| 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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| 11. |
- Burman, Erik, et al.
(författare)
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Isogeometric analysis and Augmented Lagrangian Galerkin Least Squares Methods for residual minimization in dual norm
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 417:Part B
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Tidskriftsartikel (refereegranskat)abstract
- We explore how recent advances in Isogeometric analysis, Galerkin Least-Squares methods, and Augmented Lagrangian techniques can be applied to solve nonstandard problems, for which there is no classical stability theory, such as that provided by the Lax–Milgram lemma or the Banach-Necas-Babuska theorem. In particular, we consider continuation problems where a second-order partial differential equation with incomplete boundary data is solved given measurements of the solution on a subdomain of the computational domain. The use of higher regularity spline spaces leads to simplified formulations and potentially minimal multiplier space. We show that our formulation is inf-sup stable, and given appropriate a priori assumptions, we establish optimal order convergence.
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| 12. |
- Burman, Erik, et al.
(författare)
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Shape and topology optimization using CutFEM
- 2017
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Ingår i: Simulation for Additive Manufacturing 2017, Sinam 2017. - : International Center for Numerical Methods in Engineering (CIMNE). ; , s. 208-209, s. 208-209
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 13. |
- Burman, Erik, et al.
(författare)
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Shape optimization using the cut finite element method
- 2018
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - Lausanne : Elsevier. - 0045-7825 .- 1879-2138. ; 328, s. 242-261
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 14. |
- Elfverson, Daniel, et al.
(författare)
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A new least squares stabilized Nitsche method for cut isogeometric analysis
- 2019
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 349, s. 1-16
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Tidskriftsartikel (refereegranskat)abstract
- We derive a new stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis (CutIGA). We consider C1 splines and stabilize the standard Nitsche method by adding a certain elementwise least squares terms in the vicinity of the Dirichlet boundary and an additional term on the boundary which involves the tangential gradient. We show coercivity with respect to the energy norm for functions in H2(Ω) and optimal order a priori error estimates in the energy and L2 norms. To obtain a well posed linear system of equations we combine our formulation with basis function removal which essentially eliminates basis functions with sufficiently small intersection with Ω. The upshot of the formulation is that only elementwise stabilization is added in contrast to standard procedures based on ghost penalty and related techniques and that the stabilization is consistent. In our numerical experiments we see that the method works remarkably well in even extreme cut situations using a Nitsche parameter of moderate size.
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| 15. |
- Elfverson, Daniel, et al.
(författare)
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CutIGA with basis function removal
- 2018
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Ingår i: Advanced Modeling and Simulation in Engineering Sciences. - : Springer. - 2213-7467. ; 5:6
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Tidskriftsartikel (refereegranskat)abstract
- We consider a cut isogeometric method, where the boundary of the domain is allowed to cut through the background mesh in an arbitrary fashion for a second order elliptic model problem. In order to stabilize the method on the cut boundary we remove basis functions which have small intersection with the computational domain. We determine criteria on the intersection which guarantee that the order of convergence in the energy norm is not affected by the removal. The higher order regularity of the B-spline basis functions leads to improved bounds compared to standard Lagrange elements.
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| 16. |
- Hansbo, Peter, 1959-, et al.
(författare)
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A Nitsche method for elliptic problems on composite surfaces
- 2017
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - Lausanne : Elsevier. - 0045-7825 .- 1879-2138. ; 326, s. 505-525
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Tidskriftsartikel (refereegranskat)abstract
- We develop a finite element method for elliptic partial differential equations on so called composite surfaces that are built up out of a finite number of surfaces with boundaries that fit together nicely in the sense that the intersection between any two surfaces in the composite surface is either empty, a point, or a curve segment, called an interface curve. Note that several surfaces can intersect along the same interface curve. On the composite surface we consider a broken finite element space which consists of a continuous finite element space at each subsurface without continuity requirements across the interface curves. We derive a Nitsche type formulation in this general setting and by assuming only that a certain inverse inequality and an approximation property hold we can derive stability and error estimates in the case when the geometry is exactly represented. We discuss several different realizations, including so called cut meshes, of the method. Finally, we present numerical examples.
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| 17. |
- Hansbo, Peter, et al.
(författare)
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Analysis of finite element methods for vector Laplacians on surfaces
- 2020
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Ingår i: IMA Journal of Numerical Analysis. - : Oxford University Press. - 0272-4979 .- 1464-3642. ; 40:3, s. 1652-1701
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Tidskriftsartikel (refereegranskat)abstract
- We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in R-3. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a R-3 vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and L-2 norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.
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| 18. |
- Hansbo, Peter, 1959-, et al.
(författare)
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Cut Finite Element Methods for Linear Elasticity Problems
- 2017
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Ingår i: Geometrically Unfitted Finite Element Methods and Applications. - Cham : Springer International Publishing. - 9783319714301 - 9783319714318 ; , s. 25-63
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Konferensbidrag (refereegranskat)abstract
- We formulate a cut finite element method for linear elasticity based on higher order elements on a fixed background mesh. Key to the method is a stabilization term which provides control of the jumps in the derivatives of the finite element functions across faces in the vicinity of the boundary. We then develop the basic theoretical results including error estimates and estimates of the condition number of the mass and stiffness matrices. We apply the method to the standard displacement problem, the frequency response problem, and the eigenvalue problem. We present several numerical examples including studies of thin bending dominated structures relevant for engineering applications. Finally, we develop a cut finite element method for fibre reinforced materials where the fibres are modeled as a superposition of a truss and a Euler-Bernoulli beam. The beam model leads to a fourth order problem which we discretize using the restriction of the bulk finite element space to the fibre together with a continuous/discontinuous finite element formulation. Here the bulk material stabilizes the problem and it is not necessary to add additional stabilization terms.
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| 19. |
- Hansbo, Peter, et al.
(författare)
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Intrinsic finite element modeling of curved beams
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In the mid '90s Delfour and Zolesio [4-6] established elasticity models on surfaces described using the signed distance function, an approach they called intrinsic modeling. For problems in codimension-two, e.g. one-dimensional geometries embedded in R3, an analogous description can be done using a vector distance function. In this paper we investigate the intrinsic approach for the modeling of codimension-two problems by deriving a weak formulation for a linear curved beam expressed in three dimensions from the equilibrium equations of linear elasticity. Based on this formulation we implement a finite element model using global degrees of freedom and discuss upon the effects of curvature and locking. Comparisons with classical solutions for both straight and curved cantilever beams under a tip load are given.
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| 20. |
- Hansbo, Peter, et al.
(författare)
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Variational formulation of curved beams in global coordinates
- 2014
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Ingår i: Computational Mechanics. - : Springer Science and Business Media LLC. - 0178-7675 .- 1432-0924. ; 53:4, s. 611-623
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive a variational formulation for the static analysis of a linear curved beam natively expressed in global Cartesian coordinates. Using an implicit description of the beam midline during derivation we eliminate the need for local coordinates. The only geometrical information appearing in the final expressions for the governing equations is the tangential direction. As a consequence, zero or discontinuous curvature, for example at inflection points, pose no difficulty in this formulation. Kinematic assumptions encompassing both Timoshenko and Euler-Bernoulli beam theories are considered. With the exception of truly three-dimensional formulations, models for curved beams found in the literature are typically derived in the local Frenet frame. We implement finite element methods with global degrees of freedom and discuss curvature coupling effects and locking. Numerical comparisons with classical solutions for straight and curved cantilever beams under tip load are given, as well as numerical examples illustrating curvature coupling effects.
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| 21. |
- Jonsson, Tobias, et al.
(författare)
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Cut finite element methods for elliptic problems on multipatch parametric surfaces
- 2017
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 324, s. 366-394
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Tidskriftsartikel (refereegranskat)abstract
- We develop a finite element method for the Laplace–Beltrami operator on a surface described by a set of patchwise parametrizations. The patches provide a partition of the surface and each patch is the image by a diffeomorphism of a subdomain of the unit square which is bounded by a number of smooth trim curves. A patchwise tensor product mesh is constructed by using a structured mesh in the reference domain. Since the patches are trimmed we obtain cut elements in the vicinity of the interfaces. We discretize the Laplace–Beltrami operator using a cut finite element method that utilizes Nitsche’s method to enforce continuity at the interfaces and a consistent stabilization term to handle the cut elements. Several quantities in the method are conveniently computed in the reference domain where the mappings impose a Riemannian metric. We derive a priori estimates in the energy and L2 norm and also present several numerical examples confirming our theoretical results.
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| 22. |
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| 23. |
- Jonsson, Tobias, 1991-
(författare)
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Cut isogeometric methods on trimmed multipatch surfaces
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Partial differential equations (PDE) on surfaces appear in a variety of applications, such as image processing, modeling of lubrication, fluid flows, diffusion, and transport of surfactants. In some applications, surfaces are drawn and modeled by using CAD software, giving a very precise patchwise parametric description of the surface. This thesis deals with the development of methods for finding numerical solutions to PDE posed on such parametrically described multipatch surfaces. The thesis consists of an introduction and five papers.In the first paper, we develop a general framework for the Laplace-Beltrami operator on a patchwise parametric surface. Each patch map induces a Riemannian metric, which we utilize to compute quantities in the simpler reference domain. We use the cut finite element method together with Nitsche’s method to enforce continuity over the interfaces between patches.In the second paper, we extend the framework to be able to handle geometries that consist of an arrangement of surfaces, i.e., more than two per interface. By using a Kirchhoff's condition this method avoids defining any co-normal to each surface and can deal with sharp edges. This approach is shown to be equivalent to standard Nitsche interface method for flat geometries.In the third paper, we developed a cut finite element method for elliptic problems with corner singularities. The main idea is to use an appropriate radial map that grades the finite element mesh towards the corner that counter-acts the solution's singularity.In the fourth paper, we present a new robust isogeometric method for surfaces described by CAD patches with gaps or overlaps. The main approach here is to cover all interfaces with a three-dimensional mesh and then use a hybrid variable in a Nitsche-type formulation to transfer data over the gaps. Using this hybridized approach leads to a convenient and easy to implement method with no restriction on the number of coupled patches per interface.In the fifth paper, we present a routine to the multipatch isogeometric framework for dealing with singular maps. To exemplify this, we consider a specific type of singular parametrization which essentially maps a square onto a triangle. One part of the boundary of the square will be transformed into a single point and the metric tensor becomes singular as we approach this boundary. In this work we propose a regularization procedure which is based on eigenvalue decomposition of the metric tensor.
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| 24. |
- Jonsson, Tobias, 1991-, et al.
(författare)
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Graded Parametric CutFEM and CutIGA for Elliptic Boundary Value Problems in Domains with Corners
- 2019
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 354, s. 331-350
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Tidskriftsartikel (refereegranskat)abstract
- We develop a parametric cut finite element method for elliptic boundary value problems with corner singularities where we have weighted control of higher order derivatives of the solution to a neighborhood of a point at the boundary. Our approach is based on identification of a suitable mapping that grades the mesh towards the singularity. In particular, this mapping may be chosen without identifying the opening angle at the corner. We employ cut finite elements together with Nitsche boundary conditions and stabilization in the vicinity of the boundary. We prove that the method is stable and convergent of optimal order in the energy norm and L2 norm. This is achieved by mapping to the reference domain where we employ a structured mesh.
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| 25. |
- Jonsson, Tobias, 1991-, et al.
(författare)
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Hybridized isogeometric method for elliptic problems on CAD surfaces with gaps
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 410
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Tidskriftsartikel (refereegranskat)abstract
- We develop a method for solving elliptic partial differential equations on surfaces described by CAD patches that may have gaps/overlaps. The method is based on hybridization using a three-dimensional mesh that covers the gap/overlap between patches. Thus, the hybrid variable is defined on a three-dimensional mesh, and we need to add appropriate normal stabilization to obtain an accurate solution, which we show can be done by adding a suitable term to the weak form. In practical applications, the hybrid mesh may be conveniently constructed using an octree to efficiently compute the necessary geometric information. We prove error estimates and present several numerical examples illustrating the application of the method to different problems, including a realistic CAD model.
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| 26. |
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| 27. |
- Larsson, Karl, 1981-, et al.
(författare)
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A continuous/discontinuous Galerkin method and a priori error estimates for the biharmonic problem on surfaces
- 2017
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 86:308, s. 2613-2649
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Tidskriftsartikel (refereegranskat)abstract
- We present a continuous/discontinuous Galerkin method for approximating solutions to a fourth order elliptic PDE on a surface embedded in R-3. A priori error estimates, taking both the approximation of the surface and the approximation of surface differential operators into account, are proven in a discrete energy norm and in L-2 norm. This can be seen as an extension of the formalism and method originally used by Dziuk ( 1988) for approximating solutions to the Laplace-Beltrami problem, and within this setting this is the first analysis of a surface finite element method formulated using higher order surface differential operators. Using a polygonal approximation inverted right perpendicular(h) of an implicitly defined surface inverted right perpendicular we employ continuous piecewise quadratic finite elements to approximate solutions to the biharmonic equation on inverted right perpendicular. Numerical examples on the sphere and on the torus confirm the convergence rate implied by our estimates.
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| 28. |
- Larsson, Karl, 1981-, et al.
(författare)
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Continuous piecewise linear finite elements for the Kirchhoff-Love plate equation
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff-Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the Basic Plate Triangle. Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L2 norm. Numerical results indicate that the Morley reconstruction/Basic Plate Triangle does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.
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| 29. |
- Larsson, Karl, 1981-, et al.
(författare)
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Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation
- 2012
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Ingår i: Numerische Mathematik. - : Springer Berlin/Heidelberg. - 0029-599X .- 0945-3245. ; 121:1, s. 65-97
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Tidskriftsartikel (refereegranskat)abstract
- A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L2 norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.
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| 30. |
- Larsson, Karl, 1981-
(författare)
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Finite Element Methods for Thin Structures with Applications in Solid Mechanics
- 2013
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Thin and slender structures are widely occurring both in nature and in human creations. Clever geometries of thin structures can produce strong constructions while requiring a minimal amount of material. Computer modeling and analysis of thin and slender structures have their own set of problems, stemming from assumptions made when deriving the governing equations. This thesis deals with the derivation of numerical methods suitable for approximating solutions to problems on thin geometries. It consists of an introduction and four papers.In the first paper we introduce a thread model for use in interactive simulation. Based on a three-dimensional beam model, a corotational approach is used for interactive simulation speeds in combination with adaptive mesh resolution to maintain accuracy.In the second paper we present a family of continuous piecewise linear finite elements for thin plate problems. Patchwise reconstruction of a discontinuous piecewise quadratic deflection field allows us touse a discontinuous Galerkin method for the plate problem. Assuming a criterion on the reconstructions is fulfilled we prove a priori error estimates in energy norm and L2-norm and provide numerical results to support our findings.The third paper deals with the biharmonic equation on a surface embedded in R3. We extend theory and formalism, developed for the approximation of solutions to the Laplace-Beltrami problem on an implicitly defined surface, to also cover the biharmonic problem. A priori error estimates for a continuous/discontinuous Galerkin method is proven in energy norm and L2-norm, and we support the theoretical results by numerical convergence studies for problems on a sphere and on a torus.In the fourth paper we consider finite element modeling of curved beams in R3. We let the geometry of the beam be implicitly defined by a vector distance function. Starting from the three-dimensional equations of linear elasticity, we derive a weak formulation for a linear curved beam expressed in global coordinates. Numerical results from a finite element implementation based on these equations are compared with classical results.
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| 31. |
- Larsson, Karl, 1981-
(författare)
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Finite element methods for threads and plates with real-time applications
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Thin and slender structures are widely occurring both in nature and in human creations. Clever geometries of thin structures can produce strong constructions while using a minimal amount of material. Computer modeling and analysis of thin and slender structures has its own set of problems stemming from assumptions made when deriving the equations modeling their behavior from the theory of continuum mechanics. In this thesis we consider two kinds of thin elastic structures; threads and plates. Real-time simulation of threads are of interest in various types of virtual simulations such as surgery simulation for instance. In the first paper of this thesis we develop a thread model for use in interactive applications. By viewing the thread as a continuum rather than a truly one dimensional object existing in three dimensional space we derive a thread model that naturally handles both bending, torsion and inertial effects. We apply a corotational framework to simulate large deformation in real-time. On the fly adaptive resolution is used to minimize corotational artifacts. Plates are flat elastic structures only allowing deflection in the normal direction. In the second paper in this thesis we propose a family of finite elements for approximating solutions to the Kirchhoff-Love plate equation using a continuous piecewise linear deflection field. We reconstruct a discontinuous piecewise quadratic deflection field which is applied in a discontinuous Galerkin method. Given a criterion on the reconstruction operator we prove a priori estimates in energy and L2 norms. Numerical results for the method using three possible reconstructions are presented.
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| 32. |
- Larsson, Karl, 1981-, et al.
(författare)
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Interactive simulation of a continuum mechanics based torsional thread
- 2010
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Ingår i: Vriphys 10. - Copenhagen, Denmark : Eurographics Association. - 9783905673784 ; , s. 49-58
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Konferensbidrag (refereegranskat)abstract
- This paper introduces a continuum mechanics based thread model for use in real-time simulation. The model includes both rotary inertia, shear deformation and torsion. It is based on a three-dimensional beam model, using a corotational approach for interactive simulation speeds as well as adaptive mesh resolution to maintain accuracy. Desirable aspects of this model from a numerical and implementation point of view include a true constant and symmetric mass matrix, a symmetric and easily evaluated tangent stiffness matrix, and easy implementation of time-stepping algorithms. From a modeling perspective interesting features are deformation of the thread cross section and the use of arbitrary cross sections without performance penalty.
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| 33. |
- Larsson, Karl, 1981-, et al.
(författare)
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The finite cell method with least squares stabilized Nitsche boundary conditions
- 2022
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 393
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Tidskriftsartikel (refereegranskat)abstract
- We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization, which does not lead to additional fill in. We prove a priori error estimates and bounds on the condition numbers.
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