| 1. |
- Duru, Kenneth, 1975-
(författare)
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Perfectly Matched Layers and High Order Difference Methods for Wave Equations
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The perfectly matched layer (PML) is a novel technique to simulate the absorption of waves in unbounded domains. The underlying equations are often a system of second order hyperbolic partial differential equations. In the numerical treatment, second order systems are often rewritten and solved as first order systems. There are several benefits with solving the equations in second order formulation, though. However, while the theory and numerical methods for first order hyperbolic systems are well developed, numerical techniques to solve second order hyperbolic systems are less complete.We construct a strongly well-posed PML for second order systems in two space dimensions, focusing on the equations of linear elasto-dynamics. In the continuous setting, the stability of both first order and second order formulations are linearly equivalent. We have found that if the so-called geometric stability condition is violated, approximating the first order PML with standard central differences leads to a high frequency instability at most resolutions. In the second order setting growth occurs only if growing modes are well resolved. We determine the number of grid points that can be used in the PML to ensure a discretely stable PML, for several anisotropic elastic materials.We study the stability of the PML for problems where physical boundaries are important. First, we consider the PML in a waveguide governed by the scalar wave equation. To ensure the accuracy and the stability of the discrete PML, we derived a set of equivalent boundary conditions. Second, we consider the PML for second order symmetric hyperbolic systems on a half-plane. For a class of stable boundary conditions, we derive transformed boundary conditions and prove the stability of the corresponding half-plane problem. Third, we extend the stability analysis to rectangular elastic waveguides, and demonstrate the stability of the discrete PML.Building on high order summation-by-parts operators, we derive high order accurate and strictly stable finite difference approximations for second order time-dependent hyperbolic systems on bounded domains. Natural and mixed boundary conditions are imposed weakly using the simultaneous approximation term method. Dirichlet boundary conditions are imposed strongly by injection. By constructing continuous strict energy estimates and analogous discrete strict energy estimates, we prove strict stability.
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| 2. |
- Eriksson, Sofia, 1982-
(författare)
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Stable Numerical Methods with Boundary and Interface Treatment for Applications in Aerodynamics
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In numerical simulations, problems stemming from aerodynamics pose many challenges for the method used. Some of these are addressed in this thesis, such as the fluid interacting with objects, the presence of shocks, and various types of boundary conditions.Scenarios of the kind mentioned above are described mathematically by initial boundary value problems (IBVPs). We discretize the IBVPs using high order accurate finite difference schemes on summation by parts form (SBP), combined with weakly imposed boundary conditions, a technique called simultaneous approximation term (SAT). By using the energy method, stability can be shown.The weak implementation is compared to the more commonly used strong implementation, and it is shown that the weak technique enhances the rate of convergence to steady state for problems with solid wall boundary conditions. The analysis is carried out for a linear problem and supported numerically by simulations of the fully non-linear Navier–Stokes equations.Another aspect of the boundary treatment is observed for fluid structure interaction problems. When exposed to eigenfrequencies, the coupled system starts oscillating, a phenomenon called flutter. We show that the strong implementation sometimes cause instabilities that can be mistaken for flutter.Most numerical schemes dealing with flows including shocks are first order accurate to avoid spurious oscillations in the solution. By modifying the SBP-SAT technique, a conservative and energy stable scheme is derived where the order of accuracy can be lowered locally. The new scheme is coupled to a shock-capturing scheme and it retains the high accuracy in smooth regions.For problems with complicated geometry, one strategy is to couple the finite difference method to the finite volume method. We analyze the accuracy of the latter on unstructured grids. For grids of bad quality the truncation error can be of zeroth order, indicating that the method is inconsistent, but we show that some of the accuracy is recovered.We also consider artificial boundary closures on unbounded domains. Non-reflecting boundary conditions for an incompletely parabolic problem are derived, and it is shown that they yield well-posedness. The SBP-SAT methodology is employed, and we prove that the discretized problem is stable.
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| 3. |
- Fogelklou, Oswald, 1975-
(författare)
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Computer-Assisted Proofs and Other Methods for Problems Regarding Nonlinear Differential Equations
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis treats some problems concerning nonlinear differential equations. In the first two papers computer-assisted proofs are used. The differential equations there are rewritten as fixed point problems, and the existence of solutions are proved. The problem in the first paper is one-dimensional; with one boundary condition given by an integral. The problem in the second paper is three-dimensional, and Dirichlet boundary conditions are used. Both problems have their origins in fluid dynamics. Paper III describes an inverse problem for the heat equation. Given the solution, a solution dependent diffusion coefficient is estimated by intervals at a finite set of points. The method includes the construction of set-valued level curves and two-dimensional splines. In paper IV we prove that there exists a unique, globally attracting fixed point for a differential equation system. The differential equation system arises as the number of peers in a peer-to-peer network, which is described by a suitably scaled Markov chain, goes to infinity. In the proof linearization and Dulac's criterion are used.
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| 4. |
- Kormann, Katharina, 1982-
(författare)
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Efficient and Reliable Simulation of Quantum Molecular Dynamics
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecular processes. Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields.Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices.As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem.
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| 5. |
- Pettersson, Per, 1981-
(författare)
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Uncertainty Quantification and Numerical Methods for Conservation Laws
- 2013
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Conservation laws with uncertain initial and boundary conditions are approximated using a generalized polynomial chaos expansion approach where the solution is represented as a generalized Fourier series of stochastic basis functions, e.g. orthogonal polynomials or wavelets. The stochastic Galerkin method is used to project the governing partial differential equation onto the stochastic basis functions to obtain an extended deterministic system.The stochastic Galerkin and collocation methods are used to solve an advection-diffusion equation with uncertain viscosity. We investigate well-posedness, monotonicity and stability for the stochastic Galerkin system. High-order summation-by-parts operators and weak imposition of boundary conditions are used to prove stability. We investigate the impact of the total spatial operator on the convergence to steady-state. Next we apply the stochastic Galerkin method to Burgers' equation with uncertain boundary conditions. An analysis of the truncated polynomial chaos system presents a qualitative description of the development of the solution over time. An analytical solution is derived and the true polynomial chaos coefficients are shown to be smooth, while the corresponding coefficients of the truncated stochastic Galerkin formulation are shown to be discontinuous. We discuss the problematic implications of the lack of known boundary data and possible ways of imposing stable and accurate boundary conditions.We present a new fully intrusive method for the Euler equations subject to uncertainty based on a Roe variable transformation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, it is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. A multiwavelet basis that can handle discontinuities in a robust way is used.Finally, we investigate a two-phase flow problem. Based on regularity analysis of the generalized polynomial chaos coefficients, we present a hybrid method where solution regions of varying smoothness are coupled weakly through interfaces. In this way, we couple smooth solutions solved with high-order finite difference methods with non-smooth solutions solved for with shock-capturing methods.
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| 6. |
- Rezaeiravesh, Saleh
(författare)
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Application of Uncertainty Quantification Techniques to Studies of Wall-Bounded Turbulent Flows
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Wall-bounded turbulent flows occur in many engineering applications. The quantities of interest (QoIs) of these flows can be accurately obtained through experimental measurements and scale-resolving numerical approaches, such as large eddy simulation (LES). However, due to the prohibitive computational costs imposed by the turbulent boundary layers (TBL) involved in these flows, the use of a standard wall-resolving (WR)LES is limited to low Reynolds (Re-) numbers. As an alternative, wall-modeled (WM)LES can be employed, in which the near-wall region of the TBL is modeled.This thesis evaluates the uncertainties involved in the measured QoIs of a set of experiments on TBLs, and also, investigates the predictive accuracy and sensitivity of LES, both wall-resolving and wall-modeled. For these purposes, different uncertainty quantification (UQ) techniques are employed.In particular, such techniques are applied to the forward (uncertainty propagation) and inverse (parameter estimation) problems involved in the measurement of mean velocity and wall shear stress using hot-wire anemometry and oil-film interferometry, respectively. The possibility of reducing epistemic uncertainties by a more detailed analysis is demonstrated. The metamodels constructed by combining non-intrusive generalized polynomial chaos expansion with the stochastic-collocation method are employed to investigate the sensitivity of WRLES of turbulent channel flow to grid resolution. This research further provides a set of recommendations for grid resolution. Through the use of a systematic simulation campaign, the predictive accuracy and sensitivity of WMLES of the same flow is investigated with respect to several influential factors. The metamodel technique is also used to explore the sensitivity to the grid anisotropy and wall model parameters. Based on this study, a set of best practice guidelines is obtained for WMLES of turbulent channel flow, the validity of which is confirmed in a wide range of Re-numbers. For all the UQ-based studies, variance-based sensitivity analysis is also performed.For WMLES, this thesis also introduces several developments in wall-stress modeling. The performance of algebraic wall-stress models is investigated in an a-priori framework, using accurate WRLES data. Two novel approaches based on integrating the wall model and dynamically adjusting its parameters are proposed and tested. This thesis also contributes to the development of an open-source library for WMLES based on OpenFOAM, which is used in the afore-mentioned systematic study for channel flow.
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| 7. |
- Buntin, Laura M.
(författare)
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Efficient Electromagnetic Induction Modelling : Adaptive mesh optimisation, advanced boundary methods and iterative solution techniques
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Forward modelling of electromagnetic induction data simulates the electric and magnetic fields within a computational domain for a given distribution of electromagnetic material properties and a given source of the electromagnetic field. The quantities of interest are the fields at receiver locations at the Earth's surface. Reliable results require high accuracy solutions at the receivers. First and foremost, numerical computations need to be accurate, but ideally they are also resource efficient, i.e., as fast and cheap as possible. Run time and memory demand mainly depend on the size of the numerical problem to be solved. This thesis addresses specific steps within the forward modelling procedure of electromagnetic induction data in order to improve the solution accuracy of forward modelling as well as to reduce computational resources. The solution accuracy is strongly influenced by the spatial discretisation of the computational domain, which directly correlates with the numerical problem size. To optimise the solution accuracy while keeping the numerical problem size as small as possible, a goal-oriented adaptive mesh refinement scheme for three-dimensional controlled-source electromagnetic models is developed. In addition, this thesis investigates the influence of different types of boundary methods on the solution accuracy at the receivers. To replace inhomogeneous boundary conditions in magnetotelluric total-field modelling by perfectly-matched layers (PML), a domain decomposition approach (the total and scattered field decomposition) is adapted for Earth models. By reducing boundary effects, the approach yields superior solution accuracy for specific types of models. The fastest and most memory-efficient way to solve large numerical problems are iterative solution methods. Iterative solvers, however, work poorly for numerical systems arising from domains bounded by PML. In this thesis a preconditioned iterative solution framework that efficiently solves PML-bounded magnetotelluric models is proposed.
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| 8. |
- Kronbichler, Martin, 1983-
(författare)
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Computational Techniques for Coupled Flow-Transport Problems
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis presents numerical techniques for solving problems of incompressible flow coupled to scalar transport equations using finite element discretizations in space. The two applications considered in this thesis are multi-phase flow, modeled by level set or phase field methods, and planetary mantle convection based on the Boussinesq approximation. A systematic numerical study of approximation errors in evaluating the surface tension in finite element models for two-phase flow is presented. Forces constructed from a gradient in the same discrete function space as used for the pressure are shown to give the best performance. Moreover, two approaches for introducing contact line dynamics into level set methods are proposed. Firstly, a multiscale approach extracts a slip velocity from a micro simulation based on the phase field method and imposes it as a boundary condition in the macro model. This multiscale method is shown to provide an efficient model for the simulation of contact-line driven flow. The second approach combines a level set method based on a smoothed color function with a the phase field method in different parts of the domain. Away from contact lines, the additional information in phase field models is not necessary and it is disabled from the equations by a switch function. An in-depth convergence study is performed in order to quantify the benefits from this combination. Also, the resulting hybrid method is shown to satisfy an a priori energy estimate. For the simulation of mantle convection, an implementation framework based on modern finite element and solver packages is presented. The framework is capable of running on today's large computing clusters with thousands of processors. All parts in the solution chain, from mesh adaptation over assembly to the solution of linear systems, are done in a fully distributed way. These tools are used for a parallel solver that combines higher order time and space discretizations. For treating the convection-dominated temperature equation, an advanced stabilization technique based on an artificial viscosity is used. For more efficient evaluation of finite element operators in iterative methods, a matrix-free implementation built on cell-based quadrature is proposed. We obtain remarkable speedups over sparse matrix-vector products for many finite elements which are of practical interest. Our approach is particularly efficient for systems of differential equations.
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| 9. |
- Mattsson, Ken, 1970-
(författare)
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Summation-by-Parts Operators for High Order Finite Difference Methods
- 2003
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- High order accurate finite difference methods for hyperbolic and parabolic initial boundary value problems (IBVPs) are considered. Particular focus is on time dependent wave propagating problems in complex domains. Typical applications are acoustic and electromagnetic wave propagation and fluid dynamics. To solve such problems efficiently a strictly stable, high order accurate method is required.Our recipe to obtain such schemes is to: i) Approximate the (first and second) derivatives of the IBVPs with central finite difference operators, that satisfy a summation by parts (SBP) formula. ii) Use specific procedures for implementation of boundary conditions, that preserve the SBP property. iii) Add artificial dissipation. iv) Employ a multi block structure.Stable schemes for weakly nonlinear IBVPs require artificial dissipation to absorb the energy of the unresolved modes. This led to the construction of accurate and efficient artificial dissipation operators of SBP type, that preserve the energy and error estimate of the original problem.To solve problems on complex geometries, the computational domain is broken up into a number of smooth and structured meshes, in a multi block fashion. A stable and high order accurate approximation is obtained by discretizing each subdomain using SBP operators and using the Simultaneous Approximation Term (SAT) procedure for both the (external) boundary and the (internal) interface conditions.Steady and transient aerodynamic calculations around an airfoil were performed, where the first derivative SBP operators and the new artificial dissipation operators were combined to construct high order accurate upwind schemes. The computations showed that for time dependent problems and fine structures, high order methods are necessary to accurately compute the solution, on reasonably fine grids.The construction of high order accurate SBP operators for the second derivative is one of the considerations in this thesis. It was shown that the second derivative operators could be closed with two order less accuracy at the boundaries and still yield design order of accuracy, if an energy estimate could be obtained.
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| 10. |
- Nikkar, Samira
(författare)
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Stable High Order Finite Difference Methods for Wave Propagation and Flow Problems on Deforming Domains
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We construct stable, accurate and efficient numerical schemes for wave propagation and flow problems posed on spatial geometries that are moving, deforming, erroneously described or non-simply connected. The schemes are on Summation-by-Parts (SBP) form, combined with the Simultaneous Approximation Term (SAT) technique for imposing initial and boundary conditions. The main analytical tool is the energy method, by which well-posedness, stability and conservation are investigated. To handle the deforming domains, time-dependent coordinate transformations are used to map the problem to fixed geometries.The discretization is performed in such a way that the Numerical Geometric Conservation Law (NGCL) is satisfied. Additionally, even though the schemes are constructed on fixed domains, time-dependent penalty formulations are necessary, due to the originally moving boundaries. We show how to satisfy the NGCL and present an automatic formulation for the penalty operators, such that the correct number of boundary conditions are imposed, when and where required.For problems posed on erroneously described geometries, we investigate how the accuracy of the solution is affected. It is shown that the inaccurate geometry descriptions may lead to wrong wave speeds, a misplacement of the boundary condition, the wrong boundary operator or a mismatch of data. Next, the SBP-SAT technique is extended to time-dependent coupling procedures for deforming interfaces in hyperbolic problems. We prove conservation and stability and show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the variations of the interface location while the NGCL is preserved.Moreover, dual consistent SBP-SAT schemes for the linearized incompressible Navier-Stokes equations posed on deforming domains are investigated. To simplify the derivations of the dual problem and incorporate the motions of the boundaries, the second order formulation is reduced to first order and the problem is transformed to a fixed domain. We prove energy stability and dual consistency. It is shown that the solution as well as the divergence of the solution converge with the design order of accuracy, and that functionals of the solution are superconverging.Finally, initial boundary value problems posed on non-simply connected spatial domains are investigated. The new formulation increases the accuracy of the scheme by minimizing the use of multi-block couplings. In order to show stability, the spectrum of the semi-discrete SBP-SAT formulation is studied. We show that the eigenvalues have the correct sign, which implies stability, in combination with the SBP-SAT technique in time.
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