| 1. |
- Burman, Erik, et al.
(författare)
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The Augmented Lagrangian Method as a Framework for Stabilised Methods in Computational Mechanics
- 2023
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Ingår i: Archives of Computational Methods in Engineering. - : Springer. - 1134-3060 .- 1886-1784. ; 30, s. 2579-2604
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Burman, Erik, et al.
(författare)
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CutFEM based on extended finite element spaces
- 2022
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 152, s. 331-369
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Burman, Erik, et al.
(författare)
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Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions
- 2024
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 93:345, s. 35-54
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Tidskriftsartikel (refereegranskat)abstract
- 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 ∈ Hs with s ∈ (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.
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| 4. |
- Burman, Erik, et al.
(författare)
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A cut finite element method for elliptic bulk problems with embedded surfaces
- 2019
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Ingår i: GEM - International Journal on Geomathematics. - : Springer. - 1869-2672 .- 1869-2680. ; 10:1
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Burman, Erik, et al.
(författare)
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A stabilized cut finite element method for partial differential equations on surfaces : The Laplace-Beltrami operator
- 2015
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 285, s. 188-207
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Tidskriftsartikel (refereegranskat)abstract
- We consider solving the Laplace-Beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions. The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method. (c) 2014 Elsevier B.V. All rights reserved.
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| 6. |
- Burman, Erik, et al.
(författare)
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A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 358
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Tidskriftsartikel (refereegranskat)abstract
- We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included.
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| 7. |
- Burman, Erik, et al.
(författare)
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Cut Bogner-Fox-Schmit elements for plates
- 2020
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Ingår i: Advanced Modeling and Simulation in Engineering Sciences. - : Springer. - 2213-7467. ; 7:1
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Tidskriftsartikel (refereegranskat)abstract
- We present and analyze a method for thin plates based on cut Bogner-Fox-Schmit elements, which are C1 elements obtained by taking tensor products of Hermite splines. The formulation is based on Nitsche’s method for weak enforcement of essential boundary conditions together with addition of certain stabilization terms that enable us to establish coercivity and stability of the resulting system of linear equations. We also take geometric approximation of the boundary into account and we focus our presentation on the simply supported boundary conditions which is the most sensitive case for geometric approximation of the boundary.
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| 8. |
- Burman, Erik, et al.
(författare)
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Cut finite element methods for coupled bulk–surface problems
- 2016
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 133:2, s. 203-231
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Tidskriftsartikel (refereegranskat)abstract
- We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and (Formula presented.) norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.
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| 9. |
- Burman, Erik, et al.
(författare)
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Dirichlet boundary value correction using Lagrange multipliers
- 2020
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Ingår i: BIT Numerical Mathematics. - : Springer. - 0006-3835 .- 1572-9125. ; 60:1, s. 235-260
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Tidskriftsartikel (refereegranskat)abstract
- We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to a Taylor series expansion approach previously used in the context of Nitsche's method and, in the case of inf-sup stable multiplier methods, prove a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.
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| 10. |
- Burman, Erik, et al.
(författare)
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Explicit time stepping for the wave equation using cutFEM with discrete extension
- 2022
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 44:3, s. A1254-A1289
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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