| 1. |
- O'Reilly, Ossian, 1986-
(författare)
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Numerical methods for wave propagation in solids containing faults and fluid-filled fractures
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis develops numerical methods for the simulation of wave propagation in solids containing faults and fluid-filled fractures. These techniques have applications in earthquake hazard analysis, seismic imaging of reservoirs, and volcano seismology. A central component of this work is the coupling of mechanical systems. This aspect involves the coupling of both ordinary differential equations (ODE)(s) and partial differential equations (PDE)(s) along curved interfaces. All of these problems satisfy a mechanical energy balance. This mechanical energy balance is mimicked by the numerical scheme using high-order accurate difference approximations that satisfy the principle of summation by parts, and by weakly enforcing the coupling conditions. The first part of the thesis considers the simulation of dynamic earthquake ruptures along non-planar fault geometries and the simulation of seismic wave radiation from earthquakes, when the earthquakes are idealized as point moment tensor sources. The dynamic earthquake rupture process is simulated by coupling the elastic wave equation at a fault interface to nonlinear ODEs that describe the fault mechanics. The fault geometry is complex and treated by combining structured and unstructured grid techniques. In other applications, when the earthquake source dimension is smaller than wavelengths of interest, the earthquake can be accurately described by a point moment tensor source localized at a single point. The numerical challenge is to discretize the point source with high-order accuracy and without producing spurious oscillations.The second part of the thesis presents a numerical method for wave propagation in and around fluid-filled fractures. This problem requires the coupling of the elastic wave equation to a fluid inside curved and branching fractures in the solid. The fluid model is a lubrication approximation that incorporates fluid inertia, compressibility, and viscosity. The fracture geometry can have local irregularities such as constrictions and tapered tips. The numerical method discretizes the fracture geometry by using curvilinear multiblock grids and applies implicit-explicit time stepping to isolate and overcome stiffness arising in the semi-discrete equations from viscous diffusion terms, fluid compressibility, and the particular enforcement of the fluid-solid coupling conditions. This numerical method is applied to study the interaction of waves in a fracture-conduit system. A methodology to constrain fracture geometry for oil and gas (hydraulic fracturing) and volcano seismology applications is proposed.The third part of the thesis extends the summation-by-parts methodology to staggered grids. This extension reduces numerical dispersion and enables the formulation of stable and high-order accurate multiblock discretizations for wave equations in first order form on staggered grids. Finally, the summation-by-parts methodology on staggered grids is further extended to second derivatives and used for the treatment of coordinate singularities in axisymmetric wave propagation.
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| 2. |
- Abenius, Erik, 1971-
(författare)
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Direct and Inverse Methods for Waveguides and Scattering Problems in the Time Domain
- 2005
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Numerical simulation is an important tool in understanding the electromagnetic field and how it interacts with the environment. Different topics for time-domain finite-difference (FDTD) and finite-element (FETD) methods for Maxwell's equations are treated in this thesis. Subcell models are of vital importance for the efficient modeling of small objects that are not resolved by the grid. A novel model for thin sheets using shell elements is proposed. This approach has the advantage of taking into account discontinuities in the normal component of the electric field, unlike previous models based on impedance boundary conditions (IBCs). Several results are presented to illustrate the capabilities of the shell element approach. Waveguides are of fundamental importance in many microwave applications, for example in antenna feeds. The key issues of excitation and truncation of waveguides are addressed. A complex frequency shifted form of the uniaxial perfectly matched layer (UPML) absorbing boundary condition (ABC) in FETD is developed. Prism elements are used to promote automatic grid generation and enhance the performance. Results are presented where reflection errors below -70dB are obtained for different types of waveguides, including inhomogeneous cases. Excitation and analysis via the scattering parameters are achieved using waveguide modes computed by a general frequency-domain mode solver for the vector Helmholtz equation. Huygens surfaces are used in both FDTD and FETD for excitation in waveguide ports. Inverse problems have received an increased interest due to the availability of powerful computers. An important application is non-destructive evaluation of material. A time-domain, minimization approach is presented where exact gradients are computed using the adjoint problem. The approach is applied to a general form of Maxwell's equations including dispersive media and UPML. Successful reconstruction examples are presented both using synthetic and experimental measurement data. Parameter reduction of complex geometries using simplified models is an interesting topic that leads to an inverse problem. Gradients for subcell parameters are derived and a successful reconstruction example is presented for a combined dielectric sheet and slot geometry.
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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. |
- 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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| 6. |
- 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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| 7. |
- 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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| 8. |
- Edelvik, Fredrik
(författare)
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Finite volume solvers for the Maxwell equations in time domain
- 2000
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Two unstructured finite volume solvers for the Maxwell equations in 2D and 3D are introduced. The solvers are a generalization of FD–TD to unstructured grids and they use a third-order staggered Adams–Bashforth scheme for time discretization. Analysis and experiments of this time integrator reveal that we achieve a long term stable solution on general triangular grids. A Fourier analysis shows that the 2D solver has excellent dispersion characteristics on uniform triangular grids. In 3D a spatial filter of Laplace type is introduced to enable long simulations without suffering from late time instability.The recursive convolution method proposed by Luebbers et al. to extend FD–TD to permit frequency dispersive materials is here generalized to the 3D solver. A better modelling of materials which have a strong frequency dependence in their constitutive parameters is obtained through the use of a general material model.The finite volume solvers are not intended to be stand-alone solvers but one part in two hybrid solvers with FD–TD. The numerical examples in 2D and 3D demonstrate that the hybrid solvers are superior to stand-alone FD–TD in terms of accuracy and efficiency.
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| 9. |
- 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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| 10. |
- Edlund, Johan
(författare)
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A parallel, iterative method of moments and physical optics hybrid solver for arbitrary surfaces
- 2001
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We have developed an MM–PO hybrid solver designed to deliver reasonable accuracy inexpensively in terms of both CPU-time and memory demands. The solver is based on an iterative block Gauss–Seidel process to avoid unnecessary storage and matrix computations, and can be used to solve the radiation and scattering problems for both disjunct and connected regions. It supports thin wires and dielectrica in the MM domain and has been implemented both as a serial and parallel solver.Numerical experiments have been performed on simple objects to demonstrate certain keyfeatures of the solver, and validate the positive and negative aspects of the MM/PO hybrid. Experiments have also been conducted on more complex objects such as a model aircraft, to demonstrate that the good results from the simpler objects are transferrable to the real life situation. The complex geometries have been used to conduct tests to investigate how well parallelised the code is, and the results are satisfactory.
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