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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
Keyword
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, 1-13
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, p. 1-13Article in journal (Refereed) Epub ahead of print
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
Keyword
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)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: 2018-03-20
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
Keyword
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: 2018-01-19Bibliographically 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
Keyword
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. (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
Keyword
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 ()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: 2017-04-19Bibliographically approved
Burman, E., Elfverson, D., Hansbo, P., Larson, M. G. & Larsson, K. (2017). A cut finite element method for the Bernoulli free boundary value problem. Computer Methods in Applied Mechanics and Engineering, 317, 598-618
Open this publication in new window or tab >>A cut finite element method for the Bernoulli free boundary value problem
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2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 317, p. 598-618Article in journal (Refereed) Published
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 H1 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 H1norm. Finally, we present illustrating numerical results.

Place, publisher, year, edition, pages
Elsevier, 2017
Keyword
Free boundary value problem, CutFEM, Shape optimization, Level set, Fictitious domain method
National Category
Computational Mathematics Mathematical Analysis
Identifiers
urn:nbn:se:hj:diva-34776 (URN)10.1016/j.cma.2016.12.021 (DOI)000398373500024 ()2-s2.0-85009476549 (Scopus ID)JTHProduktutvecklingIS (Local ID)JTHProduktutvecklingIS (Archive number)JTHProduktutvecklingIS (OAI)
Available from: 2017-01-18 Created: 2017-01-18 Last updated: 2017-04-19Bibliographically approved
Hansbo, P., Jonsson, T., Larson, M. G. & Larsson, K. (2017). A Nitsche method for elliptic problems on composite surfaces. Computer Methods in Applied Mechanics and Engineering, 326, 505-525
Open this publication in new window or tab >>A Nitsche method for elliptic problems on composite surfaces
2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 326, p. 505-525Article in journal (Refereed) Published
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. 

Place, publisher, year, edition, pages
Elsevier, 2017
Keyword
A priori error estimates, Composite surfaces, Laplace–Beltrami operator, Nitsche method, Inverse problems, Partial differential equations, Approximation properties, Beltrami, Composite surface, Continuity requirements, Elliptic partial differential equation, Finite element space, Priori error estimate, Finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-37566 (URN)10.1016/j.cma.2017.08.033 (DOI)000413322300022 ()2-s2.0-85029527302 (Scopus ID)
Available from: 2017-10-05 Created: 2017-10-05 Last updated: 2017-12-21Bibliographically approved
Hansbo, P. & Larson, M. G. (2017). A stabilized finite element method for the Darcy problem on surfaces. IMA Journal of Numerical Analysis, 37(3), 1274-1299
Open this publication in new window or tab >>A stabilized finite element method for the Darcy problem on surfaces
2017 (English)In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 37, no 3, p. 1274-1299Article in journal (Refereed) Published
Abstract [en]

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

Place, publisher, year, edition, pages
Oxford University Press, 2017
Keyword
Darcy problem, tangential differential calculus, surface differential equation, stabilized finite element method
National Category
Computational Mathematics Mathematical Analysis
Identifiers
urn:nbn:se:hj:diva-36717 (URN)10.1093/imanum/drw041 (DOI)000405416900008 ()2-s2.0-85021838465 (Scopus ID)
Available from: 2017-07-13 Created: 2017-07-13 Last updated: 2017-09-13Bibliographically approved
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ORCID iD: ORCID iD iconorcid.org/0000-0001-7352-1550

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