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Massing, A., Schott, B. & Wall, W. A. (2018). A stabilized Nitsche cut finite element method for the Oseen problem. Computer Methods in Applied Mechanics and Engineering, 328, 262-300
Open this publication in new window or tab >>A stabilized Nitsche cut finite element method for the Oseen problem
2018 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 262-300Article in journal (Refereed) Published
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. 

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Oseen problem, Fictitious domain method, Cut finite elements, Nitsche's method, Continuous interior penalty stabilization, Navier-Stokes equations
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-152143 (URN)10.1016/j.cma.2017.09.003 (DOI)000416218500012 ()
Funder
Swedish Research Council, 2013-4708
Available from: 2018-10-01 Created: 2018-10-01 Last updated: 2018-10-01Bibliographically 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 Rd . The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in Rd . The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then, a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighbouring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.

Keywords
surface PDE, Laplace–Beltrami, discontinuous Galerkin, cut finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-130973 (URN)10.1093/imanum/drv068 (DOI)000397147700005 ()
Available from: 2017-02-01 Created: 2017-02-01 Last updated: 2018-06-09Bibliographically approved
Hansbo, P., Larson, M. G. & Massing, A. (2017). A stabilized cut finite element method for the Darcy problem on surfaces. Computer Methods in Applied Mechanics and Engineering, 326, 298-318
Open this publication in new window or tab >>A stabilized cut finite element method for the Darcy problem on surfaces
2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 326, p. 298-318Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Surface PDE, Darcy problem, Cut finite element method, Stabilization, Condition number, A priori error estimates
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-141801 (URN)10.1016/j.cma.2017.08.007 (DOI)000413322300013 ()
Available from: 2017-11-29 Created: 2017-11-29 Last updated: 2018-06-09Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G., Massing, A. & Zahedi, S. (2016). Full gradient stabilized cut finite element methods for surface partial differential equations. Computer Methods in Applied Mechanics and Engineering, 310, 278-296
Open this publication in new window or tab >>Full gradient stabilized cut finite element methods for surface partial differential equations
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2016 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 278-296Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2016
Keywords
Surface PDE, Laplace-Beltrami operator, Cut finite element method, Stabilization, Condition number, A priori error estimates
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-127240 (URN)10.1016/j.cma.2016.06.033 (DOI)000384859400014 ()
Available from: 2016-11-14 Created: 2016-11-03 Last updated: 2018-06-09Bibliographically approved
Massing, A., Larson, M. G., Logg, A. & Rognes, M. E. (2015). A Nitsche-based cut finite element method for a fluid-structure interaction problem. Communications in Applied Mathematics and Computational Science, 10(2), 97-120
Open this publication in new window or tab >>A Nitsche-based cut finite element method for a fluid-structure interaction problem
2015 (English)In: Communications in Applied Mathematics and Computational Science, ISSN 1559-3940, E-ISSN 2157-5452, Vol. 10, no 2, p. 97-120Article in journal (Refereed) Published
Abstract [en]

We present a new composite mesh finite element method for fluid-structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh that is embedded into a fixed background fluid mesh. The embedding allows for an arbitrary overlap of the fluid meshes. The coupling between the embedded and background fluid meshes is enforced using a stabilized Nitsche formulation that allows us to establish stability and optimal-order a priori error estimates. We consider here a steady state fluid-structure interaction problem where a hyperelastic structure interacts with a viscous fluid modeled by the Stokes equations. We evaluate an iterative solution procedure based on splitting and present three-dimensional numerical examples.

Keywords
fluid-structure interaction, overlapping meshes, cut finite element method, embedded meshes, abilized finite element methods, Nitsche's method
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-110601 (URN)10.2140/camcos.2015.10.97 (DOI)000362097900001 ()
Available from: 2015-10-27 Created: 2015-10-23 Last updated: 2018-06-07Bibliographically approved
Burman, E., Claus, S. & Massing, A. (2015). A Stabilized Cut Finite Element Method for the Three Field Stokes Problem. SIAM Journal on Scientific Computing, 37(4), A1705-A1726
Open this publication in new window or tab >>A Stabilized Cut Finite Element Method for the Three Field Stokes Problem
2015 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 37, no 4, p. A1705-A1726Article in journal (Refereed) Published
Abstract [en]

We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh. The dependent variables of velocity, pressure, and extra-stress tensor are discretized on the background mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings numerically.

Keywords
three field Stokes, continuous interior penalty, fictitious domain, cut finite element method, ghost penalty, Nitsche’s method, viscoelasticity
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-130970 (URN)10.1137/140983574 (DOI)000360698000004 ()
Available from: 2017-02-01 Created: 2017-02-01 Last updated: 2018-06-09
Claus, S., Burman, E. & Massing, A. (2015). CutFEM: a stabilised Nitsche XFEM method for multi-physics problems. In: Antonio J. Gil, Rubén Sevilla (Ed.), Proceedings of the 23rd Conference on Computational Mechanics: . Paper presented at 23rd Conference on Computational Mechanics, ACME 2015, 8-10 April 2015, Swansea, Wales, UK (pp. 171-174). Swansea: Swansea University
Open this publication in new window or tab >>CutFEM: a stabilised Nitsche XFEM method for multi-physics problems
2015 (English)In: Proceedings of the 23rd Conference on Computational Mechanics / [ed] Antonio J. Gil, Rubén Sevilla, Swansea: Swansea University , 2015, p. 171-174Conference paper, Oral presentation with published abstract (Refereed)
Abstract [en]

In this communication, we will give an overview over CutFEM, a new stabilised XFEM technique. Here, different PDEs are coupled across an interface, that intersects a fixed background mesh in an arbitrary manner. The boundary conditions on the interface are enforced using Nitsche-type coupling conditions [1]. Nitsche’s method offers a flexible approach to design XFEM methods that is amenable to analysis. Classically, XFEM methods suffer from ill-conditioning if the interface intersects elements close to element nodes leaving only small parts of the element covered by the physical domain. In our method, we overcome this major difficulty, by adding ghost-penalty terms to the variational formulation over the band of elements that are cut by the interface [3, 4]. In this contribution, we will illustrate the usage of CutFEM on the three field Stokes problem.

Place, publisher, year, edition, pages
Swansea: Swansea University, 2015
Keywords
Nitsche’s method, XFEM, multi-physics problems, ghost penalty, three field Stokes
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-130971 (URN)9780956746245 (ISBN)
Conference
23rd Conference on Computational Mechanics, ACME 2015, 8-10 April 2015, Swansea, Wales, UK
Available from: 2017-02-01 Created: 2017-02-01 Last updated: 2018-06-09Bibliographically approved
Burman, E., Claus, S., Hansbo, P., Larson, M. G. & Massing, A. (2015). CutFEM: Discretizing geometry and partial differential equations. International Journal for Numerical Methods in Engineering, 104(7), 472-501
Open this publication in new window or tab >>CutFEM: Discretizing geometry and partial differential equations
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2015 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 104, no 7, p. 472-501Article in journal (Refereed) Published
Abstract [en]

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

Keywords
extended finite element method, unfitted methods, finite element methods, meshfree methods, lerkin, level sets, stability
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-110976 (URN)10.1002/nme.4823 (DOI)000362552500002 ()
Note

Special Issue: SI

Available from: 2015-11-18 Created: 2015-11-02 Last updated: 2018-06-07Bibliographically approved
Massing, A., Larson, M. G., Logg, A. & Rognes, M. E. (2014). A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem. Journal of Scientific Computing, 61(3), 604-628
Open this publication in new window or tab >>A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem
2014 (English)In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 61, no 3, p. 604-628Article in journal (Refereed) Published
Abstract [en]

We present a novel finite element method for the Stokes problem on fictitious domains. We prove inf-sup stability, optimal order convergence and uniform boundedness of the condition number of the discrete system. The finite element formulation is based on a stabilized Nitsche method with ghost penalties for the velocity and pressure to obtain stability in the presence of small cut elements. We demonstrate for the first time the applicability of the Nitsche fictitious domain method to three-dimensional Stokes problems. We further discuss a general, flexible and freely available implementation of the method and present numerical examples supporting the theoretical results.

Keywords
Fictitious domain, Stokes problem, Stabilized finite element methods, Nitsche's method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-96797 (URN)10.1007/s10915-014-9838-9 (DOI)000343821300007 ()
Available from: 2014-12-11 Created: 2014-12-03 Last updated: 2018-06-07Bibliographically approved
Massing, A., Larson, M. G., Logg, A. & Rognes, M. E. (2014). A stabilized Nitsche overlapping mesh method for the Stokes problem. Numerische Mathematik, 128(1), 73-101
Open this publication in new window or tab >>A stabilized Nitsche overlapping mesh method for the Stokes problem
2014 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, no 1, p. 73-101Article in journal (Refereed) Published
Abstract [en]

We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By extending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results.

National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-93478 (URN)10.1007/s00211-013-0603-z (DOI)000340590200003 ()
Available from: 2014-11-28 Created: 2014-09-23 Last updated: 2018-06-07Bibliographically approved
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-0803-9041

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