were used and the solver converged rapidly. Evenhere, the type of convective field highly affected the optimal solution.

This thesis presents a method for planning patient intake and assignment of treatment personnel at rehabilitative care clinics. The rehabilitation process requires patients to undergo a series of treatments spanning several weeks, requiring therapists of different disciplines. We have developed a two stage mixed-integer programming model which plans when each admitted patient will receive treatment and assigns therapists. In addition, the model provides support to decide when to admit new patients and when to hire additional staff in order to maximise the clinic’s patient throughput. Numerical results based on a real rehabilitation clinic are presented and discussed.

We study the following well-established model of reaction-diffusion type for brain tumour growth: <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cbegin%7Bequation%7D%0A%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Brcll%7D%0A%20%20%5Cpartial_%7Bt%7Du%20-%20div%20(D(x)%20%5Cnabla%20u)%20-%20f(u)%20&amp;=&amp;%200,&amp;%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%5Cmbox%7Bin%20%7D%5COmega%5Ctimes(0,T)%5C%5C%0A%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20u(0)%20&amp;%20=%20&amp;%5Cvarphi,&amp;%5Cmbox%7Bin%20%7D%5COmega%5C%5C%0AD%5Cnabla%20u%5Ccdot%20n%20&amp;=&amp;0,&amp;%20%5Cmbox%7Bon%20%7D%5Cpartial%5COmega%5Ctimes(0,T)%0A%20%5Cend%7Barray%7D%5Cright.%0A%20%5Cnonumber%0A%5Cend%7Bequation%7D" /> This equation describes the change over time of the normalised tumour cell density u as a consequence of two biological phenomena: proliferation and diffusion. We discuss a mathematical method for the inverse problem of locating the brain tumour source (origin) based on the reaction-diffusion model. Our approach consists in recovering the initial spatial distribution of the tumour cells <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Cvarphi=u(0)" /> starting from a later state <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Cpsi=u(T)" />, which can be given by a medical image. We use the nonlinear Landweber regularization method to solve the inverse problem as a sequence of well-posed forward problems. We give full 3-dimensional simulations of the tumour in time on two types of data, the 3d Shepp-Logan phantom and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repository (IBSR). These simulations are obtained using standard finite difference discretisation of the space and time-derivatives, generating a simplistic approach that performs well. We also give a variational formulation for the model to open the possibility of alternative derivations and modifications of the model. Simulations with synthetic images show the accuracy of our approach for locating brain tumour sources.

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Radial basis function (RBF) methods are meshfree, i.e., they can operate on unstructured node sets. Because the only geometric information required is the pairwise distance between the node points, these methods are highly flexible with respect to the geometry of the computational domain. The RBF approximant is a linear combination of translates of a radial function, and for PDEs the coefficients are found by applying the PDE operator to the approximant and collocating with the right hand side data. Infinitely smooth RBFs typically result in exponential convergence for smooth data, and they also have a shape parameter that determines how flat or peaked they are, and that can be used for accuracy optimization. In this thesis the focus is on global RBF collocation methods for PDEs, i.e., methods where the approximant is constructed over the whole domain at once, rather than built from several local approximations. A drawback of these methods is that they produce dense matrices that also tend to be ill-conditioned for the shape parameter range that might otherwise be optimal. One current trend is therefore to use over-determined systems and least squares approximations as this improves stability and accuracy. Another trend is to use localized RBF methods as these result in sparse matrices while maintaining a high accuracy. Global RBF collocation methods together with RBF interpolation methods, however, form the foundation for these other versions of RBF--PDE methods. Hence, understanding the behaviour and practical aspects of global collocation is still important. In this thesis an overview of global RBF collocation methods is presented, focusing on different versions of global collocation as well as on method properties such as error and convergence behaviour, approximation behaviour in the small shape parameter range, and practical aspects including how to distribute the nodes and choose the shape parameter value. Our own research illustrates these different aspects of global RBF collocation when applied to the Helmholtz equation and the Black-Scholes equation.

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