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Publications (10 of 152) Show all publications
Burman, E., Hansbo, P. & Larson, M. G. (2024). Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions. Mathematics of Computation, 93(345), 35-54
Open this publication in new window or tab >>Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions
2024 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 93, no 345, p. 35-54Article in journal (Refereed) Published
Abstract [en]

We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution u & ISIN; Hs with s & ISIN; (1, 3/2]. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2024
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-62377 (URN)10.1090/mcom/3875 (DOI)001047664700001 ()2-s2.0-85174930720 (Scopus ID);intsam;901188 (Local ID);intsam;901188 (Archive number);intsam;901188 (OAI)
Funder
Swedish Research Council, 2017-03911, 2018-05262, 2021-04925
Available from: 2023-09-04 Created: 2023-09-04 Last updated: 2023-11-06Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G. & Larsson, K. (2023). Extension operators for trimmed spline spaces. Computer Methods in Applied Mechanics and Engineering, 403, Article ID 115707.
Open this publication in new window or tab >>Extension operators for trimmed spline spaces
2023 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 403, article id 115707Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
Splines, A-stable, Approximation results, Cut isogeometric method, Discrete extension operator, Extension operators, Piecewise polynomial functions, Spline space, Stability results, Trimmed spline space, Unfitted finite element method, Finite element method, Cut isogeometric methods, Discrete extension operators, Trimmed spline spaces, Unfitted finite element methods
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-58827 (URN)10.1016/j.cma.2022.115707 (DOI)000882526600004 ()2-s2.0-85140922298 (Scopus ID)HOA;intsam;840864 (Local ID)HOA;intsam;840864 (Archive number)HOA;intsam;840864 (OAI)
Funder
eSSENCE - An eScience CollaborationSwedish Research Council, 2017-03911, 2018-05262, 2021-04925
Available from: 2022-11-08 Created: 2022-11-08 Last updated: 2022-12-02Bibliographically approved
Burman, E., Hansbo, P., Larson, M. G. & Larsson, K. (2023). Isogeometric analysis and Augmented Lagrangian Galerkin Least Squares Methods for residual minimization in dual norm. Computer Methods in Applied Mechanics and Engineering, 417(Part B), Article ID 116302.
Open this publication in new window or tab >>Isogeometric analysis and Augmented Lagrangian Galerkin Least Squares Methods for residual minimization in dual norm
2023 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 417, no Part B, article id 116302Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
Dual norm residual minimization, Error estimates, Finite element method, Galerkin Least Squares, Isogeometric analysis, Computation theory, Constrained optimization, Galerkin methods, Lagrange multipliers, Least squares approximations, Augmented Lagrangians, Lagrangian techniques, Least-squares- methods, Nonstandard problems, Residual minimization, Stability theories
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-62510 (URN)10.1016/j.cma.2023.116302 (DOI)001114119400001 ()2-s2.0-85169927833 (Scopus ID);intsam;62510 (Local ID);intsam;62510 (Archive number);intsam;62510 (OAI)
Funder
Swedish Research Council, 2021-04925, 2022-03908eSSENCE - An eScience Collaboration, EP/T033126/1, EP/V050400/1
Available from: 2023-09-20 Created: 2023-09-20 Last updated: 2024-01-09Bibliographically approved
Burman, E., Hansbo, P. & Larson, M. G. (2023). The Augmented Lagrangian Method as a Framework for Stabilised Methods in Computational Mechanics. Archives of Computational Methods in Engineering, 30, 2579-2604
Open this publication in new window or tab >>The Augmented Lagrangian Method as a Framework for Stabilised Methods in Computational Mechanics
2023 (English)In: Archives of Computational Methods in Engineering, ISSN 1134-3060, E-ISSN 1886-1784, Vol. 30, p. 2579-2604Article in journal (Refereed) Published
Abstract [en]

In this paper we will present a review of recent advances in the application of the augmented Lagrange multiplier method as a general approach for generating multiplier-free stabilised methods. The augmented Lagrangian method consists of a standard Lagrange multiplier method augmented by a penalty term, penalising the constraint equations, and is well known as the basis for iterative algorithms for constrained optimisation problems. Its use as a stabilisation methods in computational mechanics has, however, only recently been appreciated. We first show how the method generates Galerkin/Least Squares type schemes for equality constraints and then how it can be extended to develop new stabilised methods for inequality constraints. Application to several different problems in computational mechanics is given.

Place, publisher, year, edition, pages
Springer, 2023
Keywords
Constrained optimization, Iterative methods, Lagrange multipliers, Augmented lagrange multiplier methods, Augmented Lagrangian methods, Constrained optimi-zation problems, Constraint equation, Galerkin Least Squares, Iterative algorithm, Lagrange multiplier method, Penalty term, Stabilization methods, Stabilized method, Computational mechanics
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-59759 (URN)10.1007/s11831-022-09878-6 (DOI)000920400800001 ()2-s2.0-85146572606 (Scopus ID)HOA;intsam;860720 (Local ID)HOA;intsam;860720 (Archive number)HOA;intsam;860720 (OAI)
Funder
eSSENCE - An eScience CollaborationSwedish Research Council, 2017-03911, 2018-05262, 2021-04925
Available from: 2023-02-08 Created: 2023-02-08 Last updated: 2023-06-30Bibliographically approved
Hansbo, P. & Larson, M. G. (2022). A simple nonconforming tetrahedral element for the Stokes equations. Computer Methods in Applied Mechanics and Engineering, 400, Article ID 115549.
Open this publication in new window or tab >>A simple nonconforming tetrahedral element for the Stokes equations
2022 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 400, article id 115549Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2022
Keywords
Degrees of freedom (mechanics), Navier Stokes equations, Piecewise linear techniques, A-stable, Continous approximation, Lower order elements, Nonconforming element, Piecewise linear, Piecewise-linear, Simple++, Stokes equations, Stokes problem, Tetrahedral elements, Finite element method
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-58576 (URN)10.1016/j.cma.2022.115549 (DOI)000862959800004 ()2-s2.0-85138441667 (Scopus ID)HOA;;835275 (Local ID)HOA;;835275 (Archive number)HOA;;835275 (OAI)
Funder
Swedish Research Council, 2017-03911, 2018-05262, 2021-04925eSSENCE - An eScience Collaboration
Available from: 2022-10-03 Created: 2022-10-03 Last updated: 2022-10-21Bibliographically approved
Hansbo, P. & Larson, M. G. (2022). Augmented Lagrangian approach to deriving discontinuous Galerkin methods for nonlinear elasticity problems. International Journal for Numerical Methods in Engineering, 123(18), 4407-4421
Open this publication in new window or tab >>Augmented Lagrangian approach to deriving discontinuous Galerkin methods for nonlinear elasticity problems
2022 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 123, no 18, p. 4407-4421Article in journal (Refereed) Published
Abstract [en]

We use the augmented Lagrangian formalism to derive discontinuous Galerkin (DG) formulations for problems in nonlinear elasticity. In elasticity, stress is typically a symmetric function of strain, leading to symmetric tangent stiffness matrices in Newton's method when conforming finite elements are used for discretization. By use of the augmented Lagrangian framework, we can also obtain symmetric tangent stiffness matrices in DG methods. We suggest two different approaches and give examples from plasticity and from large deformation hyperelasticity.

Place, publisher, year, edition, pages
John Wiley & Sons, 2022
Keywords
antiplane shear plasticity, augmented Lagrangian, discontinuous Galerkin, finite elasticity, Nitsche's method
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-57046 (URN)10.1002/nme.7039 (DOI)000799985600001 ()HOA;;816646 (Local ID)HOA;;816646 (Archive number)HOA;;816646 (OAI)
Funder
Swedish Research Council, 2017-03911, 2018-05262, 2021-04925
Available from: 2022-06-10 Created: 2022-06-10 Last updated: 2022-09-16Bibliographically approved
Burman, E., Hansbo, P. & Larson, M. G. (2022). CutFEM based on extended finite element spaces. Numerische Mathematik, 152, 331-369
Open this publication in new window or tab >>CutFEM based on extended finite element spaces
2022 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 152, p. 331-369Article in journal (Refereed) Published
Abstract [en]

We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More, precisely the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.

Place, publisher, year, edition, pages
Springer, 2022
Keywords
65N30, 65N85
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-58559 (URN)10.1007/s00211-022-01313-z (DOI)000855517700001 ()2-s2.0-85138104101 (Scopus ID)HOA;intsam;834901 (Local ID)HOA;intsam;834901 (Archive number)HOA;intsam;834901 (OAI)
Available from: 2022-09-29 Created: 2022-09-29 Last updated: 2022-12-09Bibliographically approved
Burman, E., Hansbo, P. & Larson, M. G. (2022). Error Estimates for the Smagorinsky Turbulence Model: Enhanced Stability Through Scale Separation and Numerical Stabilization. Journal of Mathematical Fluid Mechanics, 24(1), Article ID 5.
Open this publication in new window or tab >>Error Estimates for the Smagorinsky Turbulence Model: Enhanced Stability Through Scale Separation and Numerical Stabilization
2022 (English)In: Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, E-ISSN 1422-6952, Vol. 24, no 1, article id 5Article in journal (Refereed) Published
Abstract [en]

In the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moderate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the L-2-norm for smooth flows in the pre-asymptotic high Reynolds number regime.

Place, publisher, year, edition, pages
Springer, 2022
Keywords
Navier-Stokes' equations, Trubulence modelling, LES, Smagorinsky model, Stabilized finite element
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-55150 (URN)10.1007/s00021-021-00633-8 (DOI)000718277700001 ()2-s2.0-85119322075 (Scopus ID)HOA;intsam;778862 (Local ID)HOA;intsam;778862 (Archive number)HOA;intsam;778862 (OAI)
Funder
Swedish Research Council, 2018-05262European Commission, 2017-03911
Available from: 2021-11-25 Created: 2021-11-25 Last updated: 2021-11-29Bibliographically approved
Burman, E., Hansbo, P. & Larson, M. G. (2022). Explicit time stepping for the wave equation using CutFEM with discrete extension. SIAM Journal on Scientific Computing, 44(3), A1254-A1289
Open this publication in new window or tab >>Explicit time stepping for the wave equation using CutFEM with discrete extension
2022 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 44, no 3, p. A1254-A1289Article in journal (Refereed) Published
Abstract [en]

In this paper we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples.

Place, publisher, year, edition, pages
SIAM Publications, 2022
Keywords
wave equation, explicit time stepping, CutFEM, discrete extension operator, a priori error estimates
National Category
Mathematics
Identifiers
urn:nbn:se:hj:diva-58678 (URN)10.1137/20M137937X (DOI)000862858200010 ()2-s2.0-85131259231 (Scopus ID);intsam;1705265 (Local ID);intsam;1705265 (Archive number);intsam;1705265 (OAI)
Funder
Swedish Foundation for Strategic Research, AM13-0029Swedish Research Council, 2013-4708
Available from: 2022-10-21 Created: 2022-10-21 Last updated: 2023-01-17Bibliographically approved
Hansbo, P. & Larson, M. G. (2022). Nitsche's finite element method for model coupling in elasticity. Computer Methods in Applied Mechanics and Engineering, 392, Article ID 114707.
Open this publication in new window or tab >>Nitsche's finite element method for model coupling in elasticity
2022 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 392, article id 114707Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2022
Keywords
Hybrid method, Interface stiffness, Model coupling, Nitsche's method, Stiffness, Axial stiffness, Displacement field, Elastic interfaces, Euler Bernoulli beams, Interface problems, Nitsche's methods, Two-dimensional, Finite element method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:hj:diva-55974 (URN)10.1016/j.cma.2022.114707 (DOI)000783077800009 ()2-s2.0-85124662827 (Scopus ID)HOA;;798393 (Local ID)HOA;;798393 (Archive number)HOA;;798393 (OAI)
Funder
Swedish Research Council, 2017-03911, 2018-05262eSSENCE - An eScience Collaboration
Available from: 2022-03-03 Created: 2022-03-03 Last updated: 2022-05-12Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-7352-1550

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