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  • 1. Balmus, Maximilian
    et al.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway.
    Hoffman, Johan
    Razavi, Reza
    Nordsletten, David A.
    A partition of unity approach to fluid mechanics and fluid-structure interaction2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 362, article id 112842Article in journal (Refereed)
    Abstract [en]

    For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.

  • 2.
    Bensow, Rickard E.
    et al.
    Göteborg, Sweden.
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Residual based VMS subgrid modeling for vortex flows2010In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 199, no 13-16, p. 802-809Article in journal (Refereed)
    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.

  • 3.
    Björklund, Martin
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Error estimates for finite element approximations of viscoelastic dynamics: the generalized Maxwell model2024In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 425, article id 116933Article in journal (Refereed)
    Abstract [en]

    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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  • 4. Burman, Erik
    et al.
    Elfverson, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A cut finite element method for the Bernoulli free boundary value problem2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 317, p. 598-618Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the 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.

  • 5.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Elfverson, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 462-479Article in journal (Refereed)
    Abstract [en]

    We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically 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.

  • 6.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Elfverson, Daniel
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    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 method2018In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 242-261Article in journal (Refereed)
    Abstract [en]

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

  • 7. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator2015In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 285, p. 188-207Article in journal (Refereed)
    Abstract [en]

    We consider solving the Laplace-Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions. The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method. (c) 2014 Elsevier B.V. All rights reserved.

  • 8.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, UK.
    Hansbo, Peter
    School of Engineering, Jönköping University, Jönköping, Sweden.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Extension operators for trimmed spline spaces2023In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 403, article id 115707Article in journal (Refereed)
    Abstract [en]

    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.
    Mathematics, University College London, United Kingdom.
    Hansbo, Peter
    Mechanical Engineering, Jönköping University, Sweden.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Isogeometric analysis and augmented lagrangian galerkin least squares methods for residual minimization in dual norm2023In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 417, article id 116302Article in journal (Refereed)
    Abstract [en]

    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.

  • 10. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A stable cut finite element method for partial differential equations on surfaces: The Helmholtz-Beltrami operator2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 362, article id 112803Article in journal (Refereed)
    Abstract [en]

    We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions. Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition h kappa < C, where h denotes the mesh size, kappa the wave number and C a constant depending mainly on the surface curvature kappa, but not on the surface/mesh intersection. Optimal error estimates in the H-1 and L-2-norms follow.

  • 11. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zahedi, Sara
    A stabilized cut streamline diffusion finite element method for convection-diffusion problems on surfaces2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 358, article id 112645Article in journal (Refereed)
    Abstract [en]

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

  • 12. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zahedi, Sara
    Full gradient stabilized cut finite element methods for surface partial differential equations2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 278-296Article in journal (Refereed)
    Abstract [en]

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

  • 13. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stenberg, Rolf
    Galerkin least squares finite element method for the obstacle problem2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 313, p. 362-374Article in journal (Refereed)
    Abstract [en]

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

  • 14.
    Bängtsson, Erik
    et al.
    Department of Information Technology, Uppsala University.
    Noreland, Daniel
    Department of Information Technology, Uppsala University.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    Shape optimization of an acoustic horn2003In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 192, p. 1533-1571Article in journal (Refereed)
  • 15. Cenanovic, Mirza
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cut finite element modeling of linear membranes2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 98-111Article in journal (Refereed)
    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.

  • 16. Cenanovic, Mirza
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 372, article id 113445Article in journal (Refereed)
    Abstract [en]

    In this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces. In particular, we adopt a stabilization technique which allows for first order L-2-convergence of the mean curvature vector and apply this stabilization technique also to the computation of continuous, recovered, normals using L-2-projections of the piecewise constant face normals. Finally, we use our projected normals to define an adaptive mesh refinement approach to geometry resolution where we also employ spline techniques to reconstruct the surface before refinement. We compare our results to previously proposed approaches.

  • 17. Dokken, Jørgen S.
    et al.
    Johansson, August
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway.
    Funke, Simon W.
    A multimesh finite element method for the Navier-Stokes equations based on projection methods2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 368, article id 113129Article in journal (Refereed)
    Abstract [en]

    The multimesh finite element method is a technique for solving partial differential equations on multiple non-matching meshes by enforcing interface conditions using Nitsche's method. Since the non-matching meshes can result in arbitrarily cut cells, additional stabilization terms are needed to obtain a stable method. In this contribution we extend the multimesh finite element method to the Navier-Stokes equations based on the incremental pressure-correction scheme. For each step in the pressure-correction scheme, we derive a multimesh finite element formulation with suitable stabilization terms. The proposed scheme is implemented for arbitrary many overlapping two dimensional domains, yielding expected spatial and temporal convergence rates for the Taylor-Green problem, and demonstrates good agreement for the drag and lift coefficients for the Turek-Schfifer benchmark (DFG benchmark 2D-3). Finally, we illustrate the capabilities of the proposed scheme by optimizing the layout of obstacles in a two dimensional channel.

  • 18.
    Elfverson, Daniel
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A new least squares stabilized Nitsche method for cut isogeometric analysis2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 349, p. 1-16Article in journal (Refereed)
    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.

  • 19.
    Elfverson, Daniel
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Målqvist, Axel
    Multiscale methods for problems with complex geometry2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 321, p. 103-123Article in journal (Refereed)
    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.

  • 20.
    Gürkan, Ceren
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematical Sciences, Norwegian University of Science and Technology, NO 7491, Trondheim, Norway.
    A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 348, p. 466-499Article in journal (Refereed)
    Abstract [en]

    We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of elliptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in R d , so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the background mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinuous Galerkin methods, we employ and extend ghost penalty techniques from recently developed continuous cut finite element methods, which allows for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying four abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and condition number estimates for the Poisson boundary value problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. We also demonstrate how the framework can be elegantly applied to discretize high contrast interface problems. The theoretical results are illustrated by a number of numerical experiments for various approximation orders and for two and three-dimensional test problems. (C) 2018 Published by Elsevier B.V.

  • 21. Hansbo, Peter
    et al.
    Heintz, David
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A finite element method with discontinuous rotations for the Mindlin–Reissner plate model2011In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 200, no 5-8, p. 638-648Article in journal (Refereed)
    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.

  • 22.
    Hansbo, Peter
    et al.
    Jönköping University.
    Jonsson, Tobias
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A Nitsche method for elliptic problems on composite surfaces2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 326, p. 505-525Article in journal (Refereed)
    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.

  • 23. Hansbo, Peter
    et al.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A posteriori error estimates for continuous/discontinuous Galerkin approximations of the Kirchhoff-Love plate2011In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 200, no 47-48, p. 3289-3295Article in journal (Refereed)
    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.

  • 24.
    Hansbo, Peter
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A simple nonconforming tetrahedral element for the Stokes equations2022In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 400, article id 115549Article in journal (Refereed)
    Abstract [en]

    In this paper we apply a nonconforming rotated bilinear tetrahedral element to the Stokes problem in R3. We show that the element is stable in combination with a piecewise linear, continuous, approximation of the pressure. This gives an approximation similar to the well known continuous P2–P1 Taylor–Hood element, but with fewer degrees of freedom. The element is a stable non-conforming low order element which fulfils Korn's inequality, leading to stability also in the case where the Stokes equations are written on stress form for use in the case of free surface flow.

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  • 25.
    Hansbo, Peter
    et al.
    Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Sweden.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Energy norm a posteriori error estimates for discontinuous Galerkin approximations of the linear elasticity problem2011In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 200, no 45-46, p. 3026-3030Article in journal (Refereed)
    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.

  • 26. Hansbo, Peter
    et al.
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Finite element modeling of a linear membrane shell problem using tangential differential calculus2014In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 270, p. 1-14Article in journal (Refereed)
    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.

  • 27. Hansbo, Peter
    et al.
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Locking free quadrilateral continuous/discontinuous finite element methods for the Reissner-Mindlin plate2014In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 269, p. 381-393Article in journal (Refereed)
    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.

  • 28.
    Hansbo, Peter
    et al.
    Department of Mechanical Engineering, Jönköping University, Jönköping, Sweden.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nitsche's finite element method for model coupling in elasticity2022In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 392, article id 114707Article in journal (Refereed)
    Abstract [en]

    We develop a hybridized Nitsche finite element method for a two dimensional elastic interface problems. Our approach allows for modelling of Euler–Bernoulli beams with axial stiffness embedded in an elastic bulk domain. The beams have their own displacement fields, and the elastic subdomains created by the beam network are triangulated independently and are coupled to the beams weakly by use of Nitsche's method.

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  • 29. Hansbo, Peter
    et al.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A stabilized cut finite element method for the Darcy problem on surfaces2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 326, p. 298-318Article in journal (Refereed)
    Abstract [en]

    We develop a cut finite element method for the Darcy problem on surfaces. The cut finite element method is based on embedding the surface in a three dimensional finite element mesh and using finite element spaces defined on the three dimensional mesh as trial and test functions. Since we consider a partial differential equation on a surface, the resulting discrete weak problem might be severely ill conditioned. We propose a full gradient and a normal gradient based stabilization computed on the background mesh to render the proposed formulation stable and well conditioned irrespective of the surface positioning within the mesh. Our formulation extends and simplifies the Masud-Hughes stabilized primal mixed formulation of the Darcy surface problem proposed in Hansbo and Larson (2016) on fitted triangulated surfaces. The tangential condition on the velocity and the pressure gradient is enforced only weakly, avoiding the need for any tangential projection. The presented numerical analysis accounts for different polynomial orders for the velocity, pressure, and geometry approximation which are corroborated by numerical experiments. In particular, we demonstrate both theoretically and through numerical results that the normal gradient stabilized variant results in a high order scheme.

  • 30. Hansbo, Peter
    et al.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zahedi, Sara
    A cut finite element method for coupled bulk-surface problems on time-dependent domains2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 307, p. 96-116Article in journal (Refereed)
    Abstract [en]

    In this contribution we present a new computational method for coupled bulk-surface problems on time-dependent domains. The method is based on a space-time formulation using discontinuous piecewise linear elements in time and continuous piecewise linear elements in space on a fixed background mesh. The domain is represented using a piecewise linear level set function on the background mesh and a cut finite element method is used to discretize the bulk and surface problems. In the cut finite element method the bilinear forms associated with the weak formulation of the problem are directly evaluated on the bulk domain and the surface defined by the level set, essentially using the restrictions of the piecewise linear functions to the computational domain. In addition a stabilization term is added to stabilize convection as well as the resulting algebraic system that is solved in each time step. We show in numerical examples that the resulting method is accurate and stable and results in well conditioned algebraic systems independent of the position of the interface relative to the background mesh.

  • 31. Hansbo, Peter
    et al.
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zahedi, Sara
    Characteristic cut finite element methods for convection-diffusion problems on time dependent surfaces2015In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 293, p. 431-461Article in journal (Refereed)
    Abstract [en]

    We develop a finite element method for convection-diffusion problems on a given time dependent surface, for instance modeling the evolution of a surfactant. The method is based on a characteristic-Galerkin formulation combined with a piecewise linear cut finite element method in space. The cut finite element method is constructed by embedding the surface in a background grid and then using the restriction to the surface of a finite element space defined on the background grid. The surface is allowed to cut through the background grid in an arbitrary fashion. To ensure stability and well posedness of the resulting algebraic systems of equations, independent of the position of the surface in the background grid, we add a consistent stabilization term. We prove error estimates and present confirming numerical results.

  • 32.
    Jakobsson, Håkan
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A posteriori error analysis of component mode synthesis for the elliptic eigenvalue problem2011In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 200, no 41-44, p. 2840-2847Article in journal (Refereed)
    Abstract [en]

    We develop a posteriori error estimates for the error associated with model reduction of elliptic eigenvalue problems using component mode synthesis (CMS). The estimates reflect to what degree each CMS subspace influence the overall error in the reduced solution. This allows for automatic error control through adaptive algorithms that determine suitable dimensions of each CMS subspace.

  • 33. Johansson, A.
    et al.
    Chaudhry, J. H.
    Carey, V.
    Estep, D.
    Ginting, V.
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Tavener, S.
    Adaptive finite element solution of multiscale PDE-ODE systems2015In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 287, p. 150-171Article in journal (Refereed)
    Abstract [en]

    We consider adaptive finite element methods for a multiscale system consisting of a macroscale model comprising a system of reaction-diffusion partial differential equations coupled to a microscale model comprising a system of nonlinear ordinary differential equations. A motivating example is modeling the electrical activity of the heart taking into account the chemistry inside cells in the heart. Such multiscale models are computationally challenging due to the multiple scales in time and space that are involved. We describe a mathematically consistent approach to couple the microscale and macroscale models based on introducing an intermediate "coupling scale". Since the ordinary differential equations are defined on a much finer spatial scale than the finite element discretization for the partial differential equation, we introduce a Monte Carlo approach to sampling the fine scale ordinary differential equations. We derive goal-oriented a posteriori error estimates for quantities of interest computed from the solution of the multiscale model using adjoint problems and computable residuals. We distinguish the errors in time and space for the partial differential equation and the ordinary differential equations separately and include errors due to the transfer of the solutions between the equations. The estimate also includes terms reflecting the sampling of the microscale model. Based on the accurate error estimates, we devise an adaptive solution method using a "blockwise" approach. The method and estimates are illustrated using a realistic problem.

  • 34. Johansson, August
    et al.
    Kehlet, Benjamin
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Logg, Anders
    Multimesh finite element methods: Solving PDEs on multiple intersecting meshes2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 343, p. 672-689Article in journal (Refereed)
    Abstract [en]

    We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Such multimesh finite element methods are particularly well suited to problems in which the computational domain undergoes large deformations as a result of the relative motion of the separate components of a multi-body system. In the present paper, we formulate the multimesh finite element method for the Poisson equation. Numerical examples demonstrate the optimal order convergence, the numerical robustness of the formulation and implementation in the face of thin intersections and rounding errors, as well as the applicability of the methodology. In the accompanying paper (Johansson et al., 2018), we analyze the proposed method and prove optimal order convergence and stability.

  • 35. Johansson, August
    et al.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Logg, Anders
    Multimesh finite elements with flexible mesh sizes2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 372, article id 113420Article in journal (Refereed)
    Abstract [en]

    We analyze a new framework for expressing finite element methods on arbitrarily many intersecting meshes: multimesh finite element methods. The multimesh finite element method, first presented in Johansson et al. (2019), enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Furthermore, each of these meshes may have its own mesh parameter. In the present paper we study the Poisson equation and show that the proposed formulation is stable without assumptions on the relative sizes of the mesh parameters. In particular, we prove optimal order a priori error estimates as well as optimal order estimates of the condition number. Throughout the analysis, we trace the dependence of the number of intersecting meshes. Numerical examples are included to illustrate the stability of the method. 

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  • 36.
    Jonsson, Tobias
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cut finite element methods for elliptic problems on multipatch parametric surfaces2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 324, p. 366-394Article in journal (Refereed)
    Abstract [en]

    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.

  • 37.
    Jonsson, Tobias
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Graded Parametric CutFEM and CutIGA for Elliptic Boundary Value Problems in Domains with Corners2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 354, p. 331-350Article in journal (Refereed)
    Abstract [en]

    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.

  • 38.
    Jonsson, Tobias
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Larsson, Karl
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hybridized isogeometric method for elliptic problems on CAD surfaces with gaps2023In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 410, article id 116014Article in journal (Refereed)
    Abstract [en]

    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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  • 39.
    Larson, Mats G.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zahedi, Sara
    Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden.
    Conservative cut finite element methods using macroelements2023In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 414, article id 116141Article in journal (Refereed)
    Abstract [en]

    We develop a conservative cut finite element method for an elliptic coupled bulk-interface problem. The method is based on a discontinuous Galerkin framework where stabilization is added in such a way that we retain conservation on macroelements containing one element with a large intersection with the domain and possibly a number of elements with small intersections. We derive error estimates and present confirming numerical results.

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  • 40.
    Larsson, Karl
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Kollmannsberger, Stefan
    Chair of Computational Modeling and Simulation, TUM, Germany.
    Rank, Ernst
    Chair of Computational Modeling and Simulation, TUM, Germany.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The finite cell method with least squares stabilized Nitsche boundary conditions2022In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 393, article id 114792Article in journal (Refereed)
    Abstract [en]

    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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  • 41.
    Massing, Andre
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Schott, B.
    Wall, W. A.
    A stabilized Nitsche cut finite element method for the Oseen problem2018In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 262-300Article in journal (Refereed)
    Abstract [en]

    We provide the numerical analysis for a Nitsche-based cut finite element formulation for the Oseen problem, which has been originally presented for the incompressible Navier-Stokes equations by Schott and Wall (2014) and allows the boundary of the domain to cut through the elements of an easy-to-generate background mesh. The formulation is based on the continuous interior penalty (CIP) method of Burman et al. (2006) which penalizes jumps of velocity and pressure gradients over inter-element faces to counteract instabilities arising for high local Reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure. Since the mesh does not fit the boundary, Dirichlet boundary conditions are imposed weakly by a stabilized Nitsche-type approach. The addition of CIP-like ghost-penalties in the boundary zone allows to prove that our method is inf-sup stable and to derive optimal order a priori error estimates in an energy-type norm, irrespective of how the boundary cuts the underlying mesh. All applied stabilization techniques are developed with particular emphasis on low and high Reynolds numbers. Two-and three-dimensional numerical examples corroborate the theoretical findings. Finally, the proposed method is applied to solve the transient incompressible Navier-Stokes equations on a complex geometry. 

  • 42.
    Nguyen, Quoc Khanh
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Serra-Capizzano, Stefano
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    On using a zero lower bound on the physical density in material distribution topology optimization2020In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 359, article id 112669Article in journal (Refereed)
    Abstract [en]

    The current paper studies the possibility of allowing a zero lower bound on the physical density in material distribution based topology optimization. We limit our attention to the standard test problem of minimizing the compliance of a linearly elastic structure subject to a constant forcing. First order tensor product Finite Elements discretize the problem. An elementwise constant material indicator function defines the discretized, elementwise constant, physical density by using filtering and penalization. To alleviate the ill-conditioning of the stiffness matrix, due to the variation of the elementwise constant physical density, we precondition the system. We provide a specific spectral analysis for large matrix sizes for the one-dimensional problem with Dirichlet-Neumann conditions in detail, even if most of the mathematical tools apply also in a d-dimensional setting, d >= 2. It is easy to find an elementwise constant material indicator function so that the resulting preconditioned system matrix is singular when allowing the vanishing physical densities. However, for a large class of material indicator functions, the corresponding preconditioned system matrix has a condition number of the same order as the system matrix for the case when the physical density is one in all elements. Finally, we critically report and illustrate results from numerical experiments: as a conclusion, it is indeed possible to solve large-scale topology optimization problems, allowing a vanishing physical density, without encountering ill-conditioned system matrices during the optimization.

  • 43.
    Nobis, Harrison
    et al.
    FLOW Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Royal Institute of Technology, Stockholm, Sweden.
    Schlatter, Philipp
    FLOW Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Royal Institute of Technology, Stockholm, Sweden.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden.
    Berggren, Martin
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Henningson, Dan S.
    FLOW Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Royal Institute of Technology, Stockholm, Sweden.
    Modal laminar–turbulent transition delay by means of topology optimization of superhydrophobic surfaces2023In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 403, article id 115721Article in journal (Refereed)
    Abstract [en]

    When submerged under a liquid, the microstructure of a SuperHydrophobic Surface (SHS) traps a lubricating layer of gas pockets, which has been seen to reduce the skin friction of the overlying liquid flow in both laminar and turbulent regimes. More recently, spatially homogeneous SHS have also been shown to delay laminar–turbulent transition in channel flows, where transition is triggered by modal mechanisms. In this study, we investigate, by means of topology optimization, whether a spatially inhomogeneous SHS can be designed to further delay transition in channel flows. Unsteady direct numerical simulations are conducted using the spectral element method in a 3D periodic wall-bounded channel. The effect of the SHS is modelled using a partial slip length on the walls, forming a 2D periodic optimization domain. Following a density-based approach, the optimization procedure uses the adjoint-variable method to compute gradients and a checkpointing strategy to reduce storage requirements. This methodology is adapted to optimizing over an ensemble of initial perturbations. This study presents the first application of topology optimization to laminar–turbulent transition. We show that this method can design surfaces that delay transition significantly compared to a homogeneous counterpart, by inhibiting the growth of secondary instability modes. By optimizing over an ensemble of streamwise phase-shifted perturbations, designs have been found with comparable mean transition time and lower variance.

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  • 44. Odsaeter, Lars H.
    et al.
    Wheeler, Mary F.
    Kvamsdal, Trond
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Postprocessing of non-conservative flux for compatibility with transport in heterogeneous media2017In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 315, p. 799-830Article in journal (Refereed)
    Abstract [en]

    A conservative flux postprocessing algorithm is presented for both steady-state and dynamic flow models. The postprocessed flux is shown to have the same convergence order as the original flux. An arbitrary flux approximation is projected into a conservative subspace by adding a piecewise constant correction that is minimized in a weighted L-2 norm. The application of a weighted norm appears to yield better results for heterogeneous media than the standard L-2 norm which has been considered in earlier works. We also study the effect of different flux calculations on the domain boundary. In particular we consider the continuous Galerkin finite element method for solving Darcy flow and couple it with a discontinuous Galerkin finite element method for an advective transport problem.

  • 45. Odsæter, Lars H.
    et al.
    Kvamsdal, Trond
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A simple embedded discrete fracture-matrix model for a coupled flow and transport problem in porous media2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 343, p. 572-601Article in journal (Refereed)
    Abstract [en]

    Accurate simulation of fluid flow and transport in fractured porous media is a key challenge in subsurface reservoir engineering. Due to the high ratio between its length and width, fractures can be modeled as lower dimensional interfaces embedded in the porous rock. We apply a recently developed embedded finite element method (EFEM) for the Darcy problem. This method allows for general fracture geometry, and the fractures may cut the finite element mesh arbitrarily. We present here a velocity model for EFEM and couple the Darcy problem to a transport problem for a passive solute. The main novelties of this work are a locally conservative velocity approximation derived from the EFEM solution, and the development of a lowest order upwind finite volume method for the transport problem. This numerical model is compatible with EFEM in the sense that the same computational mesh may be applied, so that we retain the same flexibility with respect to fracture geometry and meshing. Hence, our coupled solution strategy represents a simple approach in terms of formulation, implementation and meshing. We demonstrate our model by some numerical examples on both synthetic and realistic problems, including a benchmark study for single-phase flow. Despite the simplicity of the method, the results are promising.

  • 46. Selim, K
    et al.
    Logg, A
    Larson, Mats G
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    An adaptive finite element splitting method for the incompressible Navier–Stokes equations2012In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 209-212, p. 54-65Article in journal (Refereed)
    Abstract [en]

    We present an adaptive finite element method for the incompressible Navier–Stokes equations based on a standard splitting scheme (the incremental pressure correction scheme). The presented method combines the efficiency and simplicity of a splitting method with the powerful framework offered by the finite element method for error analysis and adaptivity. An a posteriori error estimate is derived which expresses the error in a goal functional of interest as a sum of contributions from spatial discretization, time discretization and a term that measures the deviation of the splitting scheme from a pure Galerkin scheme (the computational error). Numerical examples are presented which demonstrate the performance of the adaptive algorithm and high quality efficiency indices. It is further demonstrated that the computational error of the Navier–Stokes momentum equation is linear in the size of the time step while the computational error of the continuity equation is quadratic in the size of the time step.

  • 47.
    Stoter, Stein K.F.
    et al.
    Department of Mechanical Engineering, Eindhoven University of Technology, Netherlands.
    Divi, Sai C.
    Department of Mechanical Engineering, Eindhoven University of Technology, Netherlands.
    van Brummelen, E. Harald
    Department of Mechanical Engineering, Eindhoven University of Technology, Netherlands.
    Larson, Mats G.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    de Prenter, Frits
    Department of Aerospace Engineering, Delft University of Technology, Netherlands.
    Verhoosel, Clemens V.
    Department of Mechanical Engineering, Eindhoven University of Technology, Netherlands.
    Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics2023In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 412, article id 116074Article in journal (Refereed)
    Abstract [en]

    In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric explicit dynamics context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche's method. The conventional stability requirement for Nitsche's method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of ‘mass-scaling’ does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff–Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size.

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  • 48.
    Ulfsby, Tale Bakken
    et al.
    Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway.
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway.
    Sticko, Simon
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Information Technology, Uppsala University, Box 337, Uppsala, Sweden.
    Stabilized cut discontinuous Galerkin methods for advection–reaction problems on surfaces2023In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 413, article id 116109Article in journal (Refereed)
    Abstract [en]

    We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection–reaction problems on surfaces embedded in Rd. The CutDG method is based on embedding the surface into a full-dimensional background mesh and using the associated discontinuous piecewise polynomials of order k as test and trial functions. As the surface can cut through the mesh in an arbitrary fashion, we design a suitable stabilization that enables us to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. The resulting CutDG formulation is geometrically robust in the sense that all derived theoretical results hold with constants independent of any particular cut configuration. Numerical examples support our theoretical findings.

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  • 49.
    Wadbro, Eddie
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Topology Optimization of an Acoustic Horn2006In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 196, p. 420-436Article in journal (Refereed)
  • 50.
    Wadbro, Eddie
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Engström, Christian
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Topology and shape optimization of plasmonic nano-antennas2015In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 293, p. 155-169Article in journal (Refereed)
    Abstract [en]

    Metallic nano-antennas are devices used to concentrate the energy in light into regions that are much smaller than the wavelength. These structures are currently used to develop new measurement and printing techniques, such as optical microscopy with sub-wavelength resolution, and high-resolution lithography. Here, we analyze and design a nano-antenna in a two-dimensional setting with the source being a planar TE-polarized wave. The design problem is to place silver and air in a pre-specified design region to maximize the electric energy in a small given target region. At optical frequencies silver exhibits extreme dielectric properties, having permittivity with a negative real part. We prove existence and uniqueness of solutions to the governing nonstandard Helmholtz equation with absorbing boundary conditions. To solve the design optimization problem, we develop a two-stage procedure. The first stage uses a material distribution parameterization and aims at finding a conceptual design without imposing any a priori information about the number of shapes of components comprising the nano-antenna. The second design stage uses a domain variation approach and aims at finding a precise shape. Both of the above design problems are formulated as non-linear mathematical programming problems that are solved using the method of moving asymptotes. The final designs perform very well and the electric energy in the target region is several orders of magnitude larger than when there is only air in the design region. The performance of the optimized designs is verified with a high order interior penalty method.

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