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  • 1.
    Bengzon, Fredrik
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Johansson, August
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Adaptive submodeling methods for linear elasticity2005Inngår i: Multiscale Methods in Science and Engineering: Lecture Notes in Computational Science and Engineering 44, 2005, s. 171-182Konferansepaper (Fagfellevurdert)
  • 2.
    Bengzon, Fredrik
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Johansson, August
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Söderlund, Robert
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Simulation of multiphysics problems using adaptive finite elements2006Inngår i: Applied parallel computing state of the art in scientific computing: 8th International Workshop, PARA 2006, Umeå, Sweden, umeå: department of Mathematics, Umeå University , 2006, s. 1-14Konferansepaper (Fagfellevurdert)
    Abstract [en]

    Real world applications often involve several types of physics. In practice, one often solves such multiphysics problems by using already existing single physics solvers. To satisfy an overall accuracy, it is critical to understand how accurate the individual single physics solution must be. In this paper we present a framework for a posteriori error estimation of multiphysics problems and derive an algorithm for estimating the total error. We illustrate the technique by solving a coupled flow and transport problem with application in porous media flow.

  • 3.
    Bengzon, Fredrik
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A posteriori error estimates for fractional step methods in fluid mechanics2009Inngår i: Computational Methods in Marine Engineering / [ed] P. Bergan, J. Garcia, E. Onate and T. Kvamsdal, 2009Konferansepaper (Fagfellevurdert)
  • 4.
    Bengzon, Fredrik
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Adaptive finite element approximation of multiphysics problems: a fluid structure interaction model problem2010Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 84, nr 12, s. 1451-1465Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider computation of the displacement of an elastic object immersed into a viscous incompressible flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. We derive an a posteriori error estimate for this one way coupled problem using duality techniques. Based on these estimates we develop an adaptive algorithm that automatically constructs a suitable adapted mesh for the fluid and solid domains given goal quantities specified on the solid problem.

  • 5.
    Bengzon, Fredrik
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Adaptive piecewise constant discontinuous Galerkin methods for convection-diffusion problems2009Manuskript (preprint) (Annet vitenskapelig)
    Abstract [en]

    In this paper we present a discontinuous Galerkin method with  piecewise constant approximation for convection-diffusion type  equations. We show that if the discretization is carefully chosen, then the method is optimal in the L2 norm as well as in a  discrete energy norm measuring the normal flux across element  boundaries. We also derive a posteriori error estimates and  illustrate their effectiveness in combination with adaptive mesh  refinement on a few benchmark problems.

  • 6.
    Bensow, Rickard E.
    et al.
    Göteborg, Sweden.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Residual based VMS subgrid modeling for vortex flows2010Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 199, nr 13-16, s. 802-809Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This paper presents a residual based subgrid modeling approach for Large Eddy Simulations (LES) based on the variational multiscale method as a cure for the problem of preservation of vortices in numerical flow simulation. This approach combines a splitting of the non-linear term in the Navier-Stokes equations into strain and vorticity with a residual based modeling of the subgrid problems. The benefit is that certain driving phenomena, normally not present in subgrid modeling, e.g. vortex stretching, can be seen in the equations. Here, we focus on two of the subgrid terms arising from the subgrid scale problem. The effect of the two terms are illustrated in an LES of a three dimensional flow around a wing where the main feature is the formation and preservation of a tip vortex, an important phenomenon in many aerodynamic and hydrodynamical applications. We see that the addition of the new subgrid terms correctly counteracts the dissipative effect, arising from numerics and turbulence modeling, on the vortex and thus strongly improves prediction of the tip vortex. (C) 2009 Elsevier B.V. All rights reserved.

  • 7. Boiveau, Thomas
    et al.
    Burman, Erik
    Claus, Susanne
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Fictitious domain method with boundary value correction using penalty-free Nitsche method2018Inngår i: Journal of Numerical Mathematics, ISSN 1570-2820, E-ISSN 1569-3953, Vol. 26, nr 2, s. 77-95Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper, we consider a fictitious domain approach based on a Nitsche type method without penalty. To allow for high order approximation using piecewise affine approximation of the geometry we use a boundary value correction technique based on Taylor expansion from the approximate to the physical boundary. To ensure stability of the method a ghost penalty stabilization is considered in the boundary zone. We prove optimal error estimates in the H1-norm and estimates suboptimal by 𝓞(h1/2) in the L2-norm. The suboptimality is due to the lack of adjoint consistency of our formulation. Numerical results are provided to corroborate the theoretical study.

  • 8. Burman, Erik
    et al.
    Claus, Susanne
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Massing, Andre
    Center for Biomedical Computing, Simula Research Laboratory, PO Box 134, NO-1325 Lysaker, Norway.
    CutFEM: Discretizing geometry and partial differential equations2015Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, nr 7, s. 472-501Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer-aided design or image data from applied sciences. Both the treatment of boundaries and interfaces and the discretization of PDEs on surfaces are discussed and illustrated numerically.

  • 9. Burman, Erik
    et al.
    Elfverson, Daniel
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A cut finite element method for the Bernoulli free boundary value problem2017Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 317, s. 598-618Artikkel i tidsskrift (Fagfellevurdert)
    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 H-1 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 H-1 norm. Finally, we present illustrating numerical results.

  • 10.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Elfverson, Daniel
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions2019Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, s. 462-479Artikkel i tidsskrift (Fagfellevurdert)
    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 represents 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.

  • 11.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Elfverson, Daniel
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats
    Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.
    Larsson, Karl
    Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.
    Shape optimization using the cut finite element method2018Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, s. 242-261Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 12. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A cut finite element method with boundary value correction2018Inngår i: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, nr 310, s. 633-657Artikkel i tidsskrift (Fagfellevurdert)
    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 Thomee in [Math. Comp. 26 (1972), 869-879]. The cut finite element method is a fictitious domain method with Nitsche-type enforcement of Dirich-let 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.

  • 13. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A simple approach for finite element simulation of reinforced plates2018Inngår i: Finite elements in analysis and design (Print), ISSN 0168-874X, E-ISSN 1872-6925, Vol. 142, s. 51-60Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 14. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A simple finite element method for elliptic bulk problems with embedded surfaces2019Inngår i: Computational Geosciences, ISSN 1420-0597, E-ISSN 1573-1499, Vol. 23, nr 1, s. 189-199Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper, we develop a simple finite element method for simulation of embedded layers of high permeability in a matrix of lower permeability using a basic model of Darcy flow in embedded cracks. The cracks are allowed to cut through the mesh in arbitrary fashion and we take the flow in the crack into account by superposition. The fact that we use continuous elements leads to suboptimal convergence due to the loss of regularity across the crack. We therefore refine the mesh in the vicinity of the crack in order to recover optimal order convergence in terms of the global mesh parameter. The proper degree of refinement is determined based on an a priori error estimate and can thus be performed before the actual finite element computation is started. Numerical examples showing this effect and confirming the theoretical results are provided. The approach is easy to implement and beneficial for rapid assessment of the effect of crack orientation and may for example be used in an optimization loop.

  • 15. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator2015Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 285, s. 188-207Artikkel i tidsskrift (Fagfellevurdert)
    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. (c) 2014 Elsevier B.V. All rights reserved.

  • 16. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Augmented Lagrangian and Galerkin least-squares methods for membrane contact2018Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 114, nr 11, s. 1179-1191Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 17. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Augmented Lagrangian finite element methods for contact problems2019Inngår i: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 53, nr 1, s. 173-195Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 18. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization2018Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, nr 3, artikkel-id 035004Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 19. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    The Penalty-Free Nitsche Method and Nonconforming Finite Elements for the Signorini Problem2017Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, nr 6, s. 2523-2539Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 20.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Jönköping University.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Massing, Andre
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A cut discontinuous Galerkin method for the Laplace-Beltrami operator2017Inngår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 37, nr 1, s. 138-169Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We develop a discontinuous cut finite element method for the Laplace–Beltrami operator on a hypersurface embedded in Rd . The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in Rd . 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.

  • 21. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Massing, André
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions2019Inngår i: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 52, nr 6, s. 2247-2282Artikkel i tidsskrift (Fagfellevurdert)
    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 R-d 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 R-3.

  • 22. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Massing, André
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Zahedi, Sara
    Full gradient stabilized cut finite element methods for surface partial differential equations2016Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, s. 278-296Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 23. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Samvin, David
    A cut finite element method for elliptic bulk problems with embedded surfaces2019Inngår i: GEM - International Journal on Geomathematics, ISSN 1869-2672, E-ISSN 1869-2680, Vol. 10, nr 1, artikkel-id UNSP 10Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 24. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Stenberg, Rolf
    Galerkin least squares finite element method for the obstacle problem2017Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 313, s. 362-374Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 25. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Zahedi, Sara
    Cut finite element methods for coupled bulk-surface problems2016Inngår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, nr 2, s. 203-231Artikkel i tidsskrift (Fagfellevurdert)
    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 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.

  • 26. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Zahedi, Sara
    Stabilized CutFEM for the convection problem on surfaces2019Inngår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, nr 1, s. 103-139Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 27. Burman, Erik
    et al.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Oksanen, Lauri
    Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem2018Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 56, nr 6, s. 3480-3509Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Darde, Hannukainen, and Hyvonen in [SIAM T. Numer. Anal., 51 (2013), pp. 2123-2148]. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.

  • 28.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Peter, Hansbo
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cut finite elements for convection in fractured domains2019Inngår i: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 179, s. 726-734Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 29.
    Carey, V.
    et al.
    Department of Mathematics, Colorado State University, Fort collins, CO 80523.
    Estep, D.
    Department of Mathematics and Department of Statistics, Colorado State University, Fort collins, CO 80523.
    Ginting, V.
    Department of Mathematics, University of Wyoming, laramie, WY 82071.
    Johansson, August
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Tavener, S.
    Department of Mathematics , Colorado State University, Fort Collins, CO 80523.
    Adaptive finite element solution of coupled PDE-ODE systemsManuskript (preprint) (Annet vitenskapelig)
    Abstract [en]

    We consider an implicit / explicit method for solving a semilinear parabolic partial differential equation (PDE) coupled to a set of nonlinear ordinary differential equations (ODEs). More specifically the PDE of interest is the heat equation where the right hand side couple with the ODEs. For this system, a posteriori error estimates are derived using the method of dual-weighted residuals giving indicators useful for constructing adaptive algorithms.

    We distinguish the errors in time and space for the PDE and the ODEs separately and include errors due to transferring the solutions between the equations. In addition, since the ODEs in many applications are defined on a much smaller spatial scale than what can be resolved by the finite element discretization for the PDE, the error terms include possible projection errors arising when transferring the global PDE solution onto the local ODEs. Recovery errors due to passing the local ODE solutions to the PDE are also included in this analysis.

    The method is illustrated on a realistic problem consisting of a semilinear PDE and a set of nonlinear ODEs modeling the electrical activity in the heart. The method is computationally expensive, why an adaptive algorithm using blocks is used.

  • 30.
    Carey, V
    et al.
    Departments of Mathematics, Colorado State University, Fort Collins, CO 80523.
    Estep, D
    Department of mathematics and Department of Statistics, Colorado State University, Fort Collins, CO 80523 .
    Johansson, August
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Tavener, S
    Departments of Mathematics, Colorado State University, Fort Collins, CO 80523.
    Blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions2010Inngår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 32, nr 4, s. 2121-2145Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We describe and test an adaptive algorithm for evolution problems that employs a sequence of "blocks" consisting of fixed, though non-uniform, space meshes. This approach offers the advantages of adaptive mesh refinement but with reduced overhead costs associated with load balancing, re-meshing, matrix reassembly, and the solution of adjoint problems used to estimate discretization error and the effects of mesh changes. A major issue whith a blockadaptive approach is determining block discretizations from coarse scale solution information that achieve the desired accuracy. We describe several strategies to achieve this goal using adjoint-based a posteriori error estimates and we demonstrate the behavior of the proposed algorithms as well as several technical issues in a set of examples.

  • 31. Cenanovic, Mirza
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cut finite element modeling of linear membranes2016Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, s. 98-111Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We construct a cut finite element method for the membrane elasticity problem on an embedded mesh using tangential differential calculus, i.e., with the equilibrium equations pointwise projected onto the tangent plane of the surface to create a pointwise planar problem in the tangential direction. Both free membranes and membranes coupled to 3D elasticity are considered. The discretization of the membrane comes from a Galerkin method using the restriction of 3D basis functions (linear or trilinear) to the surface representing the membrane. In the case of coupling to 3D elasticity, we view the membrane as giving additional stiffness contributions to the standard stiffness matrix resulting from the discretization of the three-dimensional continuum.

  • 32. Cenanovic, Mirza
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Minimal surface computation using a finite element method on an embedded surface2015Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, nr 7, s. 502-512Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 33.
    Elfverson, Daniel
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A new least squares stabilized Nitsche method for cut isogeometric analysis2019Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 349, s. 1-16Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 34.
    Elfverson, Daniel
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    CutIGA with basis function removal2018Inngår i: Advanced Modeling and Simulation in Engineering Sciences, ISSN 2213-7467, Vol. 5, nr 6Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 35.
    Elfverson, Daniel
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Målqvist, Axel
    Multiscale methods for problems with complex geometry2017Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 321, s. 103-123Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in the energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.

  • 36. Foufas, Georgios
    et al.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
    Valuing Asian options using the finite element method and duality techniques2007Inngår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427Artikkel i tidsskrift (Fagfellevurdert)
  • 37. Hansbo, Peter
    et al.
    Heintz, David
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A finite element method with discontinuous rotations for the Mindlin–Reissner plate model2011Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 200, nr 5-8, s. 638-648Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We present a continuous-discontinuous finite element method for the Mindlin–Reissner plate model based on continuous polynomials of degree k ⩾ 2 for the transverse displacements and discontinuous polynomials of degree k − 1 for the rotations. We prove a priori convergence estimates, uniformly in the thickness of the plate, and thus show that locking is avoided. We also derive a posteriori error estimates based on duality, together with corresponding adaptive procedures for controlling linear functionals of the error. Finally, we present some numerical results.

  • 38.
    Hansbo, Peter
    et al.
    Göteborg, Sweden.
    Heintz, David
    Göteborg, Sweden.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    An adaptive finite element method for second-order plate theory2010Inngår i: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 81, nr 5, s. 584-603Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We present a discontinuous finite element method for the Kirchhoff plate model with membrane stresses. The method is based on P(2)-approximations on simplices for the out-of-plane deformations, using C(0)-continuous approximations. We derive a posteriori error estimates for linear functionals of the error and give some numerical examples. Copyright (C) 2009 John Wiley & Sons, Ltd.

  • 39.
    Hansbo, Peter
    et al.
    Jönköping University.
    Jonsson, Tobias
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A Nitsche method for elliptic problems on composite surfaces2017Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 326, s. 505-525Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 40. Hansbo, Peter
    et al.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff-Love buckling problem2015Inngår i: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 56, nr 5, s. 815-827Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Second order buckling theory involves a one-way coupled coupled problem where the stress tensor from a plane stress problem appears in an eigenvalue problem for the fourth order Kirchhoff plate. In this paper we present an a posteriori error estimate for the critical buckling load and mode corresponding to the smallest eigenvalue and associated eigenvector. A particular feature of the analysis is that we take the effect of approximate computation of the stress tensor and also provide an error indicator for the plane stress problem. The Kirchhoff plate is discretized using a continuous/discontinuous finite element method based on standard continuous piecewise polynomial finite element spaces. The same finite element spaces can be used to solve the plane stress problem.

  • 41. Hansbo, Peter
    et al.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff-Love plate2011Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 200, nr 47-48, s. 3289-3295Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We present energy norm a posteriori error estimates for continuous/discontinuous Galerkin (c/dG) approximations of the Kirchhoff-Love plate problem. The method is based on a continuous displacement field inserted into a symmetric discontinuous Galerkin formulation of the fourth order partial differential equation governing the deflection of a thin plate. We also give explicit formulas for the penalty parameter involved in the formulation. (C) 2011 Elsevier B.V. All rights reserved.

  • 42. Hansbo, Peter
    et al.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A stabilized finite element method for the Darcy problem on surfaces2017Inngår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 37, nr 3, s. 1274-1299Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider a stabilized finite element method for the Darcy problem on a surface based on the Masud-Hughes formulation. A special feature of the method is that the tangential condition of the velocity field is weakly enforced through the bilinear form, and that standard parametric continuous polynomial spaces on triangulations can be used. We prove optimal order a priori estimates that take the approximation of the geometry and the solution into account.

  • 43. Hansbo, Peter
    et al.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Continuous/discontinuous finite element modelling of Kirchhoff plate structures in R3 using tangential differential calculus2017Inngår i: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 60, nr 4, s. 693-702Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We employ surface differential calculus to derive models for Kirchhoff plates including in-plane membrane deformations. We also extend our formulation to structures of plates. For solving the resulting set of partial differential equations, we employ a finite element method based on elements that are continuous for the displacements and discontinuous for the rotations, using -elements for the discretisation of the plate as well as for the membrane deformations. Key to the formulation of the method is a convenient definition of jumps and averages of forms that are d-linear in terms of the element edge normals.

  • 44.
    Hansbo, Peter
    et al.
    Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Sweden.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Energy norm a posteriori error estimates for discontinuous Galerkin approximations of the linear elasticity problem2011Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 200, nr 45-46, s. 3026-3030Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We present a residual-based a posteriori error estimate in an energy norm of the error in a family of discontinuous Galerkin approximations of linear elasticity problems. The theory is developed in two and three spatial dimensions and general nonconvex polygonal domains are allowed. We also present some illustrating numerical examples.

  • 45. Hansbo, Peter
    et al.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Finite element modeling of a linear membrane shell problem using tangential differential calculus2014Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 270, s. 1-14Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We construct a new Galerkin finite element method for the membrane elasticity problem on a meshed surface by using two-dimensional elements extended into three dimensions. The membrane finite element model is established using a tangential differential calculus approach that avoids the use of classical differential geometric methods. The finite element method generalizes the classical flat element shell method where standard plane stress elements are used for membrane problems. This makes our method applicable to a wider range of problems and of surface descriptions, including surfaces defined by distance functions.

    (C) 2013 Elsevier B.V. All rights reserved.

  • 46. Hansbo, Peter
    et al.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Locking free quadrilateral continuous/discontinuous finite element methods for the Reissner-Mindlin plate2014Inngår i: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 269, s. 381-393Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We develop a finite element method with continuous displacements and discontinuous rotations for the Reissner-Mindlin plate model on quadrilateral elements. To avoid shear locking, the rotations must have the same polynomial degree in the parametric reference plane as the parametric derivatives of the displacements, and obey the same transformation law to the physical plane as the gradient of displacements. We prove optimal convergence, uniformly in the plate thickness, and provide numerical results that confirm our estimates.

  • 47. Hansbo, Peter
    et al.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Fredrik
    Tangential differential calculus and the finite element modeling of a large deformation elastic membrane problem2015Inngår i: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 56, nr 1, s. 87-95Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We develop a finite element method for a large deformation membrane elasticity problem on meshed curved surfaces using a tangential differential calculus approach that avoids the use of classical differential geometric methods. The method is also applied to form finding problems.

  • 48.
    Hansbo, Peter
    et al.
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cut Finite Element Methods for Linear Elasticity Problems2018Inngår i: Geometrically Unfitted Finite Element Methods and Applications: Proceedings of the UCL Workshop 2016 / [ed] Stéphane P. A. Bordas; Erik Burman; Mats G. Larson; Maxim A. Olshanskii, Springer, 2018, s. 25-63Kapittel i bok, del av antologi (Fagfellevurdert)
    Abstract [en]

    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.

  • 49.
    Hansbo, Peter
    et al.
    Tekniska Högskolan i Jönköping.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Intrinsic finite element modeling of curved beamsManuskript (preprint) (Annet vitenskapelig)
    Abstract [en]

    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.

  • 50.
    Hansbo, Peter
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Variational formulation of curved beams in global coordinates2014Inngår i: Computational Mechanics, ISSN 0178-7675, E-ISSN 1432-0924, Vol. 53, nr 4, s. 611-623Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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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