A note on the stability parameter in Nitsche’s method for unfitted boundary value problems

Prenter, Frits de; Lehrenfeld, Christoph; Massing, André

doi:10.1016/j.camwa.2018.03.032

Cited by 42 publications

(45 citation statements)

References 61 publications

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“…For the optimized case (Figure 13a), we obtain optimal convergence rates for both the velocity and pressure. When the ghost-penalty vanishes (Figure 13b), an adverse effect on the H 1 error of the velocity is observed, which is in accordance with the literature on elliptic problems [25,26,40]. When γ = 1 · 10 4 (Figure 13c), a negative impact on the accuracy of the velocity is observed, which also leads to sub-optimal convergence of the pressure in the L 2 norm.…”

Section: Steady Navier-stokes Flow In a Quarter Annulus Domainsupporting

confidence: 87%

Skeleton-stabilized immersogeometric analysis for incompressible viscous flow problems

Hoang

Verhoosel

Qin

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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A Skeleton-stabilized ImmersoGeometric Analysis technique is proposed for incompressible viscous flow problems with moderate Reynolds number. The proposed formulation fits within the framework of the finite cell method, where essential boundary conditions are imposed weakly using a Nitsche-type method. The key idea of the proposed formulation is to stabilize the jumps of high-order derivatives of variables over the skeleton of the background mesh. The formulation allows the use of identical finitedimensional spaces for the approximation of the pressure and velocity fields in immersed domains. The stability issues observed for inf-sup stable discretizations of immersed incompressible flow problems are avoided with this formulation. For B-spline basis functions of degree k with highest regularity, only the derivative of order k has to be controlled, which requires specification of only a single stabilization parameter for the pressure field. The Stokes and Navier-Stokes equations are studied numerically in two and three dimensions using various immersed test cases. Oscillation-free solutions and high-order optimal convergence rates can be obtained. The formulation is shown to be stable even in limit cases where almost every elements of the physical domain is cut, and hence it does not require the existence of interior cells. In terms of the sparsity pattern, the algebraic system has a considerably smaller stencil than counterpart approaches based on Lagrange basis functions. This important property makes the proposed skeleton-stabilized technique computationally practical. To demonstrate the stability and robustness of the method, we perform a simulation of fluid flow through a porous medium, of which the geometry is directly extracted from 3D µCT scan data.

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Section: Steady Navier-stokes Flow In a Quarter Annulus Domainsupporting

confidence: 87%

Skeleton-stabilized immersogeometric analysis for incompressible viscous flow problems

Hoang

Verhoosel

Qin

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…The first alternative rests on a complete theoretical basis; the second is common in practice but optimal order a priori bounds can not be established in general since the penalty parameter may become very large, see the discussion in [7]; and the third alternative was considered in [10] where a least squares term was added in the vicinity of the Dirichlet part of the boundary to provide the additional stability necessary to establish a priori error bounds. We refer to the overview article [5], and the recent conference proceedings [3] for an overview of current research on cut element methods.…”

Section: Introductionmentioning

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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“…An alternative is to compute the stabilization parameters based on global or element-wise generalized eigenvalue problems [16,40]. However, for the awkward cut situations, this may result in unbounded values resulting in similar problems known for penalty methods [11]. Therefore, we prefer the non-symmetric Nitsche's method herein with the main advantage that an additional stabilization is not required for imposing boundary conditions [3,41].…”

Section: Trace Femmentioning

confidence: 99%

A unified finite strain theory for membranes and ropes

Fries

Schöllhammer

2020

Computer Methods in Applied Mechanics and Engineering

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The finite strain theory is reformulated in the frame of the Tangential Differential Calculus (TDC) resulting in a unification in a threefold sense. Firstly, ropes, membranes and three-dimensional continua are treated with one set of governing equations. Secondly, the reformulated boundary value problem applies to parametrized and implicit geometries. Therefore, the formulation is more general than classical ones as it does not rely on parametrizations implying curvilinear coordinate systems and the concept of co-and contravariant base vectors. This leads to the third unification: TDC-based models are applicable to two fundamentally different numerical approaches. On the one hand, one may use the classical Surface FEM where the geometry is discretized by curved one-dimensional elements for ropes and two-dimensional surface elements for membranes. On the other hand, it also applies to recent Trace FEM approaches where the geometry is immersed in a higher-dimensional background mesh. Then, the shape functions of the background mesh are evaluated on the trace of the immersed geometry and used for the approximation. As such, the Trace FEM is a arXiv:1909.12640v1 [cs.CE] 27 Sep 2019 fictitious domain method for partial differential equations on manifolds. The numerical results show that the proposed finite strain theory yields higher-order convergence rates independent of the numerical methodology, the dimension of the manifold, and the geometric representation type.

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A note on the stability parameter in Nitsche’s method for unfitted boundary value problems

Cited by 42 publications

References 61 publications

Skeleton-stabilized immersogeometric analysis for incompressible viscous flow problems

Skeleton-stabilized immersogeometric analysis for incompressible viscous flow problems

A new least squares stabilized Nitsche method for cut isogeometric analysis

A unified finite strain theory for membranes and ropes

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