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| 2. |
- Balmus, Maximilian, et al.
(författare)
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A partition of unity approach to fluid mechanics and fluid-structure interaction
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : ELSEVIER SCIENCE SA. - 0045-7825 .- 1879-2138. ; 362
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Tidskriftsartikel (refereegranskat)abstract
- For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.
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| 3. |
- Burman, Erik, et al.
(författare)
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A cut discontinuous Galerkin method for the Laplace–Beltrami operator
- 2017
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Ingår i: IMA Journal of Numerical Analysis. - : Oxford University Press. - 0272-4979 .- 1464-3642. ; 37:1, s. 138-169
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Tidskriftsartikel (refereegranskat)abstract
- We develop a discontinuous cut finite element method for the Laplace–Beltrami operator on a hypersurface embedded in R. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in R. 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.
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| 4. |
- Burman, Erik, et al.
(författare)
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A Stabilized Cut Finite Element Method for the Three Field Stokes Problem
- 2015
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 37:4, s. A1705-A1726
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- 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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| 6. |
- Burman, E., et al.
(författare)
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A stable cut finite element method for partial differential equations on surfaces : The Helmholtz–Beltrami operator
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 362
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Tidskriftsartikel (refereegranskat)abstract
- We consider solving the surface Helmholtz equation 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. Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition hk<C, where h denotes the mesh size, k the wave number and C a constant depending mainly on the surface curvature κ, but not on the surface/mesh intersection. Optimal error estimates in the H1 and L2-norms follow.
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| 7. |
- Burman, Erik, et al.
(författare)
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Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains
- 2022
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Ingår i: Numerische Mathematik. - : Springer Science+Business Media B.V.. - 0029-599X .- 0945-3245. ; 150:2, s. 423-478
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Tidskriftsartikel (refereegranskat)abstract
- This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld and Olshanskii (ESAIM: M2AN 53(2):585–614, 2019), where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche’s method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal L2(L2) -norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.
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| 8. |
- Claus, Susanne, et al.
(författare)
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A stabilized Nitsche fictitious domain formulation for the three-field Stokes problem
- 2013
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Ingår i: Proceeding of the 26th Nordic Seminar on Computational Mechanics. - : Simula Research Laboratory. - 9788292593127 ; , s. 1-6
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Konferensbidrag (refereegranskat)abstract
- We propose a Nitsche fictitious domain method for the three-field Stokes problem where the dependent variables of velocity, pressure and extra-stress tensor are discretised with linear finite elements. To stabilise the equal order approximation, we employ a continuous interior penalty (CIP) method involving the normal gradient jumps of the velocity and pressure. On the unfitted boundary, Dirichlet boundary conditions are weakly enforced using Nitsche’s method. Adding CIP-like ghostpenalties in the vicinity of the boundary allows us to prove the inf-sup stability and optimal convergence of our method and to bound the condition number independent of the location of the boundary with respect to the computational mesh. Numerical examples corroborate the theoretical findings.
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| 9. |
- Claus, Susanne, et al.
(författare)
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CutFEM : a stabilised Nitsche XFEM method for multi-physics problems
- 2015
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Ingår i: Proceedings of the 23rd Conference on Computational Mechanics. - Swansea : Swansea University. - 9780956746245 ; , s. 171-174
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Konferensbidrag (refereegranskat)abstract
- 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.
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| 10. |
- Dokken, Jørgen S., et al.
(författare)
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A multimesh finite element method for the Navier-Stokes equations based on projection methods
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 368
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Tidskriftsartikel (refereegranskat)abstract
- The multimesh finite element method is a technique for solving partial differential equations on multiple non-matching meshes by enforcing interface conditions using Nitsche's method. Since the non-matching meshes can result in arbitrarily cut cells, additional stabilization terms are needed to obtain a stable method. In this contribution we extend the multimesh finite element method to the Navier-Stokes equations based on the incremental pressure-correction scheme. For each step in the pressure-correction scheme, we derive a multimesh finite element formulation with suitable stabilization terms. The proposed scheme is implemented for arbitrary many overlapping two dimensional domains, yielding expected spatial and temporal convergence rates for the Taylor-Green problem, and demonstrates good agreement for the drag and lift coefficients for the Turek-Schfifer benchmark (DFG benchmark 2D-3). Finally, we illustrate the capabilities of the proposed scheme by optimizing the layout of obstacles in a two dimensional channel.
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