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Burman, E., Hansbo, P. & Larson, M. G. (2018). A cut finite element method with boundary value correction. Mathematics of Computation, 87(310), 633-657
Open this publication in new window or tab >>A cut finite element method with boundary value correction
2018 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 310, p. 633-657Article in journal (Refereed) Published
Abstract [en]

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

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2018
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-38361 (URN)10.1090/mcom/3240 (DOI)000418689600004 ()2-s2.0-85038942183 (Scopus ID)JTHProduktutvecklingIS (Local ID)JTHProduktutvecklingIS (Archive number)JTHProduktutvecklingIS (OAI)
Funder
Swedish Research Council, 2011-4992Swedish Research Council, 2013-4708
Available from: 2018-01-08 Created: 2018-01-08 Last updated: 2018-01-08Bibliographically approved
Burman, E., Hansbo, P. & Larson, M. G. (2018). A simple approach for finite element simulation of reinforced plates. Finite elements in analysis and design (Print), 142, 51-60
Open this publication in new window or tab >>A simple approach for finite element simulation of reinforced plates
2018 (English)In: Finite elements in analysis and design (Print), ISSN 0168-874X, E-ISSN 1872-6925, Vol. 142, p. 51-60Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Cut finite element method, Discontinuous Galerkin, Euler–Bernoulli beam, Kirchhoff–Love plate, Reinforced plate, Galerkin methods, Plates (structural components), Reinforcement, Bernoulli, Finite element simulations, Finite element space, Kirchhoff, Optimization loop, Piecewise polynomials, Finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-38885 (URN)10.1016/j.finel.2018.01.001 (DOI)000424941600005 ()2-s2.0-85041447903 (Scopus ID)JTHProduktutvecklingIS (Local ID)JTHProduktutvecklingIS (Archive number)JTHProduktutvecklingIS (OAI)
Available from: 2018-02-20 Created: 2018-02-20 Last updated: 2018-03-02Bibliographically approved
Burman, E., Hansbo, P. & Larson, M. G. (2018). Augmented Lagrangian and Galerkin least-squares methods for membrane contact. International Journal for Numerical Methods in Engineering, 114(11), 1179-1191
Open this publication in new window or tab >>Augmented Lagrangian and Galerkin least-squares methods for membrane contact
2018 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 114, no 11, p. 1179-1191Article in journal (Refereed) Published
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. 

Place, publisher, year, edition, pages
John Wiley & Sons, 2018
Keywords
Augmented Lagrangian, Contact, Galerkin least squares, Lagrange multiplier, Membrane, Stabilization, Constrained optimization, Contacts (fluid mechanics), Finite element method, Galerkin methods, Lagrange multipliers, Membranes, Numerical methods, Optimization, Augmented Lagrangians, Elastic membranes, Numerical solution, Small deformations, Stabilized finite element methods, Least squares approximations
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-39018 (URN)10.1002/nme.5781 (DOI)000433574800002 ()2-s2.0-85042443902 (Scopus ID)JTHProduktutvecklingIS (Local ID)JTHProduktutvecklingIS (Archive number)JTHProduktutvecklingIS (OAI)
Available from: 2018-03-20 Created: 2018-03-20 Last updated: 2018-07-06Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G. & Larsson, K. (2018). Cut finite elements for convection in fractured domains. Computers & Fluids
Open this publication in new window or tab >>Cut finite elements for convection in fractured domains
2018 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747Article in journal (Refereed) Epub ahead of print
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.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
A priori error estimates, Convection problems, Fractured domains, Galerkin least squares, Mixed-dimensional domains, Fracture, Mesh generation, Piecewise linear techniques, Porous materials, Convection problem, Coupling condition, Directional derivative, Divergence operators, Modeling porous medias, Piecewise linear, Priori error estimate, Finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-41509 (URN)10.1016/j.compfluid.2018.07.022 (DOI)2-s2.0-85052134188 (Scopus ID)
Available from: 2018-09-19 Created: 2018-09-19 Last updated: 2018-09-19
Burman, E., Hansbo, P., Larson, M. G., Larsson, K. & Massing, A. (2018). Finite element approximation of the Laplace–Beltrami operator on a surface with boundary. Numerische Mathematik
Open this publication in new window or tab >>Finite element approximation of the Laplace–Beltrami operator on a surface with boundary
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2018 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245Article in journal (Refereed) Epub ahead of print
Abstract [en]

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

Place, publisher, year, edition, pages
Springer, 2018
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-41277 (URN)10.1007/s00211-018-0990-2 (DOI)XYZ ()2-s2.0-85049887016 (Scopus ID)JTHProduktutvecklingIS (Local ID)JTHProduktutvecklingIS (Archive number)JTHProduktutvecklingIS (OAI)
Available from: 2018-08-28 Created: 2018-08-28 Last updated: 2018-08-28
Burman, E., Elfverson, D., Hansbo, P., Larson, M. & Larsson, K. (2018). Shape optimization using the cut finite element method. Computer Methods in Applied Mechanics and Engineering, 328, 242-261
Open this publication in new window or tab >>Shape optimization using the cut finite element method
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2018 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 242-261Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
CutFEM, Shape optimization, Level-set, Fictitious domain method, Linear elasticity
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-37559 (URN)10.1016/j.cma.2017.09.005 (DOI)000416218500011 ()2-s2.0-85030311574 (Scopus ID)JTHProduktuvecklingIS (Local ID)JTHProduktuvecklingIS (Archive number)JTHProduktuvecklingIS (OAI)
Funder
Swedish Research Council, 2011-4992; 2013-4708
Available from: 2017-10-04 Created: 2017-10-05 Last updated: 2018-08-17Bibliographically approved
Burman, E., Hansbo, P. & Larson, M. G. (2018). Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems, 34(3), Article ID 035004.
Open this publication in new window or tab >>Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization
2018 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, no 3, article id 035004Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2018
Keywords
optimal control problem, data assimilation, source identification, finite elements, regularization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-38742 (URN)10.1088/1361-6420/aaa32b (DOI)000423854300001 ()2-s2.0-85042306705 (Scopus ID)JTHProduktutvecklingIS (Local ID)JTHProduktutvecklingIS (Archive number)JTHProduktutvecklingIS (OAI)
Available from: 2018-02-05 Created: 2018-02-05 Last updated: 2018-03-20Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G. & Zahedi, S. (2018). Stabilized CutFEM for the convection problem on surfaces. Numerische Mathematik, 1-37
Open this publication in new window or tab >>Stabilized CutFEM for the convection problem on surfaces
2018 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, p. 1-37Article in journal (Refereed) Epub ahead of print
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. 

Place, publisher, year, edition, pages
Springer, 2018
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-41510 (URN)10.1007/s00211-018-0989-8 (DOI)2-s2.0-85052521788 (Scopus ID)
Available from: 2018-09-19 Created: 2018-09-19 Last updated: 2018-09-19
Burman, E. & Hansbo, P. (2018). Stabilized nonconforming finite element methods for data assimilation in incompressible flows. Mathematics of Computation, 87(311), 1029-1050
Open this publication in new window or tab >>Stabilized nonconforming finite element methods for data assimilation in incompressible flows
2018 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 87, no 311, p. 1029-1050Article in journal (Refereed) Published
Abstract [en]

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

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2018
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-38908 (URN)10.1090/mcom/3255 (DOI)000425720100001 ()2-s2.0-85042372295 (Scopus ID)JTHProduktutvecklingIS (Local ID)JTHProduktutvecklingIS (Archive number)JTHProduktutvecklingIS (OAI)
Available from: 2018-02-23 Created: 2018-02-23 Last updated: 2018-03-20Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G. & Massing, A. (2017). A cut discontinuous Galerkin method for the Laplace–Beltrami operator. IMA Journal of Numerical Analysis, 37(1), 138-169
Open this publication in new window or tab >>A cut discontinuous Galerkin method for the Laplace–Beltrami operator
2017 (English)In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 37, no 1, p. 138-169Article in journal (Refereed) Published
Abstract [en]

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

Place, publisher, year, edition, pages
Oxford University Press, 2017
Keywords
Surface PDE, Laplace–Beltrami, discontinuous Galerkin, Cut finite element method
National Category
Computational Mathematics Mathematical Analysis
Identifiers
urn:nbn:se:hj:diva-34834 (URN)10.1093/imanum/drv068 (DOI)000397147700005 ()2-s2.0-85020561306 (Scopus ID)JTHProduktutvecklingIS (Local ID)JTHProduktutvecklingIS (Archive number)JTHProduktutvecklingIS (OAI)
Funder
Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 2011-4992; 2013-4708eSSENCE - An eScience Collaboration
Available from: 2017-01-23 Created: 2017-01-23 Last updated: 2018-10-16Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-7352-1550

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