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Publications (10 of 31) Show all publications
Burman, E., Larson, M. G., Larsson, K. & Lundholm, C. (2025). Stabilizing and solving unique continuation problems by parameterizing data and learning finite element solution operators. Computer Methods in Applied Mechanics and Engineering, 444, Article ID 118111.
Open this publication in new window or tab >>Stabilizing and solving unique continuation problems by parameterizing data and learning finite element solution operators
2025 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 444, article id 118111Article in journal (Refereed) Published
Abstract [en]

We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an autoencoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train an operator network to map the expansion coefficients representing the boundary data to the finite element (FE) solution of the PDE. Finally, we connect the autoencoder's decoder to the operator network which enables us to solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method (FEM) in the linear setting and establish an optimal error estimate in the H1-norm. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.

Place, publisher, year, edition, pages
Elsevier, 2025
Keywords
Inverse problems, Nonlinear PDE, Machine learning, Unique continuation problem
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-240021 (URN)10.1016/j.cma.2025.118111 (DOI)
Funder
Swedish Research Council, 2021-04925eSSENCE - An eScience Collaboration
Available from: 2025-06-11 Created: 2025-06-11 Last updated: 2025-06-12Bibliographically approved
Björklund, M., Larsson, K. & Larson, M. G. (2024). Error estimates for finite element approximations of viscoelastic dynamics: the generalized Maxwell model. Computer Methods in Applied Mechanics and Engineering, 425, Article ID 116933.
Open this publication in new window or tab >>Error estimates for finite element approximations of viscoelastic dynamics: the generalized Maxwell model
2024 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 425, article id 116933Article in journal (Refereed) Published
Abstract [en]

We prove error estimates for a finite element approximation of viscoelastic dynamics based on continuous Galerkin in space and time, both in energy norm and in L2 norm. The proof is based on an error representation formula using a discrete dual problem and a stability estimate involving the kinetic, elastic, and viscoelastic energies. To set up the dual error analysis and to prove the basic stability estimates, it is natural to formulate the problem as a first-order-in-time system involving evolution equations for the viscoelastic stress, the displacements, and the velocities. The equations for the viscoelastic stress can, however, be solved analytically in terms of the deviatoric strain velocity, and therefore, the viscoelastic stress can be eliminated from the system, resulting in a system for displacements and velocities.

Place, publisher, year, edition, pages
Elsevier, 2024
Keywords
Viscoelasticity, Generalized Maxwell solid, Finite element method, A priori error analysis
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-222779 (URN)10.1016/j.cma.2024.116933 (DOI)001223373200001 ()2-s2.0-85188678574 (Scopus ID)
Funder
Swedish Research Council, 2021-04925eSSENCE - An eScience CollaborationSwedish Research Council, 2017-03911
Available from: 2024-03-27 Created: 2024-03-27 Last updated: 2025-04-24Bibliographically 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
Discrete extension operators, Trimmed spline spaces, Cut isogeometric methods, Unfitted finite element methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-200655 (URN)10.1016/j.cma.2022.115707 (DOI)000882526600004 ()2-s2.0-85140922298 (Scopus ID)
Funder
Swedish Research Council, 2017-03911Swedish Research Council, 2018-05262Swedish Research Council, 2021-04925eSSENCE - An eScience Collaboration
Available from: 2022-10-30 Created: 2022-10-30 Last updated: 2023-09-05Bibliographically approved
Jonsson, T., Larson, M. G. & Larsson, K. (2023). Hybridized isogeometric method for elliptic problems on CAD surfaces with gaps. Computer Methods in Applied Mechanics and Engineering, 410, Article ID 116014.
Open this publication in new window or tab >>Hybridized isogeometric method for elliptic problems on CAD surfaces with gaps
2023 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 410, article id 116014Article in journal (Refereed) Published
Abstract [en]

We develop a method for solving elliptic partial differential equations on surfaces described by CAD patches that may have gaps/overlaps. The method is based on hybridization using a three-dimensional mesh that covers the gap/overlap between patches. Thus, the hybrid variable is defined on a three-dimensional mesh, and we need to add appropriate normal stabilization to obtain an accurate solution, which we show can be done by adding a suitable term to the weak form. In practical applications, the hybrid mesh may be conveniently constructed using an octree to efficiently compute the necessary geometric information. We prove error estimates and present several numerical examples illustrating the application of the method to different problems, including a realistic CAD model.

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
Trimmed multipatch CAD surfaces, Interfaces with gaps/overlaps, CutIGA and CutFEM, Hybridized method
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-201015 (URN)10.1016/j.cma.2023.116014 (DOI)000965103100001 ()2-s2.0-85150789500 (Scopus ID)
Funder
Swedish Research Council, 017-03911Swedish Research Council, 021-04925
Note

Originally included in thesis in manuscript form.

Available from: 2022-11-14 Created: 2022-11-14 Last updated: 2023-04-28Bibliographically 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, 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, 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
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-214395 (URN)10.1016/j.cma.2023.116302 (DOI)001114119400001 ()2-s2.0-85169927833 (Scopus ID)
Funder
Swedish Research Council, 2021-04925Swedish Research Council, 2022-03908eSSENCE - An eScience Collaboration
Note

Part B

Available from: 2023-09-19 Created: 2023-09-19 Last updated: 2023-12-29Bibliographically approved
Jonsson, T., Larson, M. G. & Larsson, K. (2022). Robust multipatch IGA with singular maps.
Open this publication in new window or tab >>Robust multipatch IGA with singular maps
2022 (English)Manuscript (preprint) (Other academic)
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-201016 (URN)
Available from: 2022-11-14 Created: 2022-11-14 Last updated: 2022-11-15
Larsson, K., Kollmannsberger, S., Rank, E. & Larson, M. G. (2022). The finite cell method with least squares stabilized Nitsche boundary conditions. Computer Methods in Applied Mechanics and Engineering, 393, Article ID 114792.
Open this publication in new window or tab >>The finite cell method with least squares stabilized Nitsche boundary conditions
2022 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 393, article id 114792Article in journal (Refereed) Published
Abstract [en]

We apply the recently developed least squares stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions to the finite cell method. The least squares stabilized Nitsche method in combination with finite cell stabilization leads to a symmetric positive definite stiffness matrix and relies only on elementwise stabilization, which does not lead to additional fill in. We prove a priori error estimates and bounds on the condition numbers.

Place, publisher, year, edition, pages
Elsevier, 2022
Keywords
Finite cell method, Dirichlet conditions, Nitsche’s method, A priori error estimates
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-193119 (URN)10.1016/j.cma.2022.114792 (DOI)000785237800003 ()2-s2.0-85126527849 (Scopus ID)
Funder
Swedish Research Council, 2021-04925Swedish Research Council, 2017-03911Swedish Foundation for Strategic Research , AM13-0029eSSENCE - An eScience Collaboration, -German Research Foundation (DFG), KO 4570/1-1German Research Foundation (DFG), RA 627/29-1
Available from: 2022-03-15 Created: 2022-03-15 Last updated: 2023-09-05Bibliographically approved
Larsson, J. & Larsson, K. (2022). The Lorentz group and the Kronecker product of matrices. European journal of physics, 43(2), Article ID 025603.
Open this publication in new window or tab >>The Lorentz group and the Kronecker product of matrices
2022 (English)In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 43, no 2, article id 025603Article in journal (Refereed) Published
Abstract [en]

The group SL(2,C)(2,C) of all complex 2 × 2 matrices with determinant one is closely related to the group L+↑ of real 4 × 4 matrices representing the restricted Lorentz transformations. This relation, sometimes called the spinor map, is of fundamental importance in relativistic quantum mechanics and has applications also in general relativity. In this paper we show how the spinor map may be expressed in terms of pure matrix algebra by including the Kronecker product between matrices in the formalism. The so-obtained formula for the spinor map may be manipulated by matrix algebra and used in the study of Lorentz transformations.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2022
National Category
Other Physics Topics
Identifiers
urn:nbn:se:umu:diva-192572 (URN)10.1088/1361-6404/ac4c88 (DOI)000756011400001 ()2-s2.0-85125752509 (Scopus ID)
Available from: 2022-02-17 Created: 2022-02-17 Last updated: 2022-03-18Bibliographically approved
Hansbo, P., Larson, M. G. & Larsson, K. (2020). Analysis of finite element methods for vector Laplacians on surfaces. IMA Journal of Numerical Analysis, 40(3), 1652-1701
Open this publication in new window or tab >>Analysis of finite element methods for vector Laplacians on surfaces
2020 (English)In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 40, no 3, p. 1652-1701Article in journal (Refereed) Published
Abstract [en]

We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in R3. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a R3 vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and L2 norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.

Place, publisher, year, edition, pages
Oxford University Press, 2020
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-174232 (URN)10.1093/imanum/drz018 (DOI)000574428700002 ()2-s2.0-85072749572 (Scopus ID)
Funder
eSSENCE - An eScience Collaboration
Available from: 2020-08-19 Created: 2020-08-19 Last updated: 2023-03-23Bibliographically approved
Elfverson, D., Larson, M. G. & Larsson, K. (2019). A new least squares stabilized Nitsche method for cut isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 349, 1-16
Open this publication in new window or tab >>A new least squares stabilized Nitsche method for cut isogeometric analysis
2019 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 349, p. 1-16Article in journal (Refereed) Published
Abstract [en]

We derive a new stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis (CutIGA). We consider C1 splines and stabilize the standard Nitsche method by adding a certain elementwise least squares terms in the vicinity of the Dirichlet boundary and an additional term on the boundary which involves the tangential gradient. We show coercivity with respect to the energy norm for functions in H2(Ω) and optimal order a priori error estimates in the energy and L2 norms. To obtain a well posed linear system of equations we combine our formulation with basis function removal which essentially eliminates basis functions with sufficiently small intersection with Ω. The upshot of the formulation is that only elementwise stabilization is added in contrast to standard procedures based on ghost penalty and related techniques and that the stabilization is consistent. In our numerical experiments we see that the method works remarkably well in even extreme cut situations using a Nitsche parameter of moderate size.

Place, publisher, year, edition, pages
Elsevier, 2019
Keywords
Fictitious domain methods, Nitsche’s method, Least squares stabilization, Isogeometric analysis
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-156840 (URN)10.1016/j.cma.2019.02.011 (DOI)000464951400001 ()2-s2.0-85062154279 (Scopus ID)
Funder
Swedish Research Council, 2013-4708Swedish Research Council, 2017-03911Swedish Foundation for Strategic Research , AM13-0029eSSENCE - An eScience Collaboration
Available from: 2019-03-01 Created: 2019-03-01 Last updated: 2023-09-05Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-7838-1307

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