| 1. |
- Berg, Jens, 1982-
(författare)
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Stable and High-Order Finite Difference Methods for Multiphysics Flow Problems
- 2013
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Partial differential equations (PDEs) are used to model various phenomena in nature and society, ranging from the motion of fluids and electromagnetic waves to the stock market and traffic jams. There are many methods for numerically approximating solutions to PDEs. Some of the most commonly used ones are the finite volume method, the finite element method, and the finite difference method. All methods have their strengths and weaknesses, and it is the problem at hand that determines which method that is suitable. In this thesis, we focus on the finite difference method which is conceptually easy to understand, has high-order accuracy, and can be efficiently implemented in computer software.We use the finite difference method on summation-by-parts (SBP) form, together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT). Together, SBP and SAT provide a technique for overcoming most of the drawbacks of the finite difference method. The SBP-SAT technique can be used to derive energy stable schemes for any linearly well-posed initial boundary value problem. The stability is not restricted by the order of accuracy, as long as the numerical scheme can be written in SBP form. The weak boundary conditions can be extended to interfaces which are used either in domain decomposition for geometric flexibility, or for coupling of different physics models.The contributions in this thesis are twofold. The first part, papers I-IV, develops stable boundary and interface procedures for computational fluid dynamics problems, in particular for problems related to the Navier-Stokes equations and conjugate heat transfer. The second part, papers V-VI, utilizes duality to construct numerical schemes which are not only energy stable, but also dual consistent. Dual consistency alone ensures superconvergence of linear integral functionals from the solutions of SBP-SAT discretizations. By simultaneously considering well-posedness of the primal and dual problems, new advanced boundary conditions can be derived. The new duality based boundary conditions are imposed by SATs, which by construction of the continuous boundary conditions ensure energy stability, dual consistency, and functional superconvergence of the SBP-SAT schemes.
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| 2. |
- Hellander, Andreas, 1982-
(författare)
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Multiscale Stochastic Simulation of Reaction-Transport Processes : Applications in Molecular Systems Biology
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Quantitative descriptions of reaction kinetics formulated at the stochastic mesoscopic level are frequently used to study various aspects of regulation and control in models of cellular control systems. For this type of systems, numerical simulation offers a variety of challenges caused by the high dimensionality of the problem and the multiscale properties often displayed by the biochemical model. In this thesis I have studied several aspects of stochastic simulation of both well-stirred and spatially heterogenous systems. In the well-stirred case, a hybrid method is proposed that reduces the dimension and stiffness of a model. We also demonstrate how both a high performance implementation and a variance reduction technique based on quasi-Monte Carlo can reduce the computational cost to estimate the probability density of the system. In the spatially dependent case, the use of unstructured, tetrahedral meshes to sample realizations of the stochastic process is proposed. Using such meshes, we then extend the reaction-diffusion framework to incorporate active transport of cellular cargo in a seamless manner. Finally, two multilevel methods for spatial stochastic simulation are considered. One of them is a space-time adaptive method combining exact stochastic, approximate stochastic and macroscopic modeling levels to reduce the simualation cost. The other method blends together mesoscale and microscale simulation methods to locally increase modeling resolution.
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| 3. |
- Ahlkrona, Josefin, 1985-
(författare)
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Computational Ice Sheet Dynamics : Error control and efficiency
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Ice sheets, such as the Greenland Ice Sheet or Antarctic Ice Sheet, have a fundamental impact on landscape formation, the global climate system, and on sea level rise. The slow, creeping flow of ice can be represented by a non-linear version of the Stokes equations, which treat ice as a non-Newtonian, viscous fluid. Large spatial domains combined with long time spans and complexities such as a non-linear rheology, make ice sheet simulations computationally challenging. The topic of this thesis is the efficiency and error control of large simulations, both in the sense of mathematical modelling and numerical algorithms. In the first part of the thesis, approximative models based on perturbation expansions are studied. Due to a thick boundary layer near the ice surface, some classical assumptions are inaccurate and the higher order model called the Second Order Shallow Ice Approximation (SOSIA) yields large errors. In the second part of the thesis, the Ice Sheet Coupled Approximation Level (ISCAL) method is developed and implemented into the finite element ice sheet model Elmer/Ice. The ISCAL method combines the Shallow Ice Approximation (SIA) and Shelfy Stream Approximation (SSA) with the full Stokes model, such that the Stokes equations are only solved in areas where both the SIA and SSA is inaccurate. Where and when the SIA and SSA is applicable is decided automatically and dynamically based on estimates of the modeling error. The ISCAL method provides a significant speed-up compared to the Stokes model. The third contribution of this thesis is the introduction of Radial Basis Function (RBF) methods in glaciology. Advantages of RBF methods in comparison to finite element methods or finite difference methods are demonstrated.
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| 4. |
- Amoignon, Olivier, 1969-
(författare)
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Numerical Methods for Aerodynamic Shape Optimization
- 2005
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Gradient-based aerodynamic shape optimization, based on Computational Fluid Dynamics analysis of the flow, is a method that can automatically improve designs of aircraft components. The prospect is to reduce a cost function that reflects aerodynamic performances.When the shape is described by a large number of parameters, the calculation of one gradient of the cost function is only feasible by recourse to techniques that are derived from the theory of optimal control. In order to obtain the best computational efficiency, the so called adjoint method is applied here on the complete mapping, from the parameters of design to the values of the cost function. The mapping considered here includes the Euler equations for compressible flow discretized on unstructured meshes by a median-dual finite-volume scheme, the primal-to-dual mesh transformation, the mesh deformation, and the parameterization. The results of the present research concern the detailed derivations of expressions, equations, and algorithms that are necessary to calculate the gradient of the cost function. The discrete adjoint of the Euler equations and the exact dual-to-primal transformation of the gradient have been implemented for 2D and 3D applications in the code Edge, a program of Computational Fluid Dynamics used by Swedish industries.Moreover, techniques are proposed here in the aim to further reduce the computational cost of aerodynamic shape optimization. For instance, an interpolation scheme is derived based on Radial Basis Functions that can execute the deformation of unstructured meshes faster than methods based on an elliptic equation.In order to improve the accuracy of the shape, obtained by numerical optimization, a moving mesh adaptation scheme is realized based on a variable diffusivity equation of Winslow type. This adaptation has been successfully applied on a simple case of shape optimization involving a supersonic flow. An interpolation technique has been derived based on a mollifier in order to improve the convergence of the coupled mesh-flow equations entering the adaptive scheme.The method of adjoint derived here has also been applied successfully when coupling the Euler equations with the boundary-layer and parabolized stability equations, with the aim to delay the laminar-to-turbulent transition of the flow. The delay of transition is an efficient way to reduce the drag due to viscosity at high Reynolds numbers.
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| 5. |
- Bängtsson, Erik, 1975-
(författare)
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Robust Preconditioners Based on the Finite Element Framework
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Robust preconditioners on block-triangular and block-factorized form for three types of linear systems of two-by-two block form are studied in this thesis. The first type of linear systems, which are dense, arise from a boundary element type of discretization of crack propagation problems. Numerical experiment show that simple algebraic preconditioning strategies results in iterative schemes that are highly competitive with a direct solution method. The second type of algebraic systems, which are sparse, indefinite and nonsymmetric, arise from a finite element (FE) discretization of the partial differential equations (PDE) that describe (visco)elastic glacial isostatic adjustment (GIA). The Schur complement approximation in the block preconditioners is constructed by assembly of local, exactly computed Schur matrices. The quality of the approximation is verified in numerical experiments. When the block preconditioners for the indefinite problem are combined with an inner iterative scheme preconditioned by a (nearly) optimal multilevel preconditioner, the resulting preconditioner is (nearly) optimal and robust with respect to problem size, material parameters, number of space dimensions, and coefficient jumps. Two approaches to mathematically formulate the PDEs for GIA are compared. In the first approach the equations are formulated in their full complexity, whereas in the second their formulation is confined to the features and restrictions of the employed FE package. Different solution methods for the algebraic problem are used in the two approaches. Analysis and numerical experiments reveal that the first strategy is more accurate and efficient than the latter. The block structure in the third type of algebraic systems is due to a fine-coarse splitting of the unknowns. The inverse of the pivot block is approximated by a sparse matrix which is assembled from local, exactly inverted matrices. Numerical experiments and analysis of the approximation show that it is robust with respect to problem size and coefficient jumps.
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| 6. |
- Cheng, Gong, 1986-
(författare)
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Numerical ice sheet modeling : Forward and inverse problems
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Ice sheets have strong influence on the climate system. Numerical simulation provides a mathematical tool to study the ice dynamics in the past and to predict their contribution to climate change in the future. Large scale ice sheets behave as incompressible non-Newtonian fluid. The evolution of ice sheet is governed by the conservation laws of mass, momentum and energy, which is formulated as a system of partial differential equations. Improving the efficiency of numerical ice sheet modeling is always a desirable feature since many of the applications have large domain and aim for long time span. With such a goal, the first part of this thesis focuses on developing efficient and accurate numerical methods for ice sheet simulation.A large variety of physical processes are involved in ice dynamics, which are described by physical laws with parameters measured from experiments and field work. These parameters are considered as the inputs of the ice sheet simulations. In certain circumstances, some parameters are unavailable or can not be measured directly. Therefore, the second part of this thesis is devoted to reveal these physical parameters by solving inverse problems.In the first part, improvements of temporal and spatial discretization methods and a sub-grid boundary treatment are purposed. We developed an adaptive time stepping method in Paper I to automatically adjust the time steps based on stability and accuracy criteria. We introduced an anisotropic Radial Basis Function method for the spatial discretization of continental scale ice sheet simulations in Paper II. We designed a sub-grid method for solving grounding line migration problem with Stokes equations in Paper VI.The second part of the thesis consists of analysis and numerical experiments on inverse problems. In Paper IV and V, we conducted sensitivity analysis and numerical examples of the inversion on time dependent ice sheet simulations. In Paper III, we solved an inverse problem for the thermal conductivity of firn pack at Lomonosovfonna, Svalbard, using the subsurface temperature measurements.
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| 7. |
- Edelvik, Fredrik, 1972-
(författare)
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Hybrid Solvers for the Maxwell Equations in Time-Domain
- 2002
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The most commonly used method for the time-domain Maxwell equations is the Finite-Difference Time-Domain method (FDTD). This is an explicit, second-order accurate method, which is used on a staggered Cartesian grid. The main drawback with the FDTD method is its inability to accurately model curved objects and small geometrical features. This is due to the Cartesian grid, which leads to a staircase approximation of the geometry and small details are not resolved at all.This thesis presents different ways to circumvent this drawback, but still take advantage of the benefits of the FDTD method. An approach to avoid staircasing errors but still retain the efficiency of the FDTD method is to use a hybrid grid. A few layers of unstructured cells are used close to curved objects and a Cartesian grid is used for the rest of the domain. For the choice of solver on the unstructured grid two different alternatives are compared: an explicit Finite-Volume Time-Domain (FVTD) solver and an implicit Finite-Element Time-Domain (FETD) solver.The hybrid solvers calculate the scattering from complex objects much more efficiently compared to using FDTD on highly resolved Cartesian grids. For the same accuracy in the solution roughly a factor of 10 in memory requirements and a factor of 20 in execution time are gained.The ability to model features that are small relative to the cell size is often important in electromagnetic simulations. In this thesis a technique to generalize a well-known subcell model for thin wires, in order to take arbitrarily oriented wires in FETD and FDTD into account, is proposed. The method gives considerable modeling flexibility compared to earlier methods and is proven stable. The results show excellent consistency and very good accuracy on different antenna configurations.The recursive convolution method is often used to model frequency dispersive materials in FDTD. This method is used to enable modeling of such materials in the unstructured FVTD and FETD solvers. The stability of both solvers is analyzed and their accuracy is demonstrated by computing the radar cross section for homogeneous as well as layered spheres with frequency dependent permittivity.
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| 8. |
- Ekström, Sven-Erik
(författare)
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A vertex-centered discontinuous Galerkin method for flow problems
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The understanding of flow problems, and finding their solution, has been important for most of human history, from the design of aqueducts to boats and airplanes. The use of physical miniature models and wind tunnels were, and still are, useful tools for design, but with the development of computers, an increasingly large part of the design process is assisted by computational fluid dynamics (CFD).Many industrial CFD codes have their origins in the 1980s and 1990s, when the low order finite volume method (FVM) was prevalent. Discontinuous Galerkin methods (DGM) have, since the turn of the century, been seen as the successor of these methods, since it is potentially of arbitrarily high order. In its lowest order form DGM is equivalent to FVM. However, many existing codes are not compatible with standard DGM and would need a complete rewrite to obtain the advantages of the higher order.This thesis shows how to extend existing vertex-centered and edge-based FVM codes to higher order, using a special kind of DGM discretization, which is different from the standard cell-centered type. Two model problems are examined to show the necessary data structures that need to be constructed, the order of accuracy for the method, and the use of an hp-adaptation scheme to resolve a developing shock. Then the method is further developed to solve the steady Euler equations, within the existing industrial Edge code, using acceleration techniques such as local time stepping and multigrid.With the ever increasing need for more efficient and accurate solvers and algorithms in CFD, the modified DGM presented in this thesis could be used to help and accelerate the adoption of high order methods in industry.
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| 9. |
- Hellander, Stefan, 1985-
(författare)
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Stochastic Simulation of Reaction-Diffusion Processes
- 2013
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Numerical simulation methods have become an important tool in the study of chemical reaction networks in living cells. Many systems can, with high accuracy, be modeled by deterministic ordinary differential equations, but other systems require a more detailed level of modeling. Stochastic models at either the mesoscopic level or the microscopic level can be used for cases when molecules are present in low copy numbers.In this thesis we develop efficient and flexible algorithms for simulating systems at the microscopic level. We propose an improvement to the Green's function reaction dynamics algorithm, an efficient microscale method. Furthermore, we describe how to simulate interactions with complex internal structures such as membranes and dynamic fibers.The mesoscopic level is related to the microscopic level through the reaction rates at the respective scale. We derive that relation in both two dimensions and three dimensions and show that the mesoscopic model breaks down if the discretization of space becomes too fine. For a simple model problem we can show exactly when this breakdown occurs.We show how to couple the microscopic scale with the mesoscopic scale in a hybrid method. Using the fact that some systems only display microscale behaviour in parts of the system, we can gain computational time by restricting the fine-grained microscopic simulations to only a part of the system.Finally, we have developed a mesoscopic method that couples simulations in three dimensions with simulations on general embedded lines. The accuracy of the method has been verified by comparing the results with purely microscopic simulations as well as with theoretical predictions.
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| 10. |
- Monokrousos, Antonios
(författare)
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Optimisation and control of boundary layer flows
- 2009
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Both optimal disturbances and optimal control are studied by means of numerical simulations for the case of the flat-plate boundary-layer flow. The optimisation method is the Lagrange multiplier technique where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearised Navier–Stokes equations. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. The optimal disturbances for spanwise wavelengths of the order of the boundary layer thickness are streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. Control is applied to the bypass-transition scenario with high levels of free-stream turbulence. In this scenario low frequency perturbations enter the boundary layer and streamwise elongated disturbances emerge due to the non-modal growth. These so-called streaks are growing in amplitude until they reach high enough energy levels and breakdown into turbulent spots via their secondary instability. When control is applied in the form of wall blowing and suction, within the region that it is active, the growth of the streaks is delayed, which implies a delay of the whole transition process. Additionally, a comparison with experimental work is performed demonstrating a remarkable agreement in the disturbance attenuation once the differences between the numerical and experimental setup are reduced.
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