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  • 51.
    Björklund, Camilla
    et al.
    University of Gothenburg, Sweden.
    Alkhede, Maria
    University of Gothenburg, Sweden.
    Kullberg, Angelika
    University of Gothenburg, Sweden.
    Reis, Maria
    University of Gothenburg, Sweden.
    Marton, Ference
    University of Gothenburg, Sweden.
    Ekdahl, Anna-Lena
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Runesson Kempe, Ulla
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Teaching finger patterns for arithmetic development to preschoolers2018Conference paper (Refereed)
    Abstract [en]

    In this paper we describe the empirical and theoretical meaning behind how finger patterns are taught to facilitate the development of preschool children’s perception of the first ten natural numbers. An intervention programme, informed by Variation theory of learning, included 65 five-year-olds and teachers at seven preschool departments in Sweden. The programme aimed at developing teaching activities and artefacts to promote children discerning necessary aspects of the first ten numbers. The design of the programme is significant to describe and evaluate as basis for forthcoming analyses of the learning outcomes, as a pedagogical approach that stands in contrast to common preschool teaching practice in Sweden is adopted.

  • 52.
    Björklund, Camilla
    et al.
    Department of Education, Communication and Learning, University of Gothenburg, Gothenburg, Sweden.
    Runesson Kempe, Ulla
    Jönköping University, School of Education and Communication, HLK, Praktiknära utbildningsforskning (PUF), Mathematics Education Research.
    Validating a theory of children’s ways of experiencing numbers2019In: Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education / [ed] U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis, 2019Conference paper (Refereed)
  • 53.
    Boeva, Veselka
    et al.
    Computer Systems and Technologies Department, Technical University of Sofia, branch Plovdiv, Plovdiv, Bulgaria.
    Ivanova, Petia
    Computer Systems and Technologies Department, Technical University of Sofia, branch Plovdiv, Plovdiv, Bulgaria.
    Lavesson, Niklas
    School of Computing Blekinge, Institute of Technology, Ronneby, Sweden.
    A Hybrid Computational Method for the Identification of Cell Cycle-regulated Genes2010Conference paper (Refereed)
    Abstract [en]

    Gene expression microarrays are the most commonly available source of high-throughput biological data. They have been widely employed in recent years for the definition of cell cycle regulated (or periodically expressed) subsets of the genome in a number of different organisms. These have driven the development of various computational methods for identifying periodical expressed genes. However, the agreement is remarkably poor when different computational methods are applied to the same data. In view of this, we are motivated to propose herein a hybrid computational method targeting the identification of periodically expressed genes, which is based on a hybrid aggregation of estimations, generated by different computational methods. The proposed hybrid method is benchmarked against three other computational methods for the identification of periodically expressed genes: statistical tests for regulation and periodicity and a combined test for regulation and periodicity. The hybrid method is shown, together with the combined test, to statistically significantly outperform the statistical test for periodicity. However, the hybrid method is also demonstrated to be significantly better than the combined test for regulation and periodicity.

  • 54.
    Boström, Henrik
    et al.
    Department of Computer and Systems Sciences, Stockholm University, Stockholm, Sweden.
    Linusson, Henrik
    Department of Information Technology, University of Borås, Borås, Sweden.
    Löfström, Tuwe
    Jönköping University, School of Engineering, JTH, Computer Science and Informatics, JTH, Jönköping AI Lab (JAIL). Department of Information Technology, University of Borås, Borås, Sweden.
    Johansson, Ulf
    Jönköping University, School of Engineering, JTH, Computer Science and Informatics, JTH, Jönköping AI Lab (JAIL).
    Accelerating difficulty estimation for conformal regression forests2017In: Annals of Mathematics and Artificial Intelligence, ISSN 1012-2443, E-ISSN 1573-7470, Vol. 81, no 1-2, p. 125-144Article in journal (Refereed)
    Abstract [en]

    The conformal prediction framework allows for specifying the probability of making incorrect predictions by a user-provided confidence level. In addition to a learning algorithm, the framework requires a real-valued function, called nonconformity measure, to be specified. The nonconformity measure does not affect the error rate, but the resulting efficiency, i.e., the size of output prediction regions, may vary substantially. A recent large-scale empirical evaluation of conformal regression approaches showed that using random forests as the learning algorithm together with a nonconformity measure based on out-of-bag errors normalized using a nearest-neighbor-based difficulty estimate, resulted in state-of-the-art performance with respect to efficiency. However, the nearest-neighbor procedure incurs a significant computational cost. In this study, a more straightforward nonconformity measure is investigated, where the difficulty estimate employed for normalization is based on the variance of the predictions made by the trees in a forest. A large-scale empirical evaluation is presented, showing that both the nearest-neighbor-based and the variance-based measures significantly outperform a standard (non-normalized) nonconformity measure, while no significant difference in efficiency between the two normalized approaches is observed. The evaluation moreover shows that the computational cost of the variance-based measure is several orders of magnitude lower than when employing the nearest-neighbor-based nonconformity measure. The use of out-of-bag instances for calibration does, however, result in nonconformity scores that are distributed differently from those obtained from test instances, questioning the validity of the approach. An adjustment of the variance-based measure is presented, which is shown to be valid and also to have a significant positive effect on the efficiency. For conformal regression forests, the variance-based nonconformity measure is hence a computationally efficient and theoretically well-founded alternative to the nearest-neighbor procedure. 

  • 55.
    Brundin, Ethel
    et al.
    Jönköping University, Jönköping International Business School, JIBS, EMM (Entrepreneurship, Marketing, Management). Jönköping University, Jönköping International Business School, JIBS, Center for Family Enterprise and Ownership.
    Wigren, Caroline
    Jönköping University, Jönköping International Business School, JIBS, EMM (Entrepreneurship, Marketing, Management).
    Isaacs, Eslyn
    Friedrich, Chris
    Visser, Kobus
    Triple Helix Networks in a Multicultural Context: Triggers and Barriers for Fostering Growth and Sustainability2008In: Journal of Developmental Entrepreneurship, ISSN 1084-9467, Vol. 13, no 1, p. 77-98Article in journal (Refereed)
    Abstract [en]

    This article deals with Triple Helix (university, industry and government co-operation) from an institutional theory perspective. The empirical context is the Western Cape Region in South Africa and the focus is entrepreneurship development. The purpose is twofold: first, the existing Triple Helix model is adapted to the South African context; and second, facilities and impediments for working according to Triple Helix in South Africa are identified. The empirical material consists of a survey and three longitudinal case studies illustrating the degree of co-operation between the three parties. The article contributes to knowledge about how the Triple Helix model works on a regional level in a developing country. The study draws the following conclusions: when co-operation is to be identified between the three actors, only two of the three are involved; one missing link in the Triple Helix model is the focus on the entrepreneur; co-operation between the three parties are incidental rather than planned and there is lack of structure. In turn, some of these conclusions may be an effect of institutional changes on a national level. For a normative legacy, the article proposes a set of suggestions for incorporating all relevant parties on a practical level.

  • 56.
    Burman, E.
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, M. G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Massing, A.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    A stable cut finite element method for partial differential equations on surfaces: The Helmholtz–Beltrami operator2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 362, article id 112803Article in journal (Refereed)
    Abstract [en]

    We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions. Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition hk<C, where h denotes the mesh size, k the wave number and C a constant depending mainly on the surface curvature κ, but not on the surface/mesh intersection. Optimal error estimates in the H1 and L2-norms follow.

  • 57.
    Burman, Erik
    et al.
    University College London, Gower Street, UK.
    Elfverson, Daniel
    Umeå University, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Umeå University, Sweden.
    Larsson, Karl
    Umeå University, Sweden.
    A cut finite element method for the Bernoulli free boundary value problem2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 317, p. 598-618Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H1 Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H1norm. Finally, we present illustrating numerical results.

  • 58.
    Burman, Erik
    et al.
    University College London, London, United Kingdom.
    Elfverson, Daniel
    Umeå University, Umeå, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå University, Umeå, Sweden.
    Larsson, Karl
    Umeå University, Umeå, Sweden.
    Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 462-479Article in journal (Refereed)
    Abstract [en]

    We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represent parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche's method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero. 

  • 59.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Elfverson, Daniel
    Umeå universitet, Institutionen för matematik och matematisk statistik.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå universitet, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Institutionen för matematik och matematisk statistik.
    Hybridized CutFEM for Elliptic Interface Problems2019In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 41, no 5, p. A3354-A3380Article in journal (Refereed)
    Abstract [en]

    We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples.

  • 60.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, Gower Street, London, United Kingdom.
    Elfverson, Daniel
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Shape and topology optimization using CutFEM2017In: Simulation for Additive Manufacturing 2017, Sinam 2017, International Center for Numerical Methods in Engineering (CIMNE), 2017, p. 208-209Conference paper (Refereed)
    Abstract [en]

    We present a shape and topology optimization method based on the cut finite element method, see [1],[2], and [3], for the optimal compliance problem in linear elasticity and problems involving restrictionson the stresses.The elastic domain is defined by a level-set function, and the evolution of the domain is obtained bymoving the level-set along a velocity field using a transport equation. The velocity field is defined tobe the largest decreasing direction of the shape derivative that resides in a certain Hilbert space and iscomputed by solving an elliptic problem, associated with the bilinear form in the Hilbert space, with theshape derivative as right hand side. The velocity field may thus be viewed as the Riesz representationof the shape derivative on the chosen Hilbert space.We thus obtain a coupled problem involving three partial differential equations: (1) the elasticity problem,(2) the elliptic problem that determines the velocity field, and (3) the transport problem for thelevelset function. The elasticity problem is solved using a cut finite element method on a fixed backgroundmesh, which completely avoids re–meshing when the domain is updated. The levelset functionand the velocity field is approximated by standard conforming elements on the background mesh. Wealso employ higher order cut approximations including isogeometric analysis for the elasticity problem.In this case the levelset function and the velocity field are represented using linear elements on a refinedmesh in order to simplify the geometric and quadrature computations on the cut elements. To obtain astable method, stabilization terms are added in the vicinity of the cut elements at the boundary, whichprovides control of the variation of the solution in the vicinity of the boundary. We present numericalexamples illustrating the performance of the method.We also study an anisotropic material model that accounts for the orientation of the layers in an additivemanufacturing process and by including the orientation in the optimization problem we determine theoptimal choice of orientation.We present numerical results including test problems and engineering applications in additive manufacturing.

    References

    [1] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: discretizing geometryand partial differential equations. Internat. J. Numer. Methods Engrg., 104(7):472–501, 2015.

    [2] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson. Shape optimization using thecut finite element method. Technical report, 2016. arXiv:1611.05673.

    [3] E. Burman, D. Elfverson, P. Hansbo, M. G. Larson, and K. Larsson. A cut finite element method forthe Bernoulli free boundary value problem. Comput. Methods Appl. Mech. Engrg., 317:598–618,2017.

  • 61.
    Burman, Erik
    et al.
    University College London, United Kingdom.
    Elfverson, Daniel
    Umeå universitet, Sweden.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats
    Umeå universitet, Sweden.
    Larsson, Karl
    Umeå universitet, Sweden.
    Shape optimization using the cut finite element method2018In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 242-261Article in journal (Refereed)
    Abstract [en]

    We present a cut finite element method for shape optimization in the case of linear elasticity. The elastic domain is defined by a level-set function, and the evolution of the domain is obtained by moving the level-set along a velocity field using a transport equation. The velocity field is the largest decreasing direction of the shape derivative that satisfies a certain regularity requirement and the computation of the shape derivative is based on a volume formulation. Using the cut finite element method no re-meshing is required when updating the domain and we may also use higher order finite element approximations. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements at the boundary, which provides control of the variation of the solution in the vicinity of the boundary. We implement and illustrate the performance of the method in the two-dimensional case, considering both triangular and quadrilateral meshes as well as finite element spaces of different order.

  • 62.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom..
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Deriving Robust Unfitted Finite Element Methods from Augmented Lagrangian Formulations2017In: Geometrically Unfitted Finite Element Methods and Applications / [ed] Bordas, Stéphane P. A.; Burman, Erik; Larson, Mats G.; Olshanskii, Maxim A., Cham: Springer International Publishing , 2017, p. 1-24Conference paper (Refereed)
    Abstract [en]

    In this paper we will discuss different coupling methods suitable for use in the framework of the recently introduced CutFEM paradigm, cf. Burman et al. (Int. J. Numer. Methods Eng. 104(7):472–501, 2015). In particular we will consider mortaring using Lagrange multipliers on the one hand and Nitsche’s method on the other. For simplicity we will first discuss these methods in the setting of uncut meshes, and end with some comments on the extension to CutFEM. We will, for comparison, discuss some different types of problems such as high contrast problems and problems with stiff coupling or adhesive contact. We will review some of the existing methods for these problems and propose some alternative methods resulting from crossovers from the Lagrange multiplier framework to Nitsche’s method and vice versa.

  • 63.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem2014In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 48, no 3, p. 859-874Article in journal (Refereed)
    Abstract [en]

    We extend the classical Nitsche type weak boundary conditions to a fictitious domain setting. An additional penalty term, acting on the jumps of the gradients over element faces in the interface zone, is added to ensure that the conditioning of the matrix is independent of how the boundary cuts the mesh. Optimal a priori error estimates in the H 1- and L 2-norms are proved as well as an upper bound on the condition number of the system matrix.

  • 64.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Stabilized nonconforming finite element methods for data assimilation in incompressible flows2018In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 311, p. 1029-1050Article in journal (Refereed)
    Abstract [en]

    We consider a stabilized nonconforming finite element method for data assimilation in incompressible flow subject to the Stokes equations. The method uses a primal dual structure that allows for the inclusion of nonstandard data. Error estimates are obtained that are optimal compared to the conditional stability of the ill-posed data assimilation problem.

  • 65.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    A cut finite element method with boundary value correction2018In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 310, p. 633-657Article in journal (Refereed)
    Abstract [en]

    In this contribution we develop a cut finite element method with boundary value correction of the type originally proposed by Bramble, Dupont, and Thomée in [Math. Comp. 26 (1972), 869-879]. The cut finite element method is a fictitious domain method with Nitsche-type enforcement of Dirichlet conditions together with stabilization of the elements at the boundary which is stable and enjoy optimal order approximation properties. A computational difficulty is, however, the geometric computations related to quadrature on the cut elements which must be accurate enough to achieve higher order approximation. With boundary value correction we may use only a piecewise linear approximation of the boundary, which is very convenient in a cut finite element method, and still obtain optimal order convergence. The boundary value correction is a modified Nitsche formulation involving a Taylor expansion in the normal direction compensating for the approximation of the boundary. Key to the analysis is a consistent stabilization term which enables us to prove stability of the method and a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.

  • 66.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    A simple approach for finite element simulation of reinforced plates2018In: Finite elements in analysis and design (Print), ISSN 0168-874X, E-ISSN 1872-6925, Vol. 142, p. 51-60Article in journal (Refereed)
    Abstract [en]

    We present a new approach for adding Bernoulli beam reinforcements to Kirchhoff plates. The plate is discretised using a continuous/discontinuous finite element method based on standard continuous piecewise polynomial finite element spaces. The beams are discretised by the CutFEM technique of letting the basis functions of the plate represent also the beams which are allowed to pass through the plate elements. This allows for a fast and easy way of assessing where the plate should be supported, for instance, in an optimization loop.

  • 67.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University.
    A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator2015In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 285, p. 188-207Article in journal (Refereed)
    Abstract [en]

    We consider solving the Laplace–Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.

    The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method. 

  • 68.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Augmented Lagrangian and Galerkin least-squares methods for membrane contact2018In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 114, no 11, p. 1179-1191Article in journal (Refereed)
    Abstract [en]

    In this paper, we propose a stabilized finite element method for the numerical solution of contact between a small deformation elastic membrane and a rigid obstacle. We limit ourselves to friction-free contact, but the formulation is readily extendable to more complex situations. 

  • 69.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Augmented Lagrangian finite element methods for contact problems2019In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 53, no 1, p. 173-195Article in journal (Refereed)
    Abstract [en]

    We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution. 

  • 70.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Dirichlet boundary value correction using Lagrange multipliers2019In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125Article in journal (Refereed)
    Abstract [en]

    We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to a Taylor series expansion approach previously used in the context of Nitsche’s method and, in the case of inf-sup stable multiplier methods, prove a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary. 

  • 71.
    Burman, Erik
    et al.
    University College London, London, UK.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G
    Umeå University, Umeå, Sweden.
    Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization2018In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, no 3, article id 035004Article in journal (Refereed)
    Abstract [en]

    Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that the stabilization methods for discretely illposed problems developed in the setting of convection-dominated convection– diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider some inverse problems for Poisson’s equation as an illustration and derive new error estimates both for the reconstruction of the solution from the measured data and reconstruction of the source term from the measured data. These estimates include both the effect of the discretization error and error in the measurements.

  • 72. Burman, Erik
    et al.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Institutionen för matematik och matematisk statistik, Umeå universitet.
    The penalty-free Nitsche Method and nonconforming finite elements for the Signorini problem2017In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, no 6, p. 2523-2539Article in journal (Refereed)
    Abstract [en]

    We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild [SIAM J. Numer. Anal., 51 (2013), pp. 1295--1307], our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm.

  • 73.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Cut finite elements for convection in fractured domains2019In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 179, p. 728-736Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain, which is a union of manifolds of different dimensions such that a d dimensional component always resides on the boundary of a d+1 dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem is formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is posed on a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.

  • 74.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Finite element approximation of the Laplace–Beltrami operator on a surface with boundary2019In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, no 1, p. 141-172Article in journal (Refereed)
    Abstract [en]

    We develop a finite element method for the Laplace–Beltrami operator on a surface with boundary and nonhomogeneous Dirichlet boundary conditions. The method is based on a triangulation of the surface and the boundary conditions are enforced weakly using Nitsche’s method. We prove optimal order a priori error estimates for piecewise continuous polynomials of order (Formula presented.) in the energy and (Formula presented.) norms that take the approximation of the surface and the boundary into account. 

  • 75.
    Burman, Erik
    et al.
    University College London, UK.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Umeå University, Sweden.
    Massing, André
    Umeå University, Sweden.
    A cut discontinuous Galerkin method for the Laplace–Beltrami operator2017In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 37, no 1, p. 138-169Article in journal (Refereed)
    Abstract [en]

    We develop a discontinuous cut finite element method for the Laplace–Beltrami operator on a hypersurface embedded in R. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in R. The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then, a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighbouring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.

  • 76.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions2019In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 52, no 6, p. 2247-2282Article in journal (Refereed)
    Abstract [en]

    We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in Rd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in R3.

  • 77.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Zahedi, Sara
    Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden.
    A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 358, article id 112645Article in journal (Refereed)
    Abstract [en]

    We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included. 

  • 78.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.
    Zahedi, Sara
    Department of Mathematics, KTH, Stockholm, Sweden.
    Full gradient stabilized cut finite element methods for surface partial differential equations2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 278-296Article in journal (Refereed)
    Abstract [en]

    We propose and analyze a new stabilized cut finite element method for the Laplace–Beltrami operator on a closed surface. The new stabilization term provides control of the full R 3 gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace–Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and L 2 norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.

  • 79.
    Burman, Erik
    et al.
    Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Samvin, David
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    A cut finite element method for elliptic bulk problems with embedded surfaces2019In: GEM - International Journal on Geomathematics, ISSN 1869-2672, E-ISSN 1869-2680, Vol. 10, no 1, article id 10Article in journal (Refereed)
    Abstract [en]

    We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractures. In particular the Laplace–Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples. 

  • 80.
    Burman, Erik
    et al.
    Department of Mathematics, University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University.
    Stenberg, Rolf
    Department of Mathematics and Systems Analysis, Aalto University.
    Galerkin least squares finite element method for the obstacle problem2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 313, p. 362-374Article in journal (Refereed)
    Abstract [en]

    We construct a consistent multiplier free method for the finite element solution of the obstacle problem. The method is based on an augmented Lagrangian formulation in which we eliminate the multiplier by use of its definition in a discrete setting. We prove existence and uniqueness of discrete solutions and optimal order a priori error estimates for smooth exact solutions. Using a saturation assumption we also prove an a posteriori error estimate. Numerical examples show the performance of the method and of an adaptive algorithm for the control of the discretization error.

  • 81.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University.
    Zahedi, Sara
    KTH Royal Institute of Technology.
    Cut finite element methods for coupled bulk–surface problems2016In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, no 2, p. 203-231Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and L2 norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

  • 82.
    Burman, Erik
    et al.
    UCL, Department of Mathematics, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå Universitet, Department of Mathematics and Mathematical Statistics, Umeå, Sweden.
    Zahedi, Sara
    The Royal Institute of Technology (KTH), Department of Mathematics, Stockholm, Sweden.
    Stabilized CutFEM for the convection problem on surfaces2019In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, no 1, p. 103-139Article in journal (Refereed)
    Abstract [en]

    We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove h3 / 2 order convergence in the natural norm associated with the method and that the full gradient enjoys h3 / 4 order of convergence in L2. We also show that the condition number of the stiffness matrix is bounded by h- 2. Finally, our results are verified by numerical examples. 

  • 83.
    Burman, Erik N.
    et al.
    UCL, Department of Mathematics, London, United Kingdom.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Materials and Manufacturing.
    Larson, Mats G.
    Umeå Universitet, Department of Mathematics and Mathematical Statistics, Umeå, Sweden.
    A cut finite element method with boundary value correction for the incompressible Stokes equations2019In: Numerical mathematics and advanced applications ENUMATH 2017, Cham: Springer, 2019, Vol. 126, p. 183-192Conference paper (Refereed)
    Abstract [en]

    We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche’s method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman et al. (Math Comput 87(310):633–657, 2018). 

  • 84.
    Burman, Maja
    Jönköping University, School of Education and Communication.
    Hur lärare stöttar elevers lärande: En kvalitativ studie om elever i svårigheter inom matematikområdet bråk.2018Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    Mathematics is a major part of a person's life and is necessary for us to work in society. This means that mathematics education should be designed to contribute to an understanding of the different parts of mathematics, so that all students will be able to develop to function in society. However, students have different capability and needs, which need to be considered when the teacher teach. Various adjustments are made in the teaching in various ways to support and guide all students' development and learning.

    The purpose of the study is, based on Shulman's view of PCK, to study how teachers' various knowledge affects the encounter with students in difficulty with fraction in mathematics. The study examines the teachers' reasoning about the difficulties with fraction, which skills the teachers consider necessary to meet and support students in difficulty as well as how teachers' work and how different adaptations can be formed. In order to achieve the purpose of the study, semi structured interviews were used to examine teachers' perceptions of mathematical difficulties in the area fraction.

    The study is also based on the socio-cultural perspective, with focus on teachers 'influence and views on students' learning.

    The study result show that the teachers believe that students are in difficulty regarding fraction of number, order of fractional numbers and understanding of what to divide. Teachers' reasoning about which students are there are difficulties, however, are spread. Some of the teachers believe that fraction is generally a difficult area within mathematics, while other teachers consider that there are only certain parts in the area of fraction that are generally difficult for the students. Further opinions are that students in difficulty with other areas of mathematics are also in difficulty in the area of fraction. The results of the study also show that teachers consider that their experience, knowledge and understanding of the students, as well as knowledge of how the teaching is best designed, are the skills that teachers need to support and guide students in difficulty with the mathematic area fraction. The study's results show, in addition to the above, that the teachers in the study believe that it is important to provide varied and level-based education to support all students in their learning, especially students in difficulty.

    The conclusion of the study is that the teachers' different knowledge of difficulties, students and teaching has a major influence on the support and guidance for students in difficulty.

  • 85.
    Burman, Maja
    et al.
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Clemborn, Frida
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    PCK: – En litteraturstudie om pedagogical content knowledge samt hur det kopplas till lärare och matematikundervisningen2017Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
  • 86.
    Carlsson, Solveig
    et al.
    Jönköping University, School of Education and Communication.
    MolidSvenningsson, Malin
    Jönköping University, School of Education and Communication.
    Matematik – en svartvit skiss eller ett färgstarkt konstverk: En fenomenografisk studie om lärares uppfattningar av vad elevers tilltro till lärande i matematik innebär och hur den främjas.2008Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

     

    The purpose of this qualitative study, based on a phenomenographical approach, was to describe teachers' perceptions of what the students' confidence in their own thinking and their own ability to teach mathematics and how they are doing to promote this. Data collection consisted of interviews with teachers from pre until year six.

    Teacher perceptions were described in the student show desire, courage, pride and responsibility, especially when they dared to get through a challenge as the first not thought to cope. A prerequisite for faith was a safe group environment.

    The promotion of confidence took the teachers based on students' experiences with positive expectations for students in a luscious and communicative teaching. Encouragement and challenges at the appropriate level, which creates understanding is characteristic for the student to feel that they succeed.

    The surrounding community is very important for the students' confidence. The teacher's dedication, curiosity and attention devoted to the pupils' knowledge to conquer new mathematical literacy was important for confidence. It appeared that the students should be aware of the mathematical knowledge and calculations behind the products that are in our surroundings.

  • 87.
    Cenanovic, Mirza
    et al.
    Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization. Jönköping University, School of Engineering, JTH, Product Development.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Mechanical Engineering. Jönköping University, School of Engineering, JTH. Research area Product Development - Simulation and Optimization.
    Larson, Mats G,
    Umeå University.
    Minimal surface computation using a finite element method on an embedded surface2015In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, no 7, p. 502-512Article in journal (Refereed)
    Abstract [en]

    We suggest a finite element method for finding minimal surfaces based on computing a discrete Laplace–Beltrami operator operating on the coordinates of the surface. The surface is a discrete representation of the zero level set of a distance function using linear tetrahedral finite elements, and the finite element discretization is carried out on the piecewise planar isosurface using the shape functions from the background three-dimensional mesh used to represent the distance function. A recently suggested stabilized scheme for finite element approximation of the mean curvature vector is a crucial component of the method.

  • 88. Chedid, Michel
    et al.
    Belov, Ilja
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Leisner, Peter
    Jönköping University, School of Engineering, JTH. Research area Materials and manufacturing - Surface technology. Jönköping University, School of Engineering, JTH. Research area Robust Embedded Systems.
    Experimental Analysis and Modelling of Textile Transmission Line for Wearable Applications2007In: International Journal of Clothing Science and Technology, ISSN 0955-6222, E-ISSN 1758-5953, Vol. 19, no 1, p. 59-71Article in journal (Refereed)
    Abstract [en]

    Purpose – The paper seeks, by means of measurement and modelling, to evaluate frequency dependent per-unit-length parameters of conductive textile transmission line (CTTL) for wearable applications and to study deterioration of these parameters when CTTL is subjected to washing.

    Design/methodology/approach – The studied transmission line is made of Nickel/Copper (Ni/Cu) plated polyester ripstop fabric and is subjected to standard 60°C cycle in a commercial off-the-shelf washing machine. The per-unit-length parameters (resistance and inductance) and characteristic impedance of the line are extracted from measurements before and after washing. Using the measurement data an equivalent circuit is created to model the degradation of the line. The circuit is then integrated in a three-dimensional transmission line matrix (TLM) model of the transmission line.

    Findings – Both an electrical equivalent circuit and a TLM model are developed describing the degradation of the conductive textile when washed. A severe deterioration of the electrical parameters of the line is noticed. Experimental and modelling results are in good agreement in the addressed frequency band.

    Research limitations/implications – Analysis is performed for frequencies up to 10?MHz. The developed TLM model can be used to conduct parametric studies of the CTTL. To counteract the degradation of the line, protective coating is to be considered in further studies.

    Originality/value – This paper extends knowledge of the subject by experimental and simulation-based characterization of the CTTL when subjected to washing cycles.

  • 89.
    Clemborn, Frida
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    ”Allt ligger ju på läraren hur en lektion blir”: En kvalitativ studie om lärares planeringsförmåga av matematikundervisning i grundskolans årskurs F-32018Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Att arbeta som lärare kräver att du erhåller vissa kunskaper. Du ska ha tillräckliga ämneskunskaper och pedagogiska kunskaper. Du ska även ha kunskaper om hur planering, utförande och utvärdering av undervisningen sker. Att skapa en lektionsplanering och integrera alla nödvändiga delar av en undervisning är komplext. Därför utvecklade Shulman (1987) sex processdelar undervisningen följer.

    Frågan är om dagens lärare använder dessa kunskaper. Syftet är därför att bidra med förståelse om vilka kunskaper lärare besitter samt hur planering av undervisning sker. Detta genom att låta lärarna planera en undervisning inom ämnet volym. Studien genomfördes genom kvalitativa intervjuer med fem stycken lärare. Resultatet av studien visar att alla lärarna följde processdelarna. Det var emellertid mycket som skiljde sig mellan lärarna. Forskning säger att erfarenhet, utbildning och fortbildning har betydelse för lärares planeringsförmåga. Resultatet av studien visar att detta i viss utsträckning har betydelse och att lärares förmågor är olika. Dock visar egen analys att det mer beror på att alla lärare är olika individer.

  • 90.
    Creignou, Nadia
    et al.
    CNRS, Aix-Marseille Université, France.
    Meier, Arne
    Institut für Theoretische Informatik, Gottfried Wilhelm Leibniz Universität, Germany.
    Müller, Julian-Steffen
    Institut für Theoretische Informatik, Gottfried Wilhelm Leibniz Universität, Germany.
    Schmidt, Johannes
    Department of Computer and Information Science, Linköping University, Sweden.
    Vollmer, Heribert
    Institut für Theoretische Informatik, Gottfried Wilhelm Leibniz Universität, Germany.
    Paradigms for Parameterized Enumeration2017In: Theory of Computing Systems, ISSN 1432-4350, E-ISSN 1433-0490, Vol. 60, no 4, p. 737-758Article in journal (Refereed)
    Abstract [en]

    The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First, we define formally different notions of efficient enumeration in the context of parameterized complexity: FPT-enumeration and delayFPT. Second, we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems. Furthermore, we study the parameterized complexity of enumerating all models of Boolean formulas having weight at least k, where k is the parameter, in the famous Schaefer’s framework. We consider propositional formulas that are conjunctions of constraints taken from a fixed finite set Γ. Given such a formula and an integer k, we are interested in enumerating all the models of the formula that have weight at least k. We obtain a dichotomy classification and prove that, according to the properties of the constraint language Γ, either one can enumerate all such models in delayFPT, or no such delayFPT enumeration algorithm exists under some complexity-theoretic assumptions.

  • 91.
    Daag, Madeleine
    Jönköping University, School of Education and Communication.
    Hur instruktioner påverkar elevers delaktighet: Matematisk problemlösning i grupp2018Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Grupparbete är ett arbetssätt som ger elever möjligheter att utveckla både sina ämneskunskaper och sin förmåga att samarbeta (Johnson & Johnson, 1999). Men grupparbete är ett komplicerat arbetssätt som kräver engagemang från samtliga deltagare för att alla elever ska kunna utvecklas. Det är inte ovanligt att en elev i gruppen väljer att inte vara aktiv. Eleven låter då resterande elever i gruppen lösa uppgiften själva (Forslund Frykedal, 2008). Det här är ett scenario som gör att alla elever inte blir delaktiga vid problemlösningen och därmed inte får samma möjligheter att utveckla sin problemlösnings- och samarbetsförmåga. Syftet med studien är att undersöka och utveckla instruktioner till problemuppgifter som ökar alla elevers delaktighet och ansvarstagande vid grupparbete. För att kunna besvara syftet används design research som metod. Genom metoden undersöks och utvecklas pedagogiska insatser.

    Studien visar att strukturerade instruktioner till problemuppgifter påverkar elevernas delaktighet positivt. Användandet av ett problemlösningsschema med flera steg gav ett positivt resultat då gruppmedlemmarna lade stor tid på problemlösningsprocessen. Det gav alla medlemmar tid att ställa frågor, framföra idéer samt bygga vidare på sina egna och andras resonemang under hela processen. I första steget i problemlösningsschemat uppmanades eleverna att se till att alla gruppmedlemmar förstod uppgiften innan problemlösningsprocessen inleddes. Studien visar att fler elever tog ansvar för att alla gruppmedlemmar skulle ha förståelse för problemet när de genom instruktionerna blev uppmuntrade till det.

  • 92.
    Daag, Madeleine
    et al.
    Jönköping University, School of Education and Communication.
    Andersson, Hanna
    Jönköping University, School of Education and Communication.
    Elever lär med hjälp av varandra: Matematisk problemlösning i små grupper2016Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
  • 93.
    Dahlin, Evelina
    et al.
    Jönköping University, School of Education and Communication.
    Hultman, Sophie
    Jönköping University, School of Education and Communication.
    Barns uppfattning av tid och klockan: En litteraturstudie om barns tidsuppfattning och svårigheter med den analoga respektive digitala klockan2018Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Syftet med den här litteraturstudien är att granska matematikdidaktisk forskning om hur elever uppfattar tid och klockan. 

    Litteraturstudien har baserats på vetenskapliga studier som har hittats genom en systematisk sökprocess. De studier som inte är presenterade i peer-granskade tidskrifter har vi själva noggrant granskat genom att jämföra dem med peer-granskade tidskrifter, genomfört författarsökningar för att se vad de tidigare skrivit och analyserat andra vetenskapliga texter som citerat från texterna. Det är krav vi haft för att texterna ska ses som användbara i den här litteraturstudien.         

    Studien visar att forskningsläget inte är helt entydigt. Dels beskrivs barns mognad när det gäller tidsuppfattning på olika sätt i olika studier. Dels verkar olika studier peka på olika svårigheter för barn när det gäller digitala och analoga klockor. Det verkar helt enkelt inte finnas någon samsyn kring vilken ålder barn utvecklar sin tidsuppfattning. Det verkar heller inte som att man enkelt kan säga att den ena tidsformen är enklare än den andra. Däremot kan vi dra slutsatsen att elevers ålder påverkar deras mognad för tidsuppfattning. Vi kan också beskriva många av de missuppfattningar som elever kan uppvisa när det gäller digital och analog representation av tid. 

  • 94.
    Debreceni, Hanna
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevers uppfattningar av mönster: En kvalitativ studie om hur elever i årskurs 1 uppfattar upprepade och växande geometriska mönster2017Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The work of patterns, through its content of structures and relationships, is an important part in the area of algebra. Teaching pupils in the lower grades about mathematical patterns, helps them to understand the more complex algebra. The study is a qualitative interview study inspired by phenomenography, where pupils in grade 1 were asked to solve a number of tasks related to repeating as well as growing geometric patterns. Based on this method, the study aims to contribute knowledge about how pupils in the younger ages perceive mathematical patterns. In connection with the purpose of this study it also contributes to the understanding about what the critical aspects connected to the two types of patterns could be. Once knowledge of the above-mentioned parts is available, teachers can design their education in a way that benefits pupils' learning.

    The result shows a number of critical aspects associated with the understanding of repeating and growing patterns. It may be critical to identify the repeating part in a repeated pattern and to continue a repeated pattern according to the same structure. Furthermore, it may be critical to distinguish a growing structure and a regular numerical connection in the growing patterns. Discovering a general relationship in a growing geometric pattern is another critical aspect that appears in the study.

    Pupils perceive patterns in many different ways, which teachers should keep in mind when planning and conducting teaching. There are important aspects that should be made visible in the education so that pupils can develop an understanding of repeating and growing patterns.

  • 95.
    Duras, Toni
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    A comparison of two estimation methods for common principal components2019In: Communications in statistics. Case studies, data analysis and applications, E-ISSN 2373-7484, Vol. 5, no 4, p. 366-393Article in journal (Refereed)
    Abstract [en]

    Common principal components (CPCs) are often estimated using maximum likelihood estimation through an algorithm called the Flury–Gautschi (FG) Algorithm. Krzanowski proposed a simpler estimation method via a principal component analysis of a weighted sum of the sample covariance matrices. These methods are compared for real-world datasets and in a Monte Carlo simulation. The real-world data is used to compare the selection of a common eigenvector model and the estimated coefficients. The simulation study investigates how the accuracy of the methods is affected by autocorrelation, the number of covariance matrices, dimensions, and sample sizes for multivariate normal and chi-square distributed data. The findings in this article support the use of Krzanowski’s method in situations where the CPC assumption is appropriate. 

  • 96.
    Duras, Toni
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    An extension of the fixed effects principal component model to a common principal component environmentManuscript (preprint) (Other academic)
  • 97.
    Duras, Toni
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Applications of common principal components in multivariate and high-dimensional analysis2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis consists of four papers, all exploring some aspect of common principal component analysis (CPCA), the generalization of PCA to multiple groups. The basic assumption of the CPC model is that the space spanned by the eigenvectors is identical across several groups, whereas eigenvalues associated with the eigenvectors can vary. CPCA is used in essentially the same areas and applications as PCA.

    The first paper compares the performance of the maximum likelihood and Krzanowski’s estimators of the CPC model for two real-world datasets and in a Monte Carlo simulation study. The simplicity and intuition of Krzanowski's estimator and the findings in this paper support and promote the use of this estimator for CPC models over the maximum likelihood estimator.

    Paper number two uses CPCA as a tool for imposing restrictions on system-wise regression models. The paper contributes to the field by proposing a variety of explicit estimators, deriving their properties and identifying the appropriate amount of smoothing that should be imposed on the estimator. 

    In the third paper, a generalization of the fixed effects PCA model to multiple populations in a CPC environment is proposed. The model includes mainly geometrical, rather than probabilistic, assumptions, and is designed to account for any possible prior information about the noise in the data to yield better estimates, obtained by minimizing a least squares criterion with respect to a specified metric.

    The fourth paper survey some properties of the orthogonal group and the associated Haar measure on it. It is demonstrated how seemingly abstract results contribute to applied statistics and, in particular, to PCA.

  • 98.
    Duras, Toni
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Aspects of common principal components2017Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    The focus of this thesis is the common principal component (CPC) model, the generalization of principal components to several populations. Common principal components refer to a group of multidimensional datasets such that their inner products share the same eigenvectors and are therefore simultaneously diagonalized by a common decorrelator matrix. Common principal component analysis is essentially applied in the same areas and analysis as its one-population counterpart. The generalization to multiple populations comes at the cost of being more mathematically involved, and many problems in the area remains to be solved.

    This thesis consists of three individual papers and an introduction chapter.In the first paper, the performance of two different estimation methods of the CPC model is compared for two real-world datasets and in a Monte Carlo simulation study. The second papers show that the orthogonal group and the Haar measure on this group plays an important role in PCA, both in single- and multi-population principal component analysis. The last paper considers using common principal component analysis as a tool for imposing restrictions on system-wise regression models. When the exogenous variables of a multi-dimensional model share common principal components, then each of the marginal models in the system is, up to their eigenvalues, identical. They henceform a class of regression models situated in between the classical seemingly unrelated regressions, where each set of explanatory variables is unique, and multivariate regression, where each marginal model shares the same common set of regressors.

  • 99.
    Duras, Toni
    et al.
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Holgersson, Thomas
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    A small excursion on the Haar measure on OpManuscript (preprint) (Other academic)
  • 100.
    Duras, Toni
    et al.
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Holgersson, Thomas
    Jönköping University, Jönköping International Business School, JIBS, Statistics.
    Common principal components with applications in regressionManuscript (preprint) (Other academic)
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