| 1. |
- Larsson, Jonas, et al.
(författare)
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An introduction to relativistic electrodynamics : Part I: Calculus with 4-vectors and 4-dyadics
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The conventional way of introducing relativity when teaching electrodynamics is to leave Gibbs' vector calculus for a more general tensor calculus. This sudden change of formalism can be quite problematic for the students and we therefore in this two-part paper consider alternate approaches. In this Part I we use a simplified tensor formalism with 4-vectors and 4-dyadics (i.e., second order tensors built by 4-vectors) but with no tensors of higher order than two. This allows for notations in good contact with the coordinate-free Gibbs' vector calculus that the students already master. Thus we use boldface notations for 4-vectors and 4-dyadics without coordinates and index algebra to formulate Lorentz transformations, Maxwell's equations, the equation of the motion of charged particles and the stress-energy conservation law. By first working with this simplified tensor formalism the students will get better prepared to learn the standard tensor calculus needed in more advanced courses.
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| 2. |
- Larsson, Jonas, et al.
(författare)
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An introduction to relativistic electrodynamics : Part II: Calculus with complex 4-vectors
- 2018
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The conventional way of introducing relativity when teaching electrodynamics is to leave Gibbs' vector calculus for a more general tensor calculus. This sudden change of formalism can be quite problematic for the students and we therefore in this two-part paper consider alternate approaches. The algebra of 2-by-2 complex matrices (sometimes presented in the form of Clifford algebra or complex quaternions) may be used for spinor related formulations of special relativity and electrodynamics. In this Part II we use this algebraic structure but with notations that fits in with the formalism of Part I. Each observer defines a product on the space of complex 4-vectors so that becomes an algebra isomorphic to with as algebra unit. The spacetime geometric equations of Part I become complex (spinor related) equations where the antisymmetric 4-dyadics have been replaced by complex 3-vectors, i.e., by elements in . For example, instead of the electromagnetic dyadic field we now have the complex field variable . Some linear algebra together with the formalism of Gibbs' vector calculus (trivially allowing for complex 3-vectors) is sufficient for dealing with the equations in their complex form.
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| 3. |
- Larsson, Jonas, et al.
(författare)
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The Lorentz group and the Kronecker product of matrices
- 2022
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Ingår i: European journal of physics. - : Institute of Physics Publishing (IOPP). - 0143-0807 .- 1361-6404. ; 43:2
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Tidskriftsartikel (refereegranskat)abstract
- The group SL(2,C)(2,C) of all complex 2 × 2 matrices with determinant one is closely related to the group L+↑ of real 4 × 4 matrices representing the restricted Lorentz transformations. This relation, sometimes called the spinor map, is of fundamental importance in relativistic quantum mechanics and has applications also in general relativity. In this paper we show how the spinor map may be expressed in terms of pure matrix algebra by including the Kronecker product between matrices in the formalism. The so-obtained formula for the spinor map may be manipulated by matrix algebra and used in the study of Lorentz transformations.
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| 4. |
- Stattin, Karl, et al.
(författare)
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The risk of different fracture types across a wide range of physical activity levels, from sedentary individuals to elite athletes
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- AbstractBackground Physical activity has been associated with a lower risk of fragility fractures, but the shape of the association is not known.Methods Individuals aged 49-68 years were drawn from the population-based Swedish Mammography Cohort (n=22,256) and Cohort of Swedish Men (n=28,749) as well as from a cohort of highly physically active participants in the Vasaloppet skiing race (n=12,984). A common measure of physical activity was created from lifestyle questionnaires and race data using generalized structural equation modeling. The median physical activity corresponded to 2-3 hours of weekly exercise or 20-40 minutes of daily walking/bicycling. The rate of any, wrist, proximal humerus, spine and hip fractures were estimated using restricted cubic splines in Cox proportional hazard models.Results During a maximal follow-up of 13 years, 8,506 fractures at any site, 2,164 wrist, 779 proximal humerus, 346 spine and 908 hip fractures occurred. The rate of any fracture was lowest close to the median physical activity and higher in both low and high levels of physical activity, hazard ratio (HR) 1.05 (95% confidence interval (CI) 1.01-1.08) and 1.11 (95% CI 1.05-1.17) for physical activity 1 SD below and 1.5 SD above the average, respectively. The rate of wrist fracture was lowest among individuals with low levels of physical activity, HR 0.92 (95% CI 0.86-0.99) for physical activity 1 SD below the average, and increased until the median level of physical activity. Proximal humerus fracture was not associated with physical activity. Spine fracture had a U-shaped association with physical activity with wide confidence intervals. Low physical activity was associated with higher rate of hip fracture, HR 1.24 (95% CI 1.12-1.36) for physical activity 1 SD below the average.Discussion In this combination of cohorts including individuals with a wide range of physical activity, from sedentary individuals to elite athletes, the associations between physical activity and fractures were non-linear and differed according to fracture site. For wrist and hip fractures, there appears to be a threshold value above which further physical activity is not associated with further changes in the rate of fracture.
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| 5. |
- Björklund, Martin, et al.
(författare)
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Error estimates for finite element approximations of viscoelastic dynamics : the generalized Maxwell model
- 2024
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 425
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Tidskriftsartikel (refereegranskat)abstract
- We prove error estimates for a finite element approximation of viscoelastic dynamics based on continuous Galerkin in space and time, both in energy norm and in L2 norm. The proof is based on an error representation formula using a discrete dual problem and a stability estimate involving the kinetic, elastic, and viscoelastic energies. To set up the dual error analysis and to prove the basic stability estimates, it is natural to formulate the problem as a first-order-in-time system involving evolution equations for the viscoelastic stress, the displacements, and the velocities. The equations for the viscoelastic stress can, however, be solved analytically in terms of the deviatoric strain velocity, and therefore, the viscoelastic stress can be eliminated from the system, resulting in a system for displacements and velocities.
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| 6. |
- Burman, Erik, et al.
(författare)
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Cut finite elements for convection in fractured domains
- 2019
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Ingår i: Computers & Fluids. - : Elsevier. - 0045-7930 .- 1879-0747. ; 179, s. 728-736
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
- Burman, Erik, et al.
(författare)
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Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions
- 2019
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 350, s. 462-479
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 8. |
- Burman, E., et al.
(författare)
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Extension operators for trimmed spline spaces
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 403
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Tidskriftsartikel (refereegranskat)abstract
- We develop a discrete extension operator for trimmed spline spaces consisting of piecewise polynomial functions of degree p with k continuous derivatives. The construction is based on polynomial extension from neighboring elements together with projection back into the spline space. We prove stability and approximation results for the extension operator. Finally, we illustrate how we can use the extension operator to construct a stable cut isogeometric method for an elliptic model problem.
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| 9. |
- Burman, Erik, et al.
(författare)
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Finite element approximation of the Laplace-Beltrami operator on a surface with boundary
- 2019
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 141:1, s. 141-172
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Tidskriftsartikel (refereegranskat)abstract
- 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 k ≥ 1 in the energy and L2 norms that take the approximation of the surface and the boundary into account.
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| 10. |
- Burman, Erik, et al.
(författare)
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Hybridized CutFEM for Elliptic Interface Problems
- 2019
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 41:5, s. A3354-A3380
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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