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Burman, E., Hansbo, P., Larson, M. G. & Samvin, D. (2019). A cut finite element method for elliptic bulk problems with embedded surfaces. GEM - International Journal on Geomathematics, 10(1), Article ID 10.
Open this publication in new window or tab >>A cut finite element method for elliptic bulk problems with embedded surfaces
2019 (English)In: GEM - International Journal on Geomathematics, ISSN 1869-2672, E-ISSN 1869-2680, Vol. 10, no 1, article id 10Article in journal (Refereed) Published
Abstract [en]

We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractures. In particular the Laplace–Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples. 

Place, publisher, year, edition, pages
Springer, 2019
Keywords
Embedded, Finite element, Fractures, Unfitted, Fracture, Porous materials, Domain geometry, Embedded surfaces, Fractured porous media, Normal component, Optimal order error estimates, Regularity assumption, Finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-43354 (URN)10.1007/s13137-019-0120-z (DOI)30873244 (PubMedID)2-s2.0-85061086676 (Scopus ID)HOA JTH 2019;JTHMaterialIS (Local ID)HOA JTH 2019;JTHMaterialIS (Archive number)HOA JTH 2019;JTHMaterialIS (OAI)
Available from: 2019-03-19 Created: 2019-03-19 Last updated: 2019-03-21Bibliographically approved
Burman, E. N., Hansbo, P. & Larson, M. G. (2019). A cut finite element method with boundary value correction for the incompressible Stokes equations. In: Numerical mathematics and advanced applications ENUMATH 2017: . Paper presented at European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017, Voss, Norway, 25 - 29 September 2017 (pp. 183-192). Cham: Springer, 126
Open this publication in new window or tab >>A cut finite element method with boundary value correction for the incompressible Stokes equations
2019 (English)In: Numerical mathematics and advanced applications ENUMATH 2017, Cham: Springer, 2019, Vol. 126, p. 183-192Conference paper, Published paper (Refereed)
Abstract [en]

We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche’s method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman et al. (Math Comput 87(310):633–657, 2018). 

Place, publisher, year, edition, pages
Cham: Springer, 2019
Series
Lecture Notes in Computational Science and Engineering, ISSN 14397358 ; 126
Keywords
Boundary conditions, Navier Stokes equations, Computational mesh, Correction techniques, Curved boundary, Dirichlet boundary condition, Discrete boundaries, Discrete domains, Incompressible Stokes equation, Piecewise affines, Finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-42799 (URN)10.1007/978-3-319-96415-7_15 (DOI)2-s2.0-85060026956 (Scopus ID)9783319964140 (ISBN)9783319964157 (ISBN)
Conference
European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017, Voss, Norway, 25 - 29 September 2017
Available from: 2019-02-01 Created: 2019-02-01 Last updated: 2019-02-15Bibliographically approved
Burman, E., Hansbo, P. & Larson, M. G. (2019). A simple finite element method for elliptic bulk problems with embedded surfaces. Computational Geosciences, 23(1), 189-199
Open this publication in new window or tab >>A simple finite element method for elliptic bulk problems with embedded surfaces
2019 (English)In: Computational Geosciences, ISSN 1420-0597, E-ISSN 1573-1499, Vol. 23, no 1, p. 189-199Article in journal (Refereed) Published
Abstract [en]

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

Place, publisher, year, edition, pages
Springer, 2019
Keywords
Cut finite element methods, Darcy equation, Embedded layer, Fracture
National Category
Computer Engineering
Identifiers
urn:nbn:se:hj:diva-42144 (URN)10.1007/s10596-018-9792-y (DOI)000459423400010 ()2-s2.0-85056316370 (Scopus ID)HOA JTH 2019;JTHProduktutvecklingIS (Local ID)HOA JTH 2019;JTHProduktutvecklingIS (Archive number)HOA JTH 2019;JTHProduktutvecklingIS (OAI)
Available from: 2018-11-26 Created: 2018-11-26 Last updated: 2019-03-14Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G. & Massing, A. (2019). Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. Mathematical Modelling and Numerical Analysis, 52(6), 2247-2282
Open this publication in new window or tab >>Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions
2019 (English)In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 52, no 6, p. 2247-2282Article in journal (Refereed) Published
Abstract [en]

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in Rd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in R3.

Place, publisher, year, edition, pages
EDP Sciences, 2019
Keywords
A priori error estimates, Arbitrary codimension, Condition number, Cut finite element method, Laplace-Beltrami operator, Stabilization, Surface PDE, Embeddings, Laplace transforms, Mesh generation, Number theory, Piecewise linear techniques, Stiffness matrix, A-priori estimates, Codimension, Condition number of the stiffness matrix, Condition numbers, Numerical results, Priori error estimate, Theoretical framework, Finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-43224 (URN)10.1051/m2an/2018038 (DOI)000457984700005 ()2-s2.0-85052126397 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
Available from: 2019-03-01 Created: 2019-03-01 Last updated: 2019-03-01Bibliographically approved
Burman, E., Elfverson, D., Hansbo, P., Larson, M. G. & Larsson, K. (2019). Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions. Computer Methods in Applied Mechanics and Engineering, 350, 462-479
Open this publication in new window or tab >>Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions
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2019 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 462-479Article in journal (Refereed) Published
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 represent parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche's method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero. 

Place, publisher, year, edition, pages
Elsevier, 2019
Keywords
Cut finite element method, Design and nondesign domain regions, Material distribution topology optimization, Elasticity, Mesh generation, Stabilization, Topology, Complicated geometry, Density parameters, Dirichlet condition, Interface conditions, Isogeometric analysis, Linear elasticity, Material distribution, Topology Optimization Method, Finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-43449 (URN)10.1016/j.cma.2019.03.016 (DOI)2-s2.0-85063489165 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
Available from: 2019-04-10 Created: 2019-04-10 Last updated: 2019-04-11Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G., Larsson, K. & Massing, A. (2019). Finite element approximation of the Laplace–Beltrami operator on a surface with boundary. Numerische Mathematik, 141(1), 141-172
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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2019 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, no 1, p. 141-172Article in journal (Refereed) Published
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, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-41277 (URN)10.1007/s00211-018-0990-2 (DOI)000457025700005 ()2-s2.0-85049887016 (Scopus ID)HOA JTH 2019; JTHMaterialIS (Local ID)HOA JTH 2019; JTHMaterialIS (Archive number)HOA JTH 2019; JTHMaterialIS (OAI)
Available from: 2018-08-28 Created: 2018-08-28 Last updated: 2019-02-20Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G. & Zahedi, S. (2019). Stabilized CutFEM for the convection problem on surfaces. Numerische Mathematik, 141(1), 103-139
Open this publication in new window or tab >>Stabilized CutFEM for the convection problem on surfaces
2019 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, no 1, p. 103-139Article in journal (Refereed) Published
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, 2019
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-41510 (URN)10.1007/s00211-018-0989-8 (DOI)000457025700004 ()2-s2.0-85052521788 (Scopus ID)HOA JTH 2019;JTHMaterialIS (Local ID)HOA JTH 2019;JTHMaterialIS (Archive number)HOA JTH 2019;JTHMaterialIS (OAI)
Funder
Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 2011-4992Swedish Research Council, 2013-4708Swedish Research Council, 2014-4804eSSENCE - An eScience Collaboration
Available from: 2018-09-19 Created: 2018-09-19 Last updated: 2019-02-20Bibliographically approved
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)JTHMaterialIS (Local ID)JTHMaterialIS (Archive number)JTHMaterialIS (OAI)
Funder
Swedish Research Council, 2011-4992Swedish Research Council, 2013-4708
Available from: 2018-01-08 Created: 2018-01-08 Last updated: 2019-02-15Bibliographically 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)JTHMaterialIS (Local ID)JTHMaterialIS (Archive number)JTHMaterialIS (OAI)
Funder
Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 2013-4708eSSENCE - An eScience Collaboration
Available from: 2018-02-20 Created: 2018-02-20 Last updated: 2019-03-25Bibliographically 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)JTHMaterialIS (Local ID)JTHMaterialIS (Archive number)JTHMaterialIS (OAI)
Available from: 2018-03-20 Created: 2018-03-20 Last updated: 2019-02-15Bibliographically approved
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ORCID iD: ORCID iD iconorcid.org/0000-0001-7352-1550

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