| 1. |
- Hartung, Kerstin, 1989-
(författare)
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Paths to improving atmospheric models across scales : The importance of the unresolved scales
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Correct representation of physical processes, the parametrizations, and their interaction with the resolved circulation is crucial for the performance of numerical models. Here, focus is put on understanding model biases and developing tools to alleviate existing biases. Atmospheric blocking can divert the typical atmospheric flow for several days up to weeks and thereby impacts the mean climate of the region experiencing blocking. Models typically underestimate the frequency of atmospheric blocking. Based on results from the global climate model EC-Earth, it is found that the atmospheric model resolution is not strongly influencing the representation of atmospheric blocking once the grid reaches about 80 km grid length in the horizontal. Updating several physical parametrizations, and thereby the model version, is the largest contributor to advancements in simulating atmospheric blocking. The importance of the topography for the large-scale atmospheric flow is further investigated with the reanalysis ERA-Interim by applying a simplified theoretical analysis. It is found that the idealized topographic forcing theory can explain some part of the observed large-scale properties of the flow, though the method does mainly produce relative results. The explained part of the large-scale structure is increased during periods of northwesterly flow and when the flow impinges the mountain ridge almost orthogonally.Small-scale processes acting in air masses transported from midlatitudes to the Arctic are also discussed. Numerical models often struggle with representing the stable conditions in the Arctic and tend to underestimate the downward longwave impact during cloudy conditions. A comparison of single-column models (SCMs) indicates that most models can capture the bimodal longwave distribution which develops from alternating cloudy and clear-sky conditions. SCMs are often used for model development as they allow to decouple the parametrized physical processes from the large-scale environment and enable many parameter sensitivity tests. A new tool is presented which can be used for the development of physical parametrizations in marine and polar conditions. It combines one-dimensional models of the atmosphere and ocean, including sea-ice, into a coupled atmosphere-ocean SCM (AOSCM). The presented setup constitutes an advantage compared to SCMs of one component because the coupling is directly modelled and the interaction between the respective boundary layers does not dependent on prescribed boundary conditions.
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| 2. |
- Holst, Henrik
(författare)
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Multiscale Methods for Wave Propagation Problems
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Simulations of wave propagation in heterogeneous media and at high frequencies are important in many applications such as seismic-, {electro-magnetic-,} acoustic-, fluid flow problems and others. These are classical multiscale problems and often too computationally expensive for direct numerical simulation. The smallest scales must be well resolved over a computational domain represented by the largest scale and this results in a very high computational cost. We develop and analyze numerical techniques based on the heterogeneous multiscale method (HMM) framework for such wave equations with highly oscillatory solutions $u^{\varepsilon}$ where $\varepsilon$ represents the size of the smallest scale. In these techniques the oscillatory microscale is approximated on small local microproblems of size $\varepsilon$ in spatial and time directions. The solution of the microproblems are then coupled to a global macroscale model in divergence form $u_{tt} = \nabla \cdot F$ where the flux $F$ is obtained from the microproblems. The oscillations can either originate from fluctuations in the velocity coefficients or from high frequency initial and boundary conditions. We have developed algorithms that couple micro and macroscales for both these cases. The choice of macroscale variables is inspired by the analytic theories of homogenization and geometrical optics respectively. In the first case local averages $u \approx u^{\varepsilon}$ are used on the macroscale. In the second case, phase $\phi$ and energy are natural macroscopic variables. There are two major goals of this research. One goal is to develop and analyze algorithms for simulating multiscale wave propagation with low computational complexity, and even independent of $\varepsilon$ for finite time problems. This is seen in many examples in one, two and three dimensions. The other goal is to use wave propagation as a model to better understand the HMM framework. An example in this direction is simulation with oscillatory wave field over long time. The dispersive effects that then occur is well approximated by a HMM method that was originally formulated for finite time where added accuracy is required but no explicit adjustment to include dispersion, an evidence of the robustness of the method.
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| 3. |
- Häggblad, Jon, 1981-
(författare)
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Boundary and Interface Conditions for Electromagnetic Wave Propagation using FDTD
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 4. |
- Arjmand, Doghonay, 1987-
(författare)
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Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations
- 2015
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 5. |
- Häggblad, Jon, 1981-
(författare)
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Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis centers on modified stencils for the Finite-Difference Time-Domain method (FDTD), or Yee scheme, when modelling curved boundaries, obstacles and holes smaller than the discretization length. The goal is to increase the accuracy while keeping the structure of the standard method, enabling improvements to existing implementations with minimal effort.We present an extension of a previously developed technique for consistent boundary approximation in the Yee scheme. We consider both Maxwell's equations and the acoustic equations in three dimensions, which require separate treatment, unlike in two dimensions.The stability properties of coefficient modifications are essential for practical usability. We present an analysis of the requirements for time-stable modifications, which we use to construct a simple and effective method for boundary approximations. The method starts from a predetermined staircase discretization of the boundary, requiring no further data on the underlying geometry that is being approximated.Not only is the standard staircasing of curved boundaries a poor approximation, it is inconsistent, giving rise to errors that do not disappear in the limit of small grid lengths. We analyze the standard staircase approximation by deriving exact solutions of the difference equations, including the staircase boundary. This facilitates a detailed error analysis, showing how staircasing affects amplitude, phase, frequency and attenuation of waves.To model obstacles and holes of smaller size than the grid length, we develop a numerical subgrid method based on locally modified stencils, where a highly resolved micro problem is used to generate effective coefficients for the Yee scheme at the macro scale.The implementations and analysis of the developed methods are validated through systematic numerical tests.
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| 6. |
- Popovic, Jelena, 1977-
(författare)
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Fast Adaptive Numerical Methods for High Frequency Waves and Interface Tracking
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main focus of this thesis is on fast numerical methods, where adaptivity is an important mechanism to lowering the methods' complexity. The application of the methods are in the areas of wireless communication, antenna design, radar signature computation, noise prediction, medical ultrasonography, crystal growth, flame propagation, wave propagation, seismology, geometrical optics and image processing. We first consider high frequency wave propagation problems with a variable speed function in one dimension, modeled by the Helmholtz equation. One significant difficulty of standard numerical methods for such problems is that the wave length is very short compared to the computational domain and many discretization points are needed to resolve the solution. The computational cost, thus grows algebraically with the frequency w. For scattering problems with impenetrable scatterer in homogeneous media, new methods have recently been derived with a provably lower cost in terms of w. In this thesis, we suggest and analyze a fast numerical method for the one dimensional Helmholtz equation with variable speed function (variable media) that is based on wave-splitting. The Helmholtz equation is split into two one-way wave equations which are then solved iteratively for a given tolerance. We show rigorously that the algorithm is convergent, and that the computational cost depends only weakly on the frequency for fixed accuracy. We next consider interface tracking problems where the interface moves by a velocity field that does not depend on the interface itself. We derive fast adaptive numerical methods for such problems. Adaptivity makes methods robust in the sense that they can handle a large class of problems, including problems with expanding interface and problems where the interface has corners. They are based on a multiresolution representation of the interface, i.e. the interface is represented hierarchically by wavelet vectors corresponding to increasingly detailed meshes. The complexity of standard numerical methods for interface tracking, where the interface is described by marker points, is O(N/dt), where N is the number of marker points on the interface and dt is the time step. The methods that we develop in this thesis have O(dt^(-1)log N) computational cost for the same order of accuracy in dt. In the adaptive version, the cost is O(tol^(-1/p)log N), where tol is some given tolerance and p is the order of the numerical method for ordinary differential equations that is used for time advection of the interface. Finally, we consider time-dependent Hamilton-Jacobi equations with convex Hamiltonians. We suggest a numerical method that is computationally efficient and accurate. It is based on a reformulation of the equation as a front tracking problem, which is solved with the fast interface tracking methods together with a post-processing step. The complexity of standard numerical methods for such problems is O(dt^(-(d+1))) in d dimensions, where dt is the time step. The complexity of our method is reduced to O(dt^(-d)|log dt|) or even to O(dt^(-d)).
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| 7. |
- Arjmand, Doghonay
(författare)
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Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations
- 2013
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 8. |
- Izzo, Federico, 1991-
(författare)
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High order trapezoidal rule-based quadratures for boundary integral methods on non-parametrized surfaces
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is concerned with computational methods for solving boundary integral equations (BIE) on surfaces defined without explicit parametrization, called Implicit Boundary Integral Methods (IBIM). Using implicit methods for describing surfaces, such as the level-set method, can be advantageous for complex geometries and problems where the surface evolves over time. In the IBIM setting, the surface integrals appearing in the BIE are written as volume integrals over domains surrounding the surface using the signed distance function. The singular integrands defined on the surface become functions singular along a straight line in the volume. Accurately integrating such functions is challenging as the special quadrature rules previously developed for BIE only deal with point singularities aligned with the grid in R2, and not line singularities in R3.In this thesis we focus on developing a framework for integrating three-dimensional functions singular along a line using the trapezoidal rule. We first split the three-dimensional problem in a composition of two-dimensional problems, where the singularity is only in a point unaligned with the grid. We then develop corrected trapezoidal rules to deal with these two-dimensional singular integrands with point singularities unaligned with the grid. Moreover we develop generalizations to such rules to Rn for a wide class of functions which can reach arbitrarily high order. Then we develop expressions and approximations of the singular layer kernels from IBIM in a way that can be used with the corrected trapezoidal rules. The expressions are related to the approximation of the surface in the target points, and the higher the order of approximation of the surface the more accurate the expressions of the kernels.We adapt and apply the quadrature methods to the computation of the electrostatic potential of macromolecules immersed in aqueous solvent. For this application, the surface represents the solute-solvent interface where the molecule and the solvent particles interact. The potential solves the linearized Poisson-Boltzmann equation, but can be written as the solution of a coupled system of BIE. The corrected trapezoidal rules developed aim to showcase IBIM as a valid and robust alternative to standard techniques for BIE for computationally intensive applications.
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| 9. |
- Leitenmaier, Lena
(författare)
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Analysis and numerical methods for multiscale problems in magnetization dynamics
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis investigates a multiscale version of the Landau-Lifshitz equation and how to solve it using the framework of Heterogeneous Multiscale Methods (HMM). The Landau-Lifshitz equation is the governing equation in micromagnetics, modeling magnetization dynamics. The considered problem involves two different scales which interact with each other: a fine scale, on which small material variations can be described, and a coarse scale for the overall magnet. Since the fast variations are much smaller than the coarse scale, the computational cost of resolving these scales in a direct numerical simulation is very high. The idea behind HMM therefore is to use a coarse macro model, involving some missing quantity, in combination with an exact micro model that provides the information necessary to complete the macro model using an averaging process, the so-called upscaling. This approach results in a computational cost that is independent of the fine scale, ε.The included papers focus on different aspects of the problem, together providing both error estimates and implementation details.Paper I investigates homogenization of the given Landau-Lifshitz equation with a rapidly oscillating material coefficient in a periodic setting. Equations for the homogenized solution and the corresponding correctors are derived and estimates for the error introduced by homogenization are given. Both the difference between actual and homogenized solution as well as corrected approximations are considered. We show convergence rates in ε up to final times T ∈ O(εσ), where 0 < σ ≤ 2, in Hq Sobolev norms. Here the choice of q is only restricted by the regularity of the solutions.In Paper II, three different ways to set up HMM are introduced, the flux, field and torque model. Each model involves a different missing quantity in the HMM macro model. In a periodic setting, the errors introduced when approximating the missing quantities are analyzed. In all three models the upscaling errors are bounded similarly and can be reduced to O(ε) when choosing the involved parameters optimally.A finite difference based implementation of the field model is studied in Paper III. Several important aspects, such as choice of time integrator, size of the micro domain, boundary conditions for the micro problem and the influence of various parameters introduced in the upscaling process are discussed. We moreover introduce the idea to use artificial damping in the micro problem to obtain a more efficient implementation.Finally, a more physical setup is considered in Paper IV. A finite element macro model that is combined with a finite difference micro model is proposed. This approach is based on a variation of the flux model introduced in Paper II. A problem setting with Neumann boundary conditions and involving several terms in the so-called effective field is considered. Numerical examples show the viability of the approach.Additionally, several geometric time integrators for the Landau-Lifshitz equation are reviewed and compared in a technical report. Their properties are investigated using numerical examples.
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| 10. |
- Malenova, Gabriela
(författare)
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Uncertainty quantification for high frequency waves
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We consider high frequency waves satisfying the scalar wave equationwith highly oscillatory initial data. The speed of propagation of the mediumas well as the phase and amplitude of the initial data is assumed to beuncertain, described by a finite number of independent random variables withknown probability distributions. We introduce quantities of interest (QoIs)aslocal averages of the squared modulus of the wave solution, or itsderivatives.The regularity of these QoIs in terms of the input random parameters and thewavelength is important for uncertainty quantification methods based oninterpolation in the stochastic space. In particular, the size of thederivativesshould be bounded and independent of the wavelength. In the contributedpapers, we show that the QoIs indeed have this property, despite the highlyoscillatory character of the waves.
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