| 1. |
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| 2. |
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| 3. |
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| 4. |
- 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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| 5. |
- Balmus, Maximilian, et al.
(författare)
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A stabilized multidomain partition of unity approach to solving incompressible viscous flow
- 2022
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 392
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Tidskriftsartikel (refereegranskat)abstract
- In this work we propose a new stabilized approach for solving the incompressible Navier-Stokes equations on fixed overlapping grids. This new approach is based on the partition of unity finite element method, which defines the solution fields as weighted sums of local fields, supported by the different grids. Here, the discrete weak formulation of the problem is re-set in cG(1)cG(1) stabilized form, which has the dual benefit of lowering grid resolution requirements for convection dominated flows and allowing for the use of velocity and pressure discretizations which do not satisfy the inf-sup condition. Additionally, we provide an outline of our implementation within an existing distributed parallel application and identify four key options to improve the code efficiency namely: the use of cache to store mapped quadrature points and basis function gradients, the intersection volume splitting algorithm, the use of lower order quadrature schemes, and tuning the partition weight associated with the interface elements. The new method is shown to have comparable accuracy to the single mesh boundary-fitted version of the same stabilized solver based on three transient flow tests including both 2D and 3D settings, as well as low and moderate Reynolds number flow conditions. Moreover, we demonstrate how the four implementation options have a synergistic effect lowering the residual assembly time by an order of magnitude compared to a naive implementation, and showing good load balancing properties.
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| 6. |
- 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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| 7. |
- 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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| 8. |
- 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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| 9. |
- 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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| 10. |
- Burman, Erik, et al.
(författare)
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Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions
- 2019
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 52:6, s. 2247-2282
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Tidskriftsartikel (refereegranskat)abstract
- We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in Rd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in R3.
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| 11. |
- Burman, Erik, et al.
(författare)
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CutFEM : Discretizing geometry and partial differential equations
- 2015
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Ingår i: International Journal for Numerical Methods in Engineering. - : Wiley. - 0029-5981 .- 1097-0207. ; 104:7, s. 472-501
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 12. |
- 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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| 13. |
- Burman, Erik, et al.
(författare)
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Finite element approximation of the Laplace-Beltrami operator on a surface with boundary
- 2019
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Ingår i: Numerische Mathematik. - : Springer. - 0029-599X .- 0945-3245. ; 141:1, s. 141-172
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Tidskriftsartikel (refereegranskat)abstract
- We develop a finite element method for the Laplace–Beltrami operator on a surface with boundary and nonhomogeneous Dirichlet boundary conditions. The method is based on a triangulation of the surface and the boundary conditions are enforced weakly using Nitsche's method. We prove optimal order a priori error estimates for piecewise continuous polynomials of order k ≥ 1 in the energy and L2 norms that take the approximation of the surface and the boundary into account.
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| 14. |
- Burman, E., et al.
(författare)
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Full gradient stabilized cut finite element methods for surface partial differential equations
- 2016
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 310, s. 278-296
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Tidskriftsartikel (refereegranskat)abstract
- 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 R3 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 L2 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.
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| 15. |
- 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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| 16. |
- 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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| 17. |
- de Prenter, Frits, et al.
(författare)
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A note on the stability parameter in Nitsche's method for unfitted boundary value problems
- 2018
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Ingår i: Computers and Mathematics with Applications. - : Elsevier. - 0898-1221 .- 1873-7668. ; 75:12, s. 4322-4336
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Tidskriftsartikel (refereegranskat)abstract
- Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. Of the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors.
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| 18. |
- 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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| 19. |
- Gürkan, Ceren, et al.
(författare)
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A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems
- 2019
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 348, s. 466-499
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Tidskriftsartikel (refereegranskat)abstract
- We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of elliptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in R d , so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the background mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinuous Galerkin methods, we employ and extend ghost penalty techniques from recently developed continuous cut finite element methods, which allows for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying four abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and condition number estimates for the Poisson boundary value problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. We also demonstrate how the framework can be elegantly applied to discretize high contrast interface problems. The theoretical results are illustrated by a number of numerical experiments for various approximation orders and for two and three-dimensional test problems. (C) 2018 Published by Elsevier B.V.
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| 20. |
- Gürkan, Ceren, et al.
(författare)
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Stabilized Cut Discontinuous Galerkin Methods for Advection-Reaction Problems
- 2020
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 42:5, s. A2620-A2654
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Tidskriftsartikel (refereegranskat)abstract
- We develop novel stabilized cut discontinuous Galerkin methods for advection-reaction problems. The domain of interest is embedded into a structured, unfitted background mesh in R-d where the domain boundary can cut through the mesh in an arbitrary fashion. To cope with robustness problems caused by small cut elements, we introduce ghost penalties in the vicinity of the embedded boundary to stabilize certain (semi-)norms associated with the advection and reaction operator. A few abstract assumptions on the ghost penalties are identified enabling us to derive geometrically robust and optimal a priori error and condition number estimates for the stationary advection-reaction problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. The theoretical results are corroborated by a number of computational studies for various approximation orders and for two- and three-dimensional test problems.
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| 21. |
- Hansbo, Peter, et al.
(författare)
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A stabilized cut finite element method for the Darcy problem on surfaces
- 2017
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 326, s. 298-318
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 22. |
- Massing, André, 1977-
(författare)
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A Cut Discontinuous Galerkin Method for Coupled Bulk-Surface Problems
- 2017
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Ingår i: Geometrically Unfitted Finite Element Methods and Applications. - Cham : Springer. - 9783319714318 - 9783319714301 ; , s. 259-279
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Konferensbidrag (refereegranskat)abstract
- We develop a cut Discontinuous Galerkin method (cutDGM) for a diffusion-reaction equation in a bulk domain which is coupled to a corresponding equation on the boundary of the bulk domain. The bulk domain is embedded into a structured, unfitted background mesh. By adding certain stabilization terms to the discrete variational formulation of the coupled bulk-surface problem, the resulting cutDGM is provably stable and exhibits optimal convergence properties as demonstrated by numerical experiments. We also show both theoretically and numerically that the system matrix is well-conditioned, irrespective of the relative position of the bulk domain in the background mesh.
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| 23. |
- Massing, André, et al.
(författare)
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A Nitsche-Based Cut Finite Element Method for a Fluid--Structure Interaction Problem
- 2015
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Ingår i: Communications in Applied Mathematics and Computational Science. - : Mathematical Sciences Publishers. - 1559-3940 .- 2157-5452. ; 10:2, s. 97-120
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 24. |
- Massing, Andre, 1977-, et al.
(författare)
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A stabilized Nitsche cut finite element method for the Oseen problem
- 2018
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 328, s. 262-300
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 25. |
- Massing, Andre, et al.
(författare)
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A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem
- 2014
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 61:3, s. 604-628
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 26. |
- Massing, Andre, et al.
(författare)
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A stabilized Nitsche overlapping mesh method for the Stokes problem
- 2014
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 128:1, s. 73-101
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 27. |
- Massing, Andre, et al.
(författare)
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Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions
- 2013
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Ingår i: SIAM Journal of Scientific Computing. - 1064-8275 .- 1095-7197 .- 1095-7200. ; 35:1, s. C23-C47
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Tidskriftsartikel (refereegranskat)abstract
- In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on nonmatching or overlapping meshes. Examples of such methods are the fictitious domain method, the extended finite element method, and Nitsche's method. In all these methods, integrals must be computed over cut cells or subsimplices, which is challenging to implement, especially in three space dimensions. In this note, we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices. As a canonical example of an overlapping mesh method, we consider Nitsche's method, which we apply to Poisson's equation and a linear elastic problem. © 2013 Society for Industrial and Applied Mathematics.
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| 28. |
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| 29. |
- Massing, Andre, 1977-, et al.
(författare)
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Towards an Implementation of Nitsche's Method on Overlapping Meshes in 3D
- 2010
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Ingår i: AIP Conference Proceedings. - : American Institute of Physics (AIP). - 9780735408340 ; , s. 783-786
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Konferensbidrag (refereegranskat)abstract
- Nitsche's method may be used to derive a systematic finite element formulation for problems with overlapping meshes that is stable and has optimal order. In this note, we formulate the method for a linear elastic model problem with discontinuous material properties. We discuss the implementation aspects, including computation of intersections of elements and integrals on the resulting polyhedra, and illustrate the method on a three dimensional test problem.
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| 30. |
- Ulfsby, Tale Bakken, et al.
(författare)
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Stabilized cut discontinuous Galerkin methods for advection–reaction problems on surfaces
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 413
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Tidskriftsartikel (refereegranskat)abstract
- We develop a novel cut discontinuous Galerkin (CutDG) method for stationary advection–reaction problems on surfaces embedded in Rd. The CutDG method is based on embedding the surface into a full-dimensional background mesh and using the associated discontinuous piecewise polynomials of order k as test and trial functions. As the surface can cut through the mesh in an arbitrary fashion, we design a suitable stabilization that enables us to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. The resulting CutDG formulation is geometrically robust in the sense that all derived theoretical results hold with constants independent of any particular cut configuration. Numerical examples support our theoretical findings.
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| 31. |
- Winter, M., et al.
(författare)
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A Nitsche cut finite element method for the Oseen problem with general Navier boundary conditions
- 2018
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : ELSEVIER SCIENCE SA. - 0045-7825 .- 1879-2138. ; 330, s. 220-252
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Tidskriftsartikel (refereegranskat)abstract
- In this work a Nitsche-based imposition of generalized Navier conditions on cut meshes for the Oseen problem is presented. Other methods from literature dealing with the generalized Navier condition impose this condition by means of substituting the tangential Robin condition in a classical Galerkin way. These methods work fine for a large slip length coefficient but lead to conditioning and stability issues when it approaches zero. We introduce a novel method for the weak imposition of the generalized Navier condition which remains well-posed and stable for arbitrary choice of slip length, including zero. The method proposed here builds on the formulation done by Juntunen and Stenberg (2009). They impose a Robin condition for the Poisson problem by means of Nitsche's method for an arbitrary combination of the Dirichlet and Neumann parts of the condition. The analysis conducted for the proposed method is done in a similar fashion as in Massing et al. (2018), but is done here for a more general type of boundary condition. The analysis proves stability for all flow regimes and all choices of slip lengths. Also an L-2-optimal estimate for the velocity error is shown, which was not conducted in the previously mentioned work. A numerical example is carried out for varying slip lengths to verify the robustness and stability of the method with respect to the choice of slip length. Even though proofs and formulations are presented for the more general case of an unfitted grid method, they can easily be reduced to the simpler case of a boundary-fitted grid with the removal of the ghost-penalty stabilization terms
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