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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, et al.
(författare)
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A numerical study of scaling relations for non-Newtonian thin film flows with applications in ice sheet modelling
- 2013
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Ingår i: Quarterly Journal of Mechanics and Applied Mathematics. - : Oxford University Press (OUP). - 0033-5614 .- 1464-3855. ; 66:4, s. 417-435
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Tidskriftsartikel (refereegranskat)abstract
- This article treats the viscous, non-Newtonian thin-film flow of ice sheets, governed by the Stokes equations, and the modelling of ice sheets with asymptotic expansion of the analytical solutions in terms of the aspect ratio, which is a small parameter measuring the shallowness of an ice sheet. An asymptotic expansion requires scalings of the field variables with the aspect ratio. There are several, conflicting, scalings in the literature used both for deriving simplified models and for analysis. We use numerical solutions of the Stokes equations for varying aspect ratios in order to compute scaling relations. Our numerically obtained results are compared with three known theoretical scaling relations: the classical scalings behind the Shallow Ice Approximation, the scalings originally used to derive the so-called Blatter-Pattyn equations, and a non-uniform scaling which takes into account a high viscosity boundary layer close to the ice surface. We find that the latter of these theories is the most appropriate one since there is indeed a boundary layer close to the ice surface where scaling relations are different than further down in the ice. This boundary layer is thicker than anticipated and there is no distinct border with the inner layer for aspect ratios appropriate for ice sheets. This makes direct application of solutions obtained by matched asymptotic expansion problematic.
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| 4. |
- Ahlkrona, Josefin, et al.
(författare)
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A numerical study of the validity of Shallow Ice Approximations
- 2012
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Improving numerical ice sheet models is a very active field of research. In part, this is because ice sheet modelling has gained societal relevance in the context of predictions of future sea level rise. Ice sheet modelling is however also a challenging mathematical and computational subject. Since the exact equations governing ice dynamics, the full Stokes equations, are computationally expensive to solve, approximations are crucially needed for many problems. Shallow ice approximations are a family of approximations derived by asymptotic expansion of the exact equations in terms of the aspect ratio, epsilon. Retaining only the zeroth order terms in this expansion yields the by far most frequently used approximation; the Shallow Ice Approximation (SIA). Including terms up to second order yields the Second Order Shallow Ice Approximation (SOSIA), which is a so-called higher order model. Here, we study the validity and accuracy of shallow ice approximations beyond previous analyses of the SIA. We perform a detailed analysis of the assumptions behind shallow ice approximations, i.e. of the order of magnitude of field variables. We do this by using a numerical solution of the exact equations for ice flow over a sloping, undulating bed. We also construct analytical solutions for the SIA and SOSIA and numerically compute the accuracy for varying epsilon by comparing to the exact solution. We find that the assumptions underlying shallow ice approximations are not entirely appropriate since they do not account for a high viscosity boundary layer developing near the ice surface as soon as small bumps are introduced at the ice base. This boundary layer is thick and has no distinct border. Other existing theories which do incorporate the boundary layer are in better, but not full, agreement with our numerical results. Our results reveal that neither the SIA nor the SOSIA is as accurate as suggested by the asymptotic expansion approach. Also, in SOSIA the ice rheology needs to be altered to avoid infinite viscosity, though both our analytical and numerical solutions show that, especially for high bump amplitudes, the accuracy of the SOSIA is highly sensitive to this alternation. However, by updating the SOSIA solution in an iterative manner, we obtain a model which utilises the advantages of shallow ice approximations, while reducing the disadvantages.
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| 5. |
- Ahlkrona, Josefin, et al.
(författare)
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Accuracy of the zeroth- and second-order shallow-ice approximation - numerical and theoretical results
- 2013
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Ingår i: Geoscientific Model Development. - : Copernicus GmbH. - 1991-959X .- 1991-9603. ; 6:6, s. 2135-2152
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Tidskriftsartikel (refereegranskat)abstract
- In ice sheet modelling, the shallow-ice approximation (SIA) and second-order shallow-ice approximation (SOSIA) schemes are approaches to approximate the solution of the full Stokes equations governing ice sheet dynamics. This is done by writing the solution to the full Stokes equations as an asymptotic expansion in the aspect ratio epsilon, i.e. the quotient between a characteristic height and a characteristic length of the ice sheet. SIA retains the zeroth-order terms and SOSIA the zeroth-, first-, and second-order terms in the expansion. Here, we evaluate the order of accuracy of SIA and SOSIA by numerically solving a two-dimensional model problem for different values of epsilon, and comparing the solutions with afinite element solution to the full Stokes equations obtained from Elmer/Ice. The SIA and SOSIA solutions are also derived analytically for the model problem. For decreasing epsilon, the computed errors in SIA and SOSIA decrease, but not always in the expected way. Moreover, they depend critically on a parameter introduced to avoid singularities in Glen's flow law in the ice model. This is because the assumptions behind the SIA and SOSIA neglect a thick, high-viscosity boundary layer near the ice surface. The sensitivity to the parameter is explained by the analytical solutions. As a verification of the comparison technique, the SIA and SOSIA solutions for a fluid with Newtonian rheology are compared to the solutions by Elmer/Ice, with results agreeing very well with theory.
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| 6. |
- 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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| 7. |
- Ahlkrona, Josefin, et al.
(författare)
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Dynamically coupling the non-linear Stokes equations with the shallow ice approximation in glaciology : Description and first applications of the ISCAL method
- 2016
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 308, s. 1-19
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Tidskriftsartikel (refereegranskat)abstract
- We propose and implement a new method, called the Ice Sheet Coupled Approximation Levels (ISCAL) method, for simulation of ice sheet flow in large domains during long time-intervals. The method couples the full Stokes (FS) equations with the Shallow Ice Approximation (SIA). The part of the domain where SIA is applied is determined automatically and dynamically based on estimates of the modeling error. For a three dimensional model problem, ISCAL computes the solution substantially faster with a low reduction in accuracy compared to a monolithic FS. Furthermore, ISCAL is shown to be able to detect rapid dynamic changes in the flow. Three different error estimations are applied and compared. Finally, ISCAL is applied to the Greenland Ice Sheet on a quasi-uniform grid, proving ISCAL to be a potential valuable tool for the ice sheet modeling community.
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| 8. |
- 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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| 9. |
- Arévalo, Carmen, et al.
(författare)
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Improving the accuracy of BDF methods for index 3 dierential algebraic equations
- 1995
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Ingår i: BIT. - 0006-3835 .- 1572-9125. ; 35:3, s. 297-308
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Tidskriftsartikel (refereegranskat)abstract
- Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, and O(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct to O(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct the O(1) errors in the algebraic variables appearing after a change of order.
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| 10. |
- Bani-Hashemian, Hossein, et al.
(författare)
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Efficient sampling in event-driven algorithms for reaction-diffusion processes
- 2011
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In event-driven algorithms for simulation of diffusing, colliding, and reacting particles, new positions and events are sampled from the cumulative distribution function (CDF) of a probability distribution. The distribution is sampled frequently and it is important for the efficiency of the algorithm that the sampling is fast. The CDF is known analytically or computed numerically. Analytical formulas are sometimes rather complicated making them difficult to evaluate. The CDF may be stored in a table for interpolation or computed directly when it is needed. Different alternatives are compared for chemically reacting molecules moving by Brownian diffusion in two and three dimensions. The best strategy depends on the dimension of the problem, the length of the time interval, the density of the particles, and the number of different reactions.
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