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.

No description available.

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.

No description available.

Simulating electromagnetic waves is of increasing importance, for example, due to the rapidly growing demand of wireless communication in the fields of antenna design, photonics and electromagnetic compatibility (EMC). Many numerical and asymptotic techniques have been developed and one of the most common is the Finite-Difference Time-Domain (FDTD) method, also known as the Yee scheme. This centered difference scheme was introduced by Yee in 1966. The success of the Yee scheme is based on its relatively high accuracy, energy conservation and superior memory efficiency from the staggered form of defining unknowns. The scheme uses a structured Cartesian grid, which is excellent for implementations on modern computer architectures. However, the structured grid results in loss of accuracy due to general geometry of boundaries and material interfaces. A natural challenge is thus to keep the overall structure of Yee scheme while modifying the coefficients in the algorithm near boundaries and interfaces in order to improve the overall accuracy. Initial results in this direction have been presented by Engquist, Gustafsson, Tornberg and Wahlund in a series of papers. Our contributions are new formulations and extensions to higher dimensions. These new formulations give improved stability properties, suitable for longer simulation times. The development of the algorithmsis supported by rigorous stability analysis. We also tackle the problem of controlling the divergence free property of the solution—which is of extra importance in three dimensions—and present results of a number of numerical tests. / QC 20101101

Understanding the mechanics of turbulent fluid flow is of key importance for industry and society as for example in aerodynamics and aero-acoustics. The massive computational cost for resolving all turbulent scales in a realistic problem makes direct numerical simulation of the underlying Navier-Stokes equations impossible. Recent advances in adaptive finite element methods offer a new powerful tool in Computational Fluid Dynamics (CFD). The computational cost for simulating turbulent flow can be minimized where the mesh is adaptively resolved, based on a posteriori error control. These adaptive methods have been implemented for efficient serial computations, but the extension to an efficient parallel solver is a challenging task. This work concerns the development of an adaptive finite element method for modern parallel computer architectures. We present efficient data structures and data decomposition methods for distributed unstructured tetrahedral meshes. Our work also concerns an efficient parallellization of local mesh refinement methods such as recursive longest edge bisection. We also address the load balance problem with the development of an a priori predictive dynamic load balancing method. Current results are encouraging with almost linear strong scaling to thousands of cores on several modern architectures. / QC 20110223

This thesis proposes a way of simulating the continuous casting process of steel using Smoothed Particle Hydrodynamics (SPH). It deals with the SPH modeling of mass, momentum and the energy equations. The interpolation kernel functions required for the SPH modeling of these equations are calculated. Solidification is modeled by some particles are used to represent fluids and others solids. Elastic forces are calculated between the particle neighbors to create deformable bodies. The fluid solidifies into the elastic body when it cools down and the elastic body melts as it is heated. In continuous casting the molten metal solidifies forming a shell when it comes in contact with the cold wall. The mold of the continuous casting is modeled with a cold oscillating wall and a symmetric wall. Once the shell is formed water is sprayed on the solidified metal. If the shell is thin and cooling is not sufficient, the elastic body melts due to the effect of the hot fluid. / Den klassiska SPH-modellen för vätskor med fri yta kompletteras med värmeledning med fasomvandling och stelning: partiklar kan byta mellan vätske-tillstånd och solid-tillstånd beroende på temperaturen. Elastiska krafter beroende på avstånd mellan partiklarna aktiveras i solid-tillståndet och slås av i fluid-tillstånd så att vätskan kan stelna och senare smälta igen om så behövs. Vid stränggjutning stelnar smältan, som fylls på via ett rör, vid kontakt med en oscillerande, kall kokill-vägg, till ett elastiskt skal. Detta kyls fortlöpande genom påsprutning av vatten utanpå kokillen och direkt på skalet, som förångas. Skalet deformeras nedanför kokillen av det hydrostatiska trycket från smältan; om det ar för tunt brister det. Som demonstration gjordes en simulering där ett skal skapas, varpå man slår av vattenkylningen på ett parti: då smälter skalet och blir tunnare och till sist brister det och all smälta rinner ut genom hålet. Noggrannheten i simuleringen lämnar en del att önska men det vore mycket svårt att bygga en så komplex modell med vanlig CFD.

Molecular Dynamics has been a field using the most advanced computer systems for decades to perform simulations. Sophisticated methods with complicated potentials and parallelized implementations have been developed starting from simple models and a few hundred atoms within one simulation in the beginning. Nowadays, we see an increasing need for large-scale parallel simulations in molecular dynamics. Computer simulations are a cost-effective research tool. Furthermore, a shift to multi-core processors provides computer systems allowing a much higher degree of parallelism in applications. It is possible to use implementations on these systems, which are based on established programming models and techniques. Nevertheless, it is necessary to analyze the limitations of such implementations on significant larger systems then common until now. Such an analysis work is an important step to the development of new programming models that make efficient use of modern multi-core processors. This thesis project has been focused on the implementation and performance analysis of a tree code that is an important algorithm for molecular dynamic simulations. The tree code algorithm has been parallelized for distributed memory computer systems using MPI. The load-balancing applies space-filling curves for work decomposition. The performance of the implementation was tested finally with input data covering a large range of parameters like the number of processors in the jobs and the number of particles per processor.