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• 1.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Non-linearstates in parallel Blasius boundary layer2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

There is large theoretical, experimental and numerical interest in studying boundary layers, which develop around any body moving through a fluid. The simplest of these boundary layers lead to the theoretical abstraction of a so-called Blasius boundary layer, which can be derived under the assumption of a flat plate and zero external pressure gradient. The Blasius solution is characterised by a slow growth of the boundary layer in the streamwise direction. For practical purposes, in particular related to studying transition scenarios, non-linear finite-amplitude states (exact coherent states, edge states), but also for turbulence, a major simplification of the problem could be attained by removing this slow streamwise growth, and instead consider a parallel boundary layer. Parallel boundary layers are found in reality, e.g. when applying suction (asymptotic suction boundary layer) or rotation (Ekman boundary layer), but not in the Blasius case. As this is only a model which is not an exact solution to the Navier-Stokes (or boundary-layer) equations, some modifications have to be introduced into the governing equations in order for such an approach to be feasible. Spalart and Yang introduced a modification term to the governing Navier-Stokes equations in 1987. In this thesis work, we adapted the amplitude of the modification term introduced by Spalart and Yang to identify the nonlinear states in the parallel Blasius boundary layer. A final application of this modification was in determining the so-called edge states for boundary layers, previously found in the asymptotic suction boundary layer

fulltext
• 2. Abdulle, Assyr
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Localized orthogonal decomposition method for the wave equation with a continuum of scales2017In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 86, no 304, p. 549-587Article in journal (Refereed)

This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an L2-projection. We derive explicit convergence rates of the method in the L∞(L2)-, W1,∞(L2)-and L∞(H1)-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.

• 3.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
GPU Monte Carlo scatter calculations for Cone Beam Computed Tomography2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

A GPU Monte Carlo code for x-ray photon transport has been implemented and extensively tested. The code is intended for scatter compensation of cone beam computed tomography images.

The code was tested to agree with other well known codes within 5% for a set of simple scenarios. The scatter compensation was also tested using an artificial head phantom. The errors in the reconstructed Hounsfield values were reduced by approximately 70%.

Several variance reduction methods have been tested, although most were found infeasible on GPUs. The code is nonetheless fast, and can simulate approximately 3 ·109 photons per minute on a NVIDIA Quadro 4000 graphics card. With the use of appropriate filtering methods, the code can be used to calculate patient specific scatter distributions for a full CBCT scan in approximately one minute, allowing scatter reduction in clinical applications.

fulltext
• 4.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Computational methods for microfluidics2013Licentiate thesis, comprehensive summary (Other academic)

This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

The first paper of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an explicit interface tracking method, wherein the interface between two fluids is decomposed into segments, which are represented locally on an Eulerian grid. The framework of this method provides a natural setting for solving the advection-diffusion equation governing the surfactant concentration on the interface. Open interfaces and moving contact lines are also incorporated into the method in a natural way, though we show that care must be taken when regularizing interface forces to the grid near the boundary of the computational domain.

In the second paper we present a boundary integral formulation for sedimenting particles in periodic Stokes flow, using the completed double layer boundary integral formulation. The long-range nature of the particle-particle interactions lead to the formulation containing sums which are not absolutely convergent if computed directly. This is solved by applying the method of Ewald summation, which in turn is computed in a fast manner by using the FFT-based spectral Ewald method. The complexity of the resulting method is O(N log N), as the system size is scaled up with the number of discretization points N. We apply the method to systems of sedimenting spheroids, which are discretized using the Nyström method and a basic quadrature rule.

The Ewald summation method used in the boundary integral method of the second paper requires a decomposition of the potential being summed. In the introductory chapters of the thesis we present an overview of the available methods for creating Ewald decompositions, and show how the methods and decompositions can be related to each other.

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• 5.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Ewald summation for the rotlet singularity of Stokes flow2016Report (Other academic)

Ewald summation is an efficient method for computing the periodic sums that appear when considering the Green's functions of Stokes flow together with periodic boundary conditions. We show how Ewald summation, and accompanying truncation error estimates, can be easily derived for the rotlet, by considering it as a superposition of electrostatic force calculations.

fulltext
• 6.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Fast and accurate integral equation methods with applications in microfluidics2016Doctoral thesis, comprehensive summary (Other academic)

This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

The first, and smaller, part of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an interface decomposition resulting in local, Eulerian grid representations. This provides a natural setting for solving the PDE governing the surfactant concentration on the interface.

The second, and larger, part of the thesis is concerned with a framework for simulating large systems of rigid particles in three-dimensional, periodic viscous flow using a boundary integral formulation. This framework can solve the underlying flow equations to high accuracy, due to the accurate nature of surface quadrature. It is also fast, due to the natural coupling between boundary integral methods and fast summation methods.

The development of the boundary integral framework spans several different fields of numerical analysis. For fast computations of large systems, a fast Ewald summation method known as Spectral Ewald is adapted to work with the Stokes double layer potential. For accurate numerical integration, a method known as Quadrature by Expansion is developed for this same potential, and also accelerated through a scheme based on geometrical symmetries. To better understand the errors accompanying this quadrature method, an error analysis based on contour integration and calculus of residues is carried out, resulting in highly accurate error estimates.

fulltext
• 7.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant2014In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 101, p. 50-63Article in journal (Refereed)

The flow behavior of many multiphase flow applications is greatly influenced by wetting properties and the presence of surfactants. We present a numerical method for two-phase flow with insoluble surfactants and contact line dynamics in two dimensions. The method is based on decomposing the interface between two fluids into segments, which are explicitly represented on a local Eulerian grid. It provides a natural framework for treating the surfactant concentration equation, which is solved locally on each segment. An accurate numerical method for the coupled interface/surfactant system is given. The system is coupled to the Navier-Stokes equations through the immersed boundary method, and we discuss the issue of force regularization in wetting problems, when the interface touches the boundary of the domain. We use the method to illustrate how the presence of surfactants influences the behavior of free and wetting drops.

fulltext
• 8.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Fast Ewald summation for free-space Stokes potentials2017In: Research in the Mathematical Sciences, ISSN 2197-9847, Vol. 4, no 1Article in journal (Refereed)

We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green’s functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems (Lindbo and Tornberg in J Comput Phys 229(23):8994–9010, 2010. doi: 10.1016/j.jcp.2010.08.026 ), with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid (Vico et al. in J Comput Phys 323:191–203, 2016. doi: 10.1016/j.jcp.2016.07.028 ). Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of $$O(N \log N)$$ O ( N log N ) for problems with N sources and targets. Comparison is made with a fast multipole method to show that the performance of the new method is competitive.

• 9.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
A fast integral equation method for solid particles in viscous flow using quadrature by expansionManuscript (preprint) (Other academic)

Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

fulltext
• 10.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
A fast integral equation method for solid particles in viscous flow using quadrature by expansion2016In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 326, p. 420-445Article in journal (Refereed)

Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

Post-print
• 11.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Adaptive Quadrature by Expansion for Layer Potential Evaluation in Two Dimensions2018In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 40, no 3, p. A1225-A1249Article in journal (Refereed)

When solving partial differential equations using boundary integral equation methods, accurate evaluation of singular and nearly singular integrals in layer potentials is crucial. A recent scheme for this is quadrature by expansion (QBX), which solves the problem by locally approximating the potential using a local expansion centered at some distance from the source boundary. In this paper we introduce an extension of the QBX scheme in two dimensions (2D) denoted AQBX—adaptive quadrature by expansion—which combines QBX with an algorithm for automated selection of parameters, based on a target error tolerance. A key component in this algorithm is the ability to accurately estimate the numerical errors in the coefficients of the expansion. Combining previous results for flat panels with a procedure for taking the panel shape into account, we derive such error estimates for arbitrarily shaped boundaries in 2D that are discretized using panel-based Gauss–Legendre quadrature. Applying our scheme to numerical solutions of Dirichlet problems for the Laplace and Helmholtz equations, and also for solving these equations, we find that the scheme is able to satisfy a given target tolerance to within an order of magnitude, making it useful for practical applications. This represents a significant simplification over the original QBX algorithm, in which choosing a good set of parameters can be hard.

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• 12.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Error estimation for quadrature by expansion in layer potential evaluation2017In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 43, no 1, p. 195-234Article in journal (Refereed)

In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

• 13.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.

In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

fulltext
• 14.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Fast Ewald summation for Stokesian particle suspensions2014In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 76, no 10, p. 669-698Article in journal (Refereed)

We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to O(N log N), where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions.

postprint
• 15.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Deep Learning models for turbulent shear flow2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

Deep neural networks trained with spatio-temporal evolution of a dynamical system may be regarded as an empirical alternative to conventional models using differential equations. In this thesis, such deep learning models are constructed for the problem of turbulent shear flow. However, as a first step, this modeling is restricted to a simplified low-dimensional representation of turbulence physics. The training datasets for the neural networks are obtained from a 9-dimensional model using Fourier modes proposed by Moehlis, Faisst, and Eckhardt [29] for sinusoidal shear flow. These modes were appropriately chosen to capture the turbulent structures in the near-wall region. The time series of the amplitudes of these modes fully describe the evolution of flow. Trained deep learning models are employed to predict these time series based on a short input seed. Two fundamentally different neural network architectures, namely multilayer perceptrons (MLP) and long short-term memory (LSTM) networks are quantitatively compared in this work. The assessment of these architectures is based on (i) the goodness of fit of their predictions to that of the 9-dimensional model, (ii) the ability of the predictions to capture the near-wall turbulence structures, and (iii) the statistical consistency of the predictions with the test data. LSTMs are observed to make predictions with an error that is around 4 orders of magnitude lower than that of the MLP. Furthermore, the flow fields constructed from the LSTM predictions are remarkably accurate in their statistical behavior. In particular, deviations of 0:45 % and 2:49 % between the true data and the LSTM predictions were obtained for the mean flow and the streamwise velocity fluctuations, respectively.

fulltext
• 16.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Implementation of the Particle Mesh Ewald method on a GPU2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

The Particle Mesh Ewald (PME) method is used for efficient long-range electrostatic calculations in molecular dynamics (MD).

In this project, PME is implemented for a single GPU alongside the existing CPU implementation, using the code base of an open source MD software GROMACS and NVIDIA CUDA toolkit. The performance of the PME GPU implementation is then studied.

The motivation for the project is examining the PME algorithm’s parallelism, and its potential benefit for performance scalability of MD simulations on various hardware.

fulltext
• 17. Arakelyan, A.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Multi-Phase Quadrature Domains and a Related Minimization Problem2016In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, p. 1-21Article in journal (Refereed)

In this paper we introduce the multi-phase version of the so-called Quadrature Domains (QD), which refers to a generalized type of mean value property for harmonic functions. The well-established and developed theory of one-phase QD was recently generalized to a two-phase version, by one of the current authors (in collaboration). Here we introduce the concept of the multi-phase version of the problem, and prove existence as well as several properties of such solutions. In particular, we discuss possibilities of multi-junction points.

• 18. Ariel, G.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Parareal multiscale methods for highly oscillatory dynamical systems2016In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 6, p. A3540-A3564Article in journal (Refereed)

We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solutions, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.

• 19.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations2015Doctoral thesis, comprehensive summary (Other academic)

This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media.

Thesis
• 20.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations2013Licentiate thesis, comprehensive summary (Other academic)

This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers.

The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large.

In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities.

Thesis_Intro
• 21. Arjmand, Doghonay
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
A time dependent approach for removing the cell boundary error in elliptic homogenization problems2016In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 314, p. 206-227Article in journal (Refereed)

This paper concerns the cell-boundary error present in multiscale algorithms for elliptic homogenization problems. Typical multiscale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. To solve the micro model, boundary conditions are required on the boundary of the microscopic domain. Imposing a naive boundary condition leads to O(epsilon/eta) error in the computation, where epsilon is the size of the microscopic variations in the media and eta is the size of the micro-domain. The removal of this error in modern multiscale algorithms still remains an important open problem. In this paper, we present a time-dependent approach which is general in terms of dimension. We provide a theorem which shows that we have arbitrarily high order convergence rates in terms of epsilon/eta in the periodic setting. Additionally, we present numerical evidence showing that the method improves the O(epsilon/eta) error to O(epsilon) in general non-periodic media.

• 22.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
Analysis of heterogeneous multiscale methods for long time wave propagation problems2014In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 12, no 3, p. 1135-1166Article in journal (Refereed)

In this paper, we analyze a multiscale method developed under the heterogeneous multiscale method (HMM) framework for numerical approximation of multiscale wave propagation problems in periodic media. In particular, we are interested in the long time O(epsilon(-2)) wave propagation, where e represents the size of the microscopic variations in the media. In large time scales, the solutions of multiscale wave equations exhibit O(1) dispersive effects which are not observed in short time scales. A typical HMM has two main components: a macromodel and a micromodel. The macromodel is incomplete and lacks a set of local data. In the setting of multiscale PDEs, one has to solve for the full oscillatory problem over local microscopic domains of size eta = O(epsilon) to upscale the parameter values which are missing in the macroscopic model. In this paper, we prove that if the microproblems are consistent with the macroscopic solutions, the HMM approximates the unknown parameter values in the macromodel up to any desired order of accuracy in terms of epsilon/eta..

• 23.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Analysis of HMM for Long Time Multiscale Wave Propagation Problems in Locally-Periodic MediaManuscript (preprint) (Other academic)

Multiscale wave propagation problems are difficult to solve numerically due to the interaction of different scales inherent in the problem. Extracting information about the average behaviour of the system requires resolving small scales in the problem. This leads to a tremendous computational burden if the size of microscopic variations are much smaller than the size of scales of interest. Heterogeneous multiscale methods (HMM) is a tool to avoid resolving the small scales everywhere. Nevertheless, it approximates the average part of the solution by upscaling the microscopic information on a small part of the domain. This leads to a substantial improvement in the computational cost. In this article, we analyze an HMM-based numerical method which approximates the long time behaviour of multiscale wave equations. In particular, we consider theoretically challenging case of locally-periodic media where fast and slow variations are allowed at the same time. We are interested in the long time regime (T=O(e^{-1})), where e represents the wavelength of the fast variations in themedia. We first use asymptotic expansions to derive effective equations describing the long time effects of the multiscale waves in multi-dimensional locally-periodic media. We then show that HMM captures these non-trivial long time eects. All the theoretical statements are general in terms of dimension. Two dimensional numericale xamples are considered to support our theoretical arguments

• 24. Arjmand, Doghonay
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
Estimates for the upscaling error in heterogeneous multiscale methods for wave propagation problems in locally periodic media2017In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 15, no 2, p. 948-976Article in journal (Refereed)

This paper concerns the analysis of a multiscale method for wave propagation problems in microscopically nonhomogeneous media. A direct numerical approximation of such problems is prohibitively expensive as it requires resolving the microscopic variations over a much larger physical domain of interest. The heterogeneous multiscale method (HMM) is an efficient framework to approximate the solutions of multiscale problems. In the HMM, one assumes an incomplete macroscopic model which is coupled to a known but expensive microscopic model. The micromodel is solved only locally to upscale the parameter values which are missing in the macro model. The resulting macroscopic model can then be solved at a cost independent of the small scales in the problem. In general, the accuracy of the HMM is related to how good the upscaling step approximates the right macroscopic quantities. The analysis of the method that we consider here was previously addressed only in purely periodic media, although the method itself is numerically shown to be applicable to more general settings. In the present study, we consider a more realistic setting by assuming a locally periodic medium where slow and fast variations are allowed at the same time. We then prove that the HMM captures the right macroscopic effects. The generality of the tools and ideas in the analysis allows us to establish convergence rates in a multidimensional setting. The theoretical findings here imply an improved convergence rate in one dimension, which also justifies the numerical observations from our earlier study.

• 25.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
ENSTA ParisTech.
A Finite Element Heterogenous Multiscale Method with Improved Control Over the Modeling ErrorManuscript (preprint) (Other academic)

Multiscale partial dierential equations (PDEs) are difficult to solve by traditional numerical methods due to the need to resolve the small wavelengths in the media over the entire computational domain. We develop and analyze a Finite Element Heterogeneous Multiscale Method (FE-HMM) for approximating the homogenized solutions of multiscale PDEs of elliptic, parabolic,and hyperbolic type. Typical multiscale methods require a coupling between a micro and a macromodel. Inspired from the homogenization theory, traditional FE-HMM schemes use elliptic PDEs as the micro model. We use, however, the second order wave equation as our micro model independent of the type of the problem on the macro level. This allows us to control the modeling error originating by the coupling between the dierent scales. In a spatially fully discrete a priori error analysis we prove that the modeling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media. We provide numerical examples in one and two dimensions confirming the theoretical results. Further examples show that the method captures the effective solutions in general non-periodic settings as well

• 26.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Numerical simulation of an inertial spheroidal particle in Stokes flow2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

Particle suspensions occur in many situations in nature and industry. In this master’s thesis, the motion of a single rigid spheroidal particle immersed in Stokes flow is studied numerically using a boundary integral method and a new specialized quadrature method known as quadrature by expansion (QBX). This method allows the spheroid to be massless or inertial, and placed in any kind of underlying Stokesian flow.

A parameter study of the QBX method is presented, together with validation cases for spheroids in linear shear flow and quadratic flow. The QBX method is able to compute the force and torque on the spheroid as well as the resulting rigid body motion with small errors in a short time, typically less than one second per time step on a regular desktop computer. Novel results are presented for the motion of an inertial spheroid in quadratic flow, where in contrast to linear shear flow the shear rate is not constant. It is found that particle inertia induces a translational drift towards regions in the fluid with higher shear rate.

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• 27.
Kings Coll London, Kings Hlth Partners, Sch Imaging Sci & Biomed Engn, Dept Biomed Engn, London SE1 7EH, England..
Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden.;Norwegian Univ Sci & Technol, Dept Math Sci, NO-7491 Trondheim, Norway.. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. Kings Coll London, Kings Hlth Partners, Sch Imaging Sci & Biomed Engn, Dept Biomed Engn, London SE1 7EH, England.. Kings Coll London, Kings Hlth Partners, Sch Imaging Sci & Biomed Engn, Dept Biomed Engn, London SE1 7EH, England.;Univ Michigan, Dept Biomed Engn & Cardiac Surg, Ann Arbor, MI 48109 USA..
A partition of unity approach to fluid mechanics and fluid-structure interaction2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 362, article id 112842Article in journal (Refereed)

For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.

• 28.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Parallelisation and Performance Analysis of a TreeSPH Code for Galaxy Simulations2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

In cosmological simulations, the Lagrangian method Smoothed Particle Hydrodynamics is often applied to cover gas dynamics and combined with tree algorithms for long-range potentials like the Barnes-Hut method to include self-gravity and derive the nearest neighbour lists efficiently. In this thesis, a so-called TreeSPH code is parallelized by using MPI and subsequently the performance is analysed. For the domain decomposition to the processes, the structure of an octree is examined and space filling curves are applied to achieve well-working dynamical load balancing. For an efficient parallel SPH calculation, a novel method with a localised boundary handling is proposed to reduce communication overhead

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• 29. Bartuschat, D.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Two computational models for simulating the tumbling motion of elongated particles in fluids2016In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 127, p. 17-35Article in journal (Refereed)

Suspensions with fiber-like particles in the low Reynolds number regime are modeled by two different approaches that both use a Lagrangian representation of individual particles. The first method is the well-established formulation based on Stokes flow that is formulated as integral equations. It uses a slender body approximation for the fibers to represent the interaction between them directly without explicitly computing the flow field. The second is a new technique using the 3D lattice Boltzmann method on parallel supercomputers. Here the flow computation is coupled to a computational model of the dynamics of rigid bodies using fluid-structure interaction techniques. Both methods can be applied to simulate fibers in fluid flow. They are carefully validated and compared against each other, exposing systematically their strengths and weaknesses regarding their accuracy, the computational cost, and possible model extensions.

• 30.
University of Heidelberg.
University of Warwick. University of Münster. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. FU Berlin. University of Freiburg. University of Münster. TU Dresden. University of Münster.
Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management2012In: Progress in Industrial Mathematics at ECMI 2010 / [ed] Michael Günther and Andreas Bartel and Markus Brunk and Sebastian Schöps and Michael Striebel, Springer Berlin/Heidelberg, 2012, p. 561-567Conference paper (Refereed)

This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated–unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet–Neumann method, using the dune-grid-glue module in the numerics software DUNE.

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• 31. Bayer, C.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces2015In: Applied Mathematics Research eXpress, ISSN 1687-1200, E-ISSN 1687-1197, no 2, p. 329-417Article in journal (Refereed)

The difference of the values of observables for the time-independent Schrödinger equation, with matrix-valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states, and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio and the probability to be in excited states) for this difference of microcanonical observables, assuming that molecular dynamics space-time averages converge, with a rate related to the maximal Lyapunov exponent. The error estimate is uniform in the number of particles and the analysis does not assume a uniform lower bound on the spectral gap of the electron operator and consequently the probability to be in excited states can be large. A numerical method to determine the probability to be in excited states is then presented, based on Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem.

• 32. Bayer, Christian
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
On nonasymptotic optimal stopping criteria in monte carlo simulations2014In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 36, no 2, p. A869-A885Article in journal (Refereed)

We consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.

• 33.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Suturing in Surgical Simulations2019Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

The goal of this project is to develop virtual surgical simulation software in order to simulate the suturing and knot tying processes associated with surgical thread. State equations are formulated using Lagrangian mechanics, which is useful for the conservation of energy. Solver methods are developed with theory based in Differential Algebraic Equations (DAEs) which concern governing Ordinary Differential Equations (ODEs) that are constraint with Algebraic Equations (AE). An implicit integration scheme and Newton's method is used to solve the system in each step. Furthermore, a collision response process based on the Linear Complementarity Problem (LCP) is implemented to handle collisions and measure their forces. Models have been developed to represent the different types of objects. A spline model is used to represent the suture and mass-spring model for the tissue. They were both selected for their efficiency and base on real physical properties. The spline model was also chosen as it is continuous and can be evaluated at any point along the length. Other objects are also defined such as rigid bodies. The Lagrangian multiplier method is used to define the constraints in the model. This allows for the construction of complex models. An important constraint is the suturing constraint, which is created when a sufficient force is applied by the suture tip on to the tissue. This constraint allows only a sliding point along the suture to pass through a specific point on the tissue. This results in a virtual suturing model which can be built on for use in surgical simulations. Further investigations would be interesting to increase performance, accuracy and scope of the simulator.

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• 34. Beeumen, R. V.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems2016In: Numerical Linear Algebra with Applications, ISSN 1070-5325, E-ISSN 1099-1506Article in journal (Refereed)

Summary: We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter-dependent matrix. In several applications, M(λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi-based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169-195]. In this paper, the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems.

• 35.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Konstruktion av en optimal balk med Tikhonovregularisering2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

In this bachelor thesis the problem of numerically minimizing the elastic energy of a uniformly loaded, simply supported beam is studied. In this problem, the beam is assumed to be divided into a given number of equally long pieces with constant flexural rigidity and the minimization is done with respect to the flexural rigidities of the pieces. For a fine division of the beam there is a risk that the numerical method becomes unstable. The aim of this study is to investigate whether Tikhonov regularization can stabilize the numerical method in the case of a fine division. In the solution of the beam problem the Bernoulli-Euler equation and Lagrange’s multiplier method areused to derive the function to be minimized. Minimization is done iteratively by the gradient method. The problem is solved for a division of the beam into three and 24 pieces. In the case of 24 pieces, the iterations in the gradient method do not converge and negative values of the flexural rigidities are obtained. Tikhonov regularization is shown to stabilize the iterations so that convergence is attained and the flexural rigidities remain positive.

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• 36. Björk, T.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Monte Carlo Euler approximations of HJM term structure financial models2013In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, no 2, p. 341-383Article in journal (Refereed)

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

• 37.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Investigation of Outflow Boundary Conditions for Convection-Dominated Incompressible Fluid Flows in a Spectral Element Framework2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

In this thesis we implement and study the effects of different convective outflow boundary conditions for the high order spectral element solver Nek5000 in the context of solving convective-dominated fluid flow problems. By numerical testing we show that the convective boundary conditions preserve the spatial and temporal convergence rates of the solver. We also study highly convective test cases such as a single vortex propagating through the outflow boundary, and the typical Kármán vortex shedding problem to analyze the accuracy and stability. A detailed comparison with the natural boundary condition that corresponds to the variational form of the incompressible Navier–Stokes equations (the Nek5000 “O” condition), and a stabilized version of it (by Dong et al. (2014)), are also presented.

Our results show a major advantage of using the convective boundary conditions over the natural counterpart in solving convective problems, both according to stability and accuracy. Analytic and numerical results show that the natural condition has big stability problems for high Reynolds numbers, which make the use of stabilization methods or damping regions crucial. But, the (Dong) stabilized natural condition does not improve accuracy, and damping regions are computationally expensive. The convective conditions show very good accuracy if its convection speed is approximated accurately, and our results indicate that it can be used without damping regions efficiently. Our results also show that the magnitude of reflections significantly depends on the amplitude of the disturbances that move through the boundary. The convective boundary condition can handle large disturbances without producing significant reflections, while the natural one or a stabilized version of it in general can not.

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• 38.
KTH, Centres, Science for Life Laboratory, SciLifeLab.
KTH, School of Computer Science and Communication (CSC), Computational Science and Technology (CST). KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Multirate method for co-simulation of electrical-chemical systems in multiscale modeling2017In: Journal of Computational Neuroscience, ISSN 0929-5313, E-ISSN 1573-6873, Vol. 42, no 3, p. 245-256Article in journal (Refereed)

Multiscale modeling by means of co-simulation is a powerful tool to address many vital questions in neuroscience. It can for example be applied in the study of the process of learning and memory formation in the brain. At the same time the co-simulation technique makes it possible to take advantage of interoperability between existing tools and multi-physics models as well as distributed computing. However, the theoretical basis for multiscale modeling is not sufficiently understood. There is, for example, a need of efficient and accurate numerical methods for time integration. When time constants of model components are different by several orders of magnitude, individual dynamics and mathematical definitions of each component all together impose stability, accuracy and efficiency challenges for the time integrator. Following our numerical investigations in Brocke et al. (Frontiers in Computational Neuroscience, 10, 97, 2016), we present a new multirate algorithm that allows us to handle each component of a large system with a step size appropriate to its time scale. We take care of error estimates in a recursive manner allowing individual components to follow their discretization time course while keeping numerical error within acceptable bounds. The method is developed with an ultimate goal of minimizing the communication between the components. Thus it is especially suitable for co-simulations. Our preliminary results support our confidence that the multirate approach can be used in the class of problems we are interested in. We show that the dynamics ofa communication signal as well as an appropriate choice of the discretization order between system components may have a significant impact on the accuracy of the coupled simulation. Although, the ideas presented in the paper have only been tested on a single model, it is likely that they can be applied to other problems without loss of generality. We believe that this work may significantly contribute to the establishment of a firm theoretical basis and to the development of an efficient computational framework for multiscale modeling and simulations.

• 39.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Large-scale time parallelization for molecular dynamics problems2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

As modern supercomputers draw their power from the sheer number of cores, an efficient parallelization of programs is crucial for achieving good performance. When one tries to solve differential equations in parallel this is usually done by parallelizing the computation of one single time step. As the speedup of such parallelization schemes is usually limited, e.g. by the spatial size of the problem, additional parallelization in time may be useful to achieve better scalability.

This thesis will introduce two well-known schemes for time-parallelization, namely the waveform relaxation method and the parareal algorithm. These methods are then applied to a molecular dynamics problem which is a useful test example as the number of required time steps is high while the number of unknowns is relatively low. Afterwards it is investigated how these methods can be adapted to large-scale computations.

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• 40. Burman, E.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Full gradient stabilized cut finite element methods for surface partial differential equations2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 278-296Article in journal (Refereed)

We propose and analyze a new stabilized cut finite element method for the Laplace–Beltrami operator on a closed surface. The new stabilization term provides control of the full R3 gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace–Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and L2 norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.

• 41. Burman, Erik
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Cut finite element methods for coupled bulk–surface problems2016In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, no 2, p. 203-231Article in journal (Refereed)

We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and (Formula presented.) norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

• 42.
UCL, Dept Math, London WC1E 6BT, England..
Jonkoping Univ, Dept Mech Engn, SE-55111 Jonkoping, Sweden.. Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden.. Umea Univ, Dept Math & Math Stat, SE-90187 Umea, Sweden.. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
A stabilized cut streamline diffusion finite element method for convection-diffusion problems on surfaces2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 358, article id UNSP 112645Article in journal (Refereed)

We develop a stabilized cut finite element method for the stationary convection-diffusion problem on a surface embedded in R-d. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included.

• 43.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
An accurate integral equation method for Stokes flow with piecewise smooth boundariesManuscript (preprint) (Other academic)

Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive numerical method to solve the Stokes equations, as the problem can be reformulated into a problem that must be solved only over the boundary of the domain. When the boundary is at least C1 smooth, the boundary integral kernel is a compact operator, and traditional Nyström methods can be used to obtain highly accurate solutions. In the case of Lipschitz continuous boundaries however, obtaining accurate solutions using the standard Nyström method can require high resolution. We adapt a technique known as recursively compressed inverse preconditioning to accurately solve the Stokes equations without requiring any more resolution than is needed to resolve the boundary. Combined with a periodic fast summation method we construct a method that is O(N log N ) where N is the number of quadrature points on the boundary. We demonstrate the robustness of this method by extending an existing boundary integral method for viscous drops to handle the movement of drops near corners.

• 44.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Stable and contact-free time stepping for dense rigid particle suspensions2020In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 92, no 2, p. 94-113Article in journal (Refereed)

We consider suspensions of rigid bodies in a two-dimensional viscous fluid. Even with high-fidelity numerical methods, unphysical contact between particles occurs because of spatial and temporal discretization errors. We extend a time stepping method that avoids overlap by imposing a minimum separation distance between all pairs of bodies. In its original form, the method discretizes interactions between different particles explicitly. Therefore, to avoid stiffness, a large minimum separation distance is used. In this paper, we introduce a new implicit time stepping method that is able to simulate dense suspensions with large time step sizes and a small minimum separation distance. The method is tested on various unbounded and bounded flows, and rheological properties of the resulting suspensions are computed.

• 45.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Modeling, Simulation and Dynamic control of a Wave Energy Converter2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

The energy in ocean waves is a renewable energy resource not yet fully exploited. Research in converting ocean energy to useful electricity has been ongoing for about 40 years, but no one has so far succeed to do it at sufficiently low cost. CorPower Ocean has developed a method, which in theory can achieve this. It uses a light buoy and a control strategy called Phase Control.

The purpose of this thesis is to develop a mathematical model of this method - using LinearWave Theory to derive the hydrodynamic forces - and from the simulated results analyze the energy output of the method. In the process we create a program that will help realizing and improving the method further.

The model is implemented and simulated in the simulation program Simulink. On the basis of the simulated results, we can concludes that the CorPower Ocean method is promising. The outcome shows that the energy output increases - up to five times- compared to conventional methods.

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• 46.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Long Time Integration of Molecular Dynamics at Constant Temperature with the Symplectic Euler Method2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

Simulations of particle systems at constant temperature may be used to estimate several of the system’s physical properties, and some require integration over very long time to be accurate. To achieve sufficient accuracy in finite time the choice of numerical scheme is important and we suggest to use the symplectic Euler method combined with a step in an Ornstein-Uhlenbeck process. This scheme is computationally very cheap and is often used in applications of molecular dynamics. This thesis strives to motivate the usage of the scheme due to the lack of theoretical results and comparisons to alternative methods. We conduct three numerical experiments to evaluate the scheme. The design of each experiment aims to expose weaknesses or strengths of the method. For both model problems and more realistic experiments are the results positive in favor of the method; the symplectic Euler method combined with an Ornstein- Uhlenbeck step does perform well over long times.

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• 47. Chaudhry, Q. A.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Surface reactions on the cytoplasmatic membranes - Mathematical modeling of reaction and diffusion systems in a cell2014In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 262, p. 244-260Article in journal (Refereed)

A human cell consists schematically of an outer cellular membrane, a cytoplasm containing a large number of organelles (mitochondria, endoplasmatic reticulum etc.), a nuclear membrane and finally the cellular nucleus containing DNA. The organelles create a complex and dense system of membranes or sub-domains throughout the cytoplasm. The mathematical description leads to a system of reaction-diffusion equations in a complex geometrical domain, dominated by thin membranous structures with similar physical and chemical properties. In a previous model, we considered only spatially distributed reaction and diffusion processes. However, from experiments it is known that membrane bound proteins play an important role in the metabolism of certain substances. In the present paper we develop a homogenization strategy which includes both volume and surface reactions. The homogenized system is a reaction-diffusion system in the cytoplasm which is coupled to the surrounding cell components by correspondingly modified transfer conditions. The approach is verified by application to a system modeling the cellular uptake and intracellular dynamics of carcinogenic polycyclic aromatic hydrocarbons.

• 48. Chaudhry, Qasim A.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Study of intracellular reaction and diffusion mechanism of carcinogenic PAHs: Using non-standard compartment modeling approach2013In: Toxicology Letters, ISSN 0378-4274, E-ISSN 1879-3169, Vol. 221, p. S182-S182Article in journal (Other academic)
• 49. Chen, C.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
An implicit boundary integral method for interfaces evolving by Mullins-Sekerka dynamics2017In: Springer Proceedings in Mathematics and Statistics, Springer New York LLC , 2017, p. 1-21Conference paper (Refereed)

We present an algorithm for computing the nonlinear interface dynamics of the Mullins-Sekerka model for interfaces that are defined implicitly (e.g. by a level set function) using integral equations. The computation of the dynamics involves solving Laplace’s equation with Dirichlet boundary conditions on multiply connected and unbounded domains and propagating the interface using a normal velocity obtained from the solution of the PDE at each time step. Our method is based on a simple formulation for implicit interfaces, which rewrites boundary integrals as volume integrals over the entire space. The resulting algorithm thus inherits the benefits of both level set methods and boundary integral methods to simulate the nonlocal front propagation problem with possible topological changes. We present numerical results in both two and three dimensions to demonstrate the effectiveness of the algorithm.

• 50. Chen, Chieh
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Implicit boundary integral methods for the Helmholtz equation in exterior domains2017In: RESEARCH IN THE MATHEMATICAL SCIENCES, ISSN 2197-9847, Vol. 4, article id UNSP 19Article in journal (Refereed)
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