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  • 1. Abarbanel, Saul
    et al.
    Ditkowski, Adi
    Gustafsson, Bertil
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    On error bounds of finite difference approximations to partial differential equations: Temporal behavior and rate of convergence2000Report (Other academic)
  • 2. Abarbanel, Saul
    et al.
    Ditkowski, Adi
    Gustafsson, Bertil
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    On error bounds of finite difference approximations to partial differential equations: Temporal behavior and rate of convergence2000In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 15, 79-116 p.Article in journal (Refereed)
  • 3.
    Abbas, Qaisar
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Weak Boundary and Interface Procedures for Wave and Flow Problems2011Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    In this thesis, we have analyzed the accuracy and stability aspects of weak boundary and interface conditions (WBCs) for high order finite difference methods on Summations-By-Parts (SBP) form. The numerical technique has been applied to wave propagation and flow problems.

    The advantage of WBCs over strong boundary conditions is that stability of the numerical scheme can be proven. The boundary procedures in the advection-diffusion equation for a boundary layer problem is analyzed. By performing Navier-Stokes calculations, it is shown that most of the conclusions from the model problem carries over to the fully nonlinear case.

    The work was complemented to include the new idea of using WBCs on multiple grid points in a region, where the data is known, instead of at a single point. It was shown that we can achieve high accuracy, an increased rate of convergence to steady-state and non-reflecting boundary conditions by using this approach.

    Using the SBP technique and WBCs, we have worked out how to construct conservative and energy stable hybrid schemes for shocks using two different approaches. In the first method, we combine a high order finite difference scheme with a second order MUSCL scheme. In the second method, a procedure to locally change the order of accuracy of the finite difference schemes is developed. The main purpose is to obtain a higher order accurate scheme in smooth regions and a low order non-oscillatory scheme in the vicinity of shocks.

    Furthermore, we have analyzed the energy stability of the MUSCL scheme, by reformulating the scheme in the framework of SBP and artificial dissipation operators. It was found that many of the standard slope limiters in the MUSCL scheme do not lead to a negative semi-definite dissipation matrix, as required to get pointwise stability.

    Finally, high order simulations of shock diffracting over a convex wall with two facets were performed. The numerical study is done for a range of Reynolds numbers. By monitoring the velocities at the solid wall, it was shown that the computations were resolved in the boundary layer. Schlieren images from the computational results were obtained which displayed new interesting flow features.

    List of papers
    1. Weak versus strong no-slip boundary conditions for the Navier-Stokes equations
    Open this publication in new window or tab >>Weak versus strong no-slip boundary conditions for the Navier-Stokes equations
    2010 (English)In: Engineering Applications of Computational Fluid Mechanics, ISSN 1994-2060, Vol. 4, 29-38 p.Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-112977 (URN)000276898600003 ()
    Available from: 2010-01-24 Created: 2010-01-24 Last updated: 2018-01-12Bibliographically approved
    2. A weak boundary procedure for high order finite difference approximations of hyperbolic problems
    Open this publication in new window or tab >>A weak boundary procedure for high order finite difference approximations of hyperbolic problems
    2011 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2011-019
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-159353 (URN)
    Available from: 2011-09-23 Created: 2011-09-28 Last updated: 2011-11-04Bibliographically approved
    3. Accurate and stable calculations involving shocks using a new hybrid scheme
    Open this publication in new window or tab >>Accurate and stable calculations involving shocks using a new hybrid scheme
    2009 (English)In: Proc. 19th AIAA CFD Conference, AIAA , 2009Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    AIAA, 2009
    Series
    Conference Proceeding Series, 2009-3985
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-110133 (URN)
    Available from: 2009-11-04 Created: 2009-11-04 Last updated: 2018-01-12Bibliographically approved
    4. A stable and conservative method for locally adapting the design order of finite difference schemes
    Open this publication in new window or tab >>A stable and conservative method for locally adapting the design order of finite difference schemes
    2011 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, 4216-4231 p.Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-134055 (URN)10.1016/j.jcp.2010.11.020 (DOI)000290185000007 ()
    Available from: 2010-11-20 Created: 2010-11-21 Last updated: 2018-01-12Bibliographically approved
    5. Energy stability of the MUSCL scheme
    Open this publication in new window or tab >>Energy stability of the MUSCL scheme
    2010 (English)In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, 61-68 p.Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    Berlin: Springer-Verlag, 2010
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-132925 (URN)10.1007/978-3-642-11795-4_5 (DOI)978-3-642-11794-7 (ISBN)
    Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2011-11-26Bibliographically approved
    6. The Effect of Reynolds Number in High Order Accurate Calculations with Shock Diffraction
    Open this publication in new window or tab >>The Effect of Reynolds Number in High Order Accurate Calculations with Shock Diffraction
    2010 (English)In: Proc. 7th South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2010, 416-423 p.Conference paper, Published paper (Refereed)
    Place, publisher, year, edition, pages
    South African Association for Theoretical and Applied Mechanics, 2010
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-113571 (URN)978-0-620-49192-1 (ISBN)
    Available from: 2010-01-29 Created: 2010-01-29 Last updated: 2018-01-12Bibliographically approved
  • 4.
    Abbas, Qaisar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    A weak boundary procedure for high order finite difference approximations of hyperbolic problems2011Report (Other academic)
  • 5.
    Abbas, Qaisar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Weak versus strong no-slip boundary conditions for the Navier-Stokes equations2010In: Engineering Applications of Computational Fluid Mechanics, ISSN 1994-2060, Vol. 4, 29-38 p.Article in journal (Refereed)
  • 6.
    Abbas, Qaisar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Nordström, Jan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Weak versus Strong No-Slip Boundary Conditions for the Navier-Stokes Equations2008In: Proc. 6th South African Conference on Computational and Applied Mechanics, South African Association for Theoretical and Applied Mechanics , 2008, 52-62 p.Conference paper (Other academic)
  • 7.
    Abbas, Qaisar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    van der Weide, Edwin
    Nordström, Jan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Accurate and stable calculations involving shocks using a new hybrid scheme2009In: Proc. 19th AIAA CFD Conference, AIAA , 2009Conference paper (Refereed)
  • 8.
    Abbas, Qaisar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    van der Weide, Edwin
    Nordström, Jan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Energy stability of the MUSCL scheme2010In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, 61-68 p.Conference paper (Refereed)
  • 9.
    Abenius, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Direct and Inverse Methods for Waveguides and Scattering Problems in the Time Domain2005Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    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.

    List of papers
    1. Hybrid time domain solvers for the Maxwell equations in 2D
    Open this publication in new window or tab >>Hybrid time domain solvers for the Maxwell equations in 2D
    Show others...
    2002 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 53, 2185-2199 p.Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-44333 (URN)10.1002/nme.380 (DOI)
    Projects
    GEMS
    Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2018-01-11Bibliographically approved
    2. Waveguide Truncation Using UPML in the Finite-Element Time-Domain Method
    Open this publication in new window or tab >>Waveguide Truncation Using UPML in the Finite-Element Time-Domain Method
    2005 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2005-026
    National Category
    Computational Mathematics
    Identifiers
    urn:nbn:se:uu:diva-80256 (URN)
    Projects
    GEMS
    Available from: 2007-02-03 Created: 2007-02-03 Last updated: 2014-09-03Bibliographically approved
    3. Solving inverse electromagnetic problems using FDTD and gradient-based minimization
    Open this publication in new window or tab >>Solving inverse electromagnetic problems using FDTD and gradient-based minimization
    2006 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 68, 650-673 p.Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-80802 (URN)10.1002/nme.1731 (DOI)000241713800003 ()
    Projects
    GEMS
    Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2018-01-13Bibliographically approved
    4. Thin Sheet Modeling Using Shell Elements in the Finite-Element Time-Domain Method
    Open this publication in new window or tab >>Thin Sheet Modeling Using Shell Elements in the Finite-Element Time-Domain Method
    2006 (English)In: IEEE Transactions on Antennas and Propagation, ISSN 0018-926X, E-ISSN 1558-2221, Vol. 54, 28-34 p.Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-80325 (URN)10.1109/TAP.2005.861554 (DOI)000235016700004 ()
    Projects
    GEMS
    Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2018-01-13Bibliographically approved
    5. Modeling of inhomogeneous waveguides using hybrid methods
    Open this publication in new window or tab >>Modeling of inhomogeneous waveguides using hybrid methods
    2005 (English)Manuscript (preprint) (Other academic)
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-93590 (URN)
    Projects
    GEMS
    Available from: 2005-10-11 Created: 2005-10-11 Last updated: 2018-01-13Bibliographically approved
  • 10. Abenius, Erik
    et al.
    Andersson, Ulf
    Edelvik, Fredrik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Eriksson, Lasse
    Ledfelt, Gunnar
    Hybrid time domain solvers for the Maxwell equations in 2D2002In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 53, 2185-2199 p.Article in journal (Refereed)
  • 11.
    Abenius, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Edelvik, Fredrik
    Thin Sheet Modeling Using Shell Elements in the Finite-Element Time-Domain Method2006In: IEEE Transactions on Antennas and Propagation, ISSN 0018-926X, E-ISSN 1558-2221, Vol. 54, 28-34 p.Article in journal (Refereed)
  • 12.
    Abenius, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Edelvik, Fredrik
    Johansson, Christer
    Waveguide Truncation Using UPML in the Finite-Element Time-Domain Method2005Report (Other academic)
  • 13.
    Abenius, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Johansson, Christer
    A General Approach for Time-Domain Simulation of Waveguides in 3D2004In: Proc. EMB 04, Computational Electromagnetics: Methods and Applications, Göteborg, Sweden: Department of Electromagnetics, Chalmers University of Technology , 2004, 220-226 p.Conference paper (Other academic)
  • 14.
    Abenius, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Johansson, Christer
    Modeling of inhomogeneous waveguides using hybrid methods2005Manuscript (preprint) (Other academic)
  • 15.
    Abenius, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Strand, Bo
    Solving inverse electromagnetic problems using FDTD and gradient-based minimization2006In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 68, 650-673 p.Article in journal (Refereed)
  • 16. Abenius, Erik
    et al.
    Strand, Bo
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Alestra, Stephane
    Inverse Electromagnetic Scattering Using the Finite-Difference Time-Domain Method2000In: Proc. Millennium Conference on Antennas and Propagation, Noordwijk, The Netherlands: ESA Publications , 2000, 4- p.Conference paper (Refereed)
  • 17. Acosta, Oscar
    et al.
    Frimmel, Hans
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computerized Image Analysis.
    Fenster, Aaron
    Ourselin, Sébastien
    Filtering and restoration of structures in 3D ultrasound images2007In: Proc. 4th International Symposium on Biomedical Imaging, Piscataway, NJ: IEEE , 2007, 888-891 p.Conference paper (Refereed)
  • 18. Acosta, Oscar
    et al.
    Frimmel, Hans
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computerized Image Analysis.
    Fenster, Aaron
    Salvado, Olivier
    Ourselin, Sébastien
    Pyramidal flux in an anisotropic diffusion scheme for enhancing structures in 3D images2008In: Medical Imaging 2008: Image Processing, Bellingham, WA, 2008, 691429:1-12 p.Conference paper (Refereed)
  • 19. Adams, Robin
    et al.
    Fincher, Sally
    Pears, Arnold
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computer Systems.
    Börstler, Jürgen
    Boustedt, Jonas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing.
    Dalenius, Peter
    Eken, Gunilla
    Heyer, Tim
    Jacobsson, Andreas
    Lindberg, Vanja
    Molin, Bengt
    Moström, Jan Erik
    Wiggberg, Mattias
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computer Systems.
    What is the word for "Engineering" in Swedish: Swedish students' conceptions of their discipline2007Report (Other academic)
  • 20.
    Ahlkrona, Josefin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing.
    Implementing Higher Order Dynamics into the Ice Sheet Model SICOPOLIS2011Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Ice sheet modeling is an important tool both for reconstructing past ice sheets and predicting their future evolution, but is complex and computationally costly. It involves modeling a system including the ice sheet, ice shelves and ice streams, which all have different dynamical behavior. The governing equations are non-linear, and to capture a full glacial cycle more than 100,000 years need to be simulated. To reduce the problem size, approximations of the equations are introduced. The most common approximation, the Shallow Ice Approximation (SIA), works well in the ice bulk but fails in e.g. the modeling of ice streams and the ice sheet/ice shelf coupling. In recent years more accurate models, so-called higher order models, have been constructed to address these problems. However, these models are generally constructed in an ad hoc fashion, lacking rigor. In this thesis, so-called Second Order Shallow Ice Approximation (SOSIA) equations for pressure, vertical shear stress and velocity are implemented into the ice sheet model SICOPOLIS. The SOSIA is a rigorous model derived by Baral in 1999 [3]. The numerical solution for a simple model problem is compared to an analytical solution, and benchmark experiments, comparing the model to other higher order models, are carried out. The numerical and analytical solution agree well, but the results regarding vertical shear stress and velocity differ from other models. It is concluded that there are problems with the model implemented, most likely in the treatment of the relation between stress and strain rate.

     

  • 21.
    Amoignon, Olivier
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Adjoint-based aerodynamic shape optimization2003Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    An adjoint system of the Euler equations of gas dynamics is derived in order to solve aerodynamic shape optimization problems with gradient-based methods. The derivation is based on the fully discrete flow model and involves differentiation and transposition of the system of equations obtained by an unstructured and node-centered finite-volume discretization. Solving the adjoint equations allows an efficient calculation of gradients, also when the subject of optimization is described by hundreds or thousands of design parameters.

    Such a fine geometry description may cause wavy or otherwise irregular designs during the optimization process. Using the one-to-one mapping defined by a Poisson problem is a known technique that produces smooth design updates while keeping a fine resolution of the geometry. This technique is extended here to combine the smoothing effect with constraints on the geometry, by defining the design updates as solutions of a quadratic programming problem associated with the Poisson problem.

    These methods are applied to airfoil shape optimization for reduction of the wave drag, that is, the drag caused by gas dynamic effects that occur close to the speed of sound. A second application concerns airfoil design optimization to delay the laminar-to-turbulent transition point in the boundary layer in order to reduce the drag. The latter application has been performed by the author with collaborators, also using gradient-based optimization. Here, the growth of convectively unstable disturbances are modeled by successively solving the Euler equations, the boundary layer equations, and the parabolized stability equations.

  • 22.
    Amoignon, Olivier
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Moving mesh adaptation scheme for aerodynamic shape optimization2006Report (Other academic)
  • 23.
    Amoignon, Olivier
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Numerical Methods for Aerodynamic Shape Optimization2005Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    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.

    List of papers
    1. Adjoint of a median-dual finite-volume scheme: Application to transonic aerodynamic shape optimization
    Open this publication in new window or tab >>Adjoint of a median-dual finite-volume scheme: Application to transonic aerodynamic shape optimization
    2006 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2006-013
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-80174 (URN)
    Projects
    Design Optimization
    Available from: 2007-09-16 Created: 2007-09-16 Last updated: 2018-01-13Bibliographically approved
    2. Shape optimization for delay of laminar-turbulent transition
    Open this publication in new window or tab >>Shape optimization for delay of laminar-turbulent transition
    Show others...
    2006 (English)In: AIAA Journal, ISSN 0001-1452, E-ISSN 1533-385X, Vol. 44, 1009-1024 p.Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-80640 (URN)10.2514/1.12431 (DOI)000237492000010 ()
    Projects
    Design Optimization
    Available from: 2007-01-31 Created: 2007-01-31 Last updated: 2018-01-13Bibliographically approved
    3. Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization
    Open this publication in new window or tab >>Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization
    2007 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 36, 1119-1136 p.Article in journal (Refereed) Published
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-22158 (URN)10.1016/j.compfluid.2006.11.002 (DOI)000246537700008 ()
    Projects
    Design Optimization
    Available from: 2007-04-16 Created: 2007-04-16 Last updated: 2018-01-12Bibliographically approved
    4. Moving mesh adaptation scheme for aerodynamic shape optimization
    Open this publication in new window or tab >>Moving mesh adaptation scheme for aerodynamic shape optimization
    2006 (English)Report (Other academic)
    Series
    Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2006-014
    National Category
    Computational Mathematics Computer Sciences
    Identifiers
    urn:nbn:se:uu:diva-80175 (URN)
    Projects
    Design Optimization
    Available from: 2007-09-17 Created: 2007-09-17 Last updated: 2018-01-13Bibliographically approved
  • 24.
    Amoignon, Olivier
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Berggren, Martin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Adjoint of a median-dual finite-volume scheme: Application to transonic aerodynamic shape optimization2006Report (Other academic)
  • 25.
    Amoignon, Olivier
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Berggren, Martin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Discrete adjoint-based shape optimization for an edge-based finite-volume solver2003In: Computational Fluid and Solid Mechanics: 2003, Elsevier Science , 2003, 2190-2193 p.Conference paper (Refereed)
  • 26.
    Amoignon, Olivier
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Pralits, Jan
    Hanifi, Ardeshir
    Berggren, Martin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Henningson, Dan
    Shape optimization for delay of laminar-turbulent transition2006In: AIAA Journal, ISSN 0001-1452, E-ISSN 1533-385X, Vol. 44, 1009-1024 p.Article in journal (Refereed)
  • 27.
    Anbo, Anders
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing.
    Identification of extreme load cases for a surface drill rig by means of MBS simulations2010Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    This master thesis is Atlas Copco Craelius’ first step in incorporating numericalmethods in load case analysis during the development process. Atlas Copco needs toconstantly evolve and refine their methods in the design process to remain as thenumber one manufacturer of mining and construction equipment. Poor knowledge ofloads results either in structural failures or in oversized structures, both very costlyfor Atlas Copco.The main goal of this thesis is to examine the potential in MBS software by using it toidentify extreme load cases in one of Atlas Copco Craelius’ surface drill rigs, Mustang5. The MBS-software ADAMS View is used to build a model of the Mustang 5 drill rigand evaluate simulation results. The rig model is subject to motions which representreal case scenarios. The feed positioning possibilities are examined thoroughly since it was expected that the positioning has impact on the load levels. 25 different feedpositioning are simulated.The main conclusion is that the load levels are highly dependent on the feedpositioning. For example, the load levels in the boom raising cylinder can increaseseven times, if the rig is operated with the most unfavorable positioning compared tothe recommended. It could also be concluded that not only one positioning isextreme in terms of loads; it depends on which part of the boom system is beingobserved. Thus, several positioning cases have to be taken into consideration in orderto optimize the design of parts in the boom system.

  • 28. Andersson, Paul
    et al.
    Berggren, Martin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Henningson, Dan S.
    Optimal disturbances and bypass transition in boundary layers1999In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 11, 134-150 p.Article in journal (Refereed)
  • 29. Appelö, Daniel
    et al.
    Hagstrom, Thomas
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Perfectly matched layers for hyperbolic systems: General formulation, well-posedness, and stability2006In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 67, 1-23 p.Article in journal (Refereed)
  • 30. Appelö, Daniel
    et al.
    Kreiss, Gunilla
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Application of a perfectly matched layer to the nonlinear wave equation2007In: Wave motion, ISSN 0165-2125, E-ISSN 1878-433X, Vol. 44, 531-548 p.Article in journal (Refereed)
  • 31. Appleton, Owen
    et al.
    Cameron, David
    Cernák, Jozef
    Dóbé, Péter
    Ellert, Mattias
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, High Energy Physics.
    Frågåt, Thomas
    Grønager, Michael
    Johansson, Daniel
    Jönemo, Johan
    Kleist, Josva
    Kocan, Marek
    Konstantinov, Aleksandr
    Kónya, Balázs
    Márton, Iván
    Mohn, Bjarte
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, High Energy Physics.
    Möller, Steffen
    Müller, Henning
    Nagy, Zsombor
    Nilsen, Jon K.
    Ould-Saada, Farid
    Pajchel, Katarina
    Qiang, Weizhong
    Read, Alexander
    Rosendahl, Peter
    Röczei, Gábor
    Savko, Martin
    Skou Andersen, Martin
    Smirnova, Oxana
    Stefán, Péter
    Szalai, Ferenc
    Taga, Adrian
    Toor, Salman Z.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
    Wäänänen, Anders
    Zhou, Xin
    The next-generation ARC middleware2010In: Annales des télécommunications, ISSN 0003-4347, E-ISSN 1958-9395, Vol. 65, 771-776 p.Article in journal (Refereed)
  • 32.
    Arnell, Robert
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Surface Biotechnology. Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Physical and Analytical Chemistry, Surface Biotechnology.
    Forssén, Patrik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing.
    Fornstedt, Torgny
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Surface Biotechnology. Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Physical and Analytical Chemistry, Surface Biotechnology.
    Accurate and rapid estimation of adsorption isotherms in liquid chromatography using the inverse method on plateaus2005In: Journal of Chromatography A, ISSN 0021-9673, E-ISSN 1873-3778, Vol. 1099, 167-174 p.Article in journal (Refereed)
  • 33.
    Arnell, Robert
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Physical and Analytical Chemistry, Surface Biotechnology.
    Forssén, Patrik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing.
    Fornstedt, Torgny
    Uppsala University, Disciplinary Domain of Science and Technology, Chemistry, Department of Physical and Analytical Chemistry, Surface Biotechnology.
    Tuneable Peak Deformations in Chiral Liquid Chromatography2007In: Analytical Chemistry, ISSN 0003-2700, E-ISSN 1520-6882, Vol. 79, 5838-5847 p.Article in journal (Refereed)
  • 34.
    Axelsson, Owe
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    An additive matrix preconditioning method with application for domain decomposition and two-level matrix partitionings2010In: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2010, 76-83 p.Conference paper (Refereed)
  • 35.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Blaheta, Radim
    Neytcheva, Maya
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    A black-box generalized conjugate gradient minimum residual method based on variable preconditioners and local element approximations2007Report (Other academic)
  • 36.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Blaheta, Radim
    Neytcheva, Maya
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Preconditioning of boundary value problems using elementwise Schur complements2006Report (Other academic)
  • 37.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Blaheta, Radim
    Neytcheva, Maya
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Preconditioning of boundary value problems using elementwise Schur complements2009In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 31, 767-789 p.Article in journal (Refereed)
  • 38.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Chen, Xiaojun
    Matrix Computations and Nonlinear Equations2011Conference proceedings (editor) (Refereed)
  • 39.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Glushkov, Evgeny
    Glushkova, Natalya
    The local Green's function method in singularly perturbed convection-diffusion problems2009In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 78, no 265, 153-170 p.Article in journal (Refereed)
  • 40.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Condition number analysis for various forms of block matrix preconditioners2010In: Electronic Transactions on Numerical Analysis, ISSN 1068-9613, E-ISSN 1068-9613, Vol. 36, 168-194 p.Article in journal (Refereed)
  • 41.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Double Sobolev gradient preconditioning for nonlinear elliptic problems2007In: Numerical Methods for Partial Differential Equations, ISSN 0749-159X, E-ISSN 1098-2426, Vol. 23, 1018-1036 p.Article in journal (Refereed)
  • 42.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Equivalent operator preconditioning for elliptic problems2009In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 50, 297-380 p.Article, review/survey (Refereed)
  • 43.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Equivalent operator preconditioning for linear elliptic problems2008Report (Other academic)
  • 44.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Mesh independent convergence rates via differential operator pairs2008In: Large-Scale Scientific Computing, Berlin: Springer-Verlag , 2008, 3-15 p.Conference paper (Refereed)
  • 45.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Mesh independent superlinear PCG rates via compact-equivalent operators2006Report (Other academic)
  • 46.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Mesh independent superlinear PCG rates via compact-equivalent operators2007In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 45, 1495-1516 p.Article in journal (Refereed)
  • 47.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Symmetric part preconditioning of the CG method for Stokes type saddle-point systems2007In: Numerical Functional Analysis and Optimization, ISSN 0163-0563, E-ISSN 1532-2467, Vol. 28, 1027-1049 p.Article in journal (Refereed)
  • 48.
    Axelsson, Owe
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Karátson, János
    Symmetric part preconditioning of the CGM for Stokes type saddle-point systems2006Report (Other academic)
  • 49. Axelsson, Owe
    et al.
    Neytcheva, Maya
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    A general approach to analyse preconditioners for two-by-two block matrices2010Report (Other academic)
  • 50. Axelsson, Owe
    et al.
    Neytcheva, Maya
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Numerical Analysis.
    Eigenvalue estimates for preconditioned saddle point matrices2004Report (Other academic)
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