Cut finite element methods for elliptic problems on multipatch parametric surfaces

Jonsson, Tobias; Larson, Mats G.; Larsson, Karl

doi:10.1016/j.cma.2017.06.018

Cited by 31 publications

(28 citation statements)

References 20 publications

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“…Both of these terms can be computed elementwise using the standard assembly of the stiffness matrix. Note also that the stabilization terms do not cause fill in which is the case with standard ghost penalty approaches [4,15,16].…”

Section: Galerkin Orthogonality It Holdsmentioning

confidence: 99%

A new least squares stabilized Nitsche method for cut isogeometric analysis

Elfverson

Larson

Larsson

2019

Computer Methods in Applied Mechanics and Engineering

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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 C 1 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 H 2 (Ω) and optimal order a priori error estimates in the energy and L 2 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.

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Section: Galerkin Orthogonality It Holdsmentioning

confidence: 99%

A new least squares stabilized Nitsche method for cut isogeometric analysis

Elfverson

Larson

Larsson

2019

Computer Methods in Applied Mechanics and Engineering

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“…Previous Work The possibility of discretizing partial differential equations on the cut part of elements was first proposed by Olshanskii, Reusken, and Grande [20], and has been further developed by this group in, e.g., [11,12,21]. The work presented here builds on the CutFEM developments by the authors and their coworkers concerning stabilization of cut elements in [1,4,5,16,18] and concerning beam, plate, and membrane formulations in [6,14,15]. Related work has been presented by Fries et al [9,10] and by Rank and coworkers [22,23], see also [7,24].…”

Section: Introductionmentioning

confidence: 99%

Cut Finite Element Methods for Linear Elasticity Problems

Hansbo

Larson

Larsson

2017

Lecture Notes in Computational Science and Engineering

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We formulate a cut finite element method for linear elasticity based on higher order elements on a fixed background mesh. Key to the method is a stabilization term which provides control of the jumps in the derivatives of the finite element functions across faces in the vicinity of the boundary. We then develop the basic theoretical results including error estimates and estimates of the condition number of the mass and stiffness matrices. We apply the method to the standard displacement problem, the frequency response problem, and the eigenvalue problem. We present several numerical examples including studies of thin bending dominated structures relevant for engineering applications. Finally, we develop a cut finite element method for fibre reinforced materials where the fibres are modeled as a superposition of a truss and a Euler-Bernoulli beam. The beam model leads to a fourth order problem which we discretize using the restriction of the bulk finite element space to the fibre together with a continuous/discontinuous finite element formulation. Here the bulk material stabilizes the problem and it is not necessary to add additional stabilization terms. *

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“…Although, we remark that the use of a level-set domain description is not an inherent limitation of CutFEMs, see for example[84] where a parametric description of the domain boundary is used.…”

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confidence: 99%

Projection-based reduced order models for a cut finite element method in parametrized domains

Karatzas

Ballarin

Rozza

2020

Computers & Mathematics with Applications

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This work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method. Such methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modeling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and transport of embedded solutions to a common background is analyzed in detail. Recent improvements go under the names of Ghost-Cell finite difference methods, Cut-Cell finite volume approach, Immersed Interface, Ghost Fluid, Volume Penalty methods, for which we refer to the review paper [1] and references within. In particular, for what concerns incompressible flows in arbitrary smooth domains, the Immersed Boundary Smooth Extension method has shown high-order convergence for the incompressible Navier-Stokes equations [4].More in detail, extended mesh finite element methods using cut elements are examined in [5,6] for stationary Stokes flow systems, as well as for Navier-Stokes. An analysis for high Reynolds numbers, independent of the local Reynolds, has been carried out in [7,8,9]. XFEM approaches in 2D and 3D Navier-Stokes are reported in [10]. Higher Reynolds number aerodynamics problems in unbounded domains, thin vortex ring and Lattice Green function Immersed Boundary methods are studied in [11,12]. Furthermore, embedded and immersed methods have been used in solving fluid structure

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Cut finite element methods for elliptic problems on multipatch parametric surfaces

Cited by 31 publications

References 20 publications

A new least squares stabilized Nitsche method for cut isogeometric analysis

A new least squares stabilized Nitsche method for cut isogeometric analysis

Cut Finite Element Methods for Linear Elasticity Problems

Projection-based reduced order models for a cut finite element method in parametrized domains

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