| 1. |
- Hansbo, Peter, et al.
(författare)
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STABILIZED FINITE ELEMENT APPROXIMATION OF THE MEAN CURVATURE VECTOR ON CLOSED SURFACES
- 2015
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 53:4, s. 1806-1832
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Tidskriftsartikel (refereegranskat)abstract
- The mean curvature vector of a surface is obtained by letting the Laplace-Beltrami operator act on the embedding of the surface in R-3. In this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in L-2. The stabilization involves the jump in the tangent gradient in the direction of the outer co-normals at each edge in the surface mesh. We consider both standard meshed surfaces and so-called cut surfaces that are level sets of piecewise linear distance functions. We prove a priori error estimates and verify the theoretical results numerically.
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| 2. |
- Burman, E., et al.
(författare)
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Extension operators for trimmed spline spaces
- 2023
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 403
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- 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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| 4. |
- Burman, Erik N., et al.
(författare)
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A cut finite element method with boundary value correction for the incompressible Stokes equations
- 2019
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Ingår i: Numerical mathematics and advanced applications ENUMATH 2017. - Cham : Springer. - 9783319964140 - 9783319964157 ; , s. 183-192
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Konferensbidrag (refereegranskat)abstract
- We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche’s method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman et al. (Math Comput 87(310):633–657, 2018).
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| 5. |
- Cenanovic, Mirza, et al.
(författare)
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Finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 372
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces. In particular, we adopt a stabilization technique which allows for first order L2-convergence of the mean curvature vector and apply this stabilization technique also to the computation of continuous, recovered, normals using L2-projections of the piecewise constant face normals. Finally, we use our projected normals to define an adaptive mesh refinement approach to geometry resolution where we also employ spline techniques to reconstruct the surface before refinement. We compare our results to previously proposed approaches.
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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. |
- Odsæter, Lars H., et al.
(författare)
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A simple embedded discrete fracture-matrix model for a coupled flow and transport problem in porous media
- 2019
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 343, s. 572-601
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Tidskriftsartikel (refereegranskat)abstract
- Accurate simulation of fluid flow and transport in fractured porous media is a key challenge in subsurface reservoir engineering. Due to the high ratio between its length and width, fractures can be modeled as lower dimensional interfaces embedded in the porous rock. We apply a recently developed embedded finite element method (EFEM) for the Darcy problem. This method allows for general fracture geometry, and the fractures may cut the finite element mesh arbitrarily. We present here a velocity model for EFEM and couple the Darcy problem to a transport problem for a passive solute. The main novelties of this work are a locally conservative velocity approximation derived from the EFEM solution, and the development of a lowest order upwind finite volume method for the transport problem. This numerical model is compatible with EFEM in the sense that the same computational mesh may be applied, so that we retain the same flexibility with respect to fracture geometry and meshing. Hence, our coupled solution strategy represents a simple approach in terms of formulation, implementation and meshing. We demonstrate our model by some numerical examples on both synthetic and realistic problems, including a benchmark study for single-phase flow. Despite the simplicity of the method, the results are promising.
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