A stabilised finite element method for the plate obstacle problem

Gustafsson, Tom; Stenberg, Rolf; Videman, Juha H.

doi:10.1007/s10543-018-0728-7

Cited by 7 publications

(17 citation statements)

References 41 publications

(82 reference statements)

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“…The stabilisation parameter is chosen as α = 10 −5 , and the marking and adaptive refinement strategies are as in [19]. For this problem the contact set is one-dimensional.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…0,E , with V n (u h ) and M nn (u h ) denoting the jumps over interior edges of the Kirchhoff shear force and the normal moment, see [18,19] for more details.…”

Section: Nitsche's Methodsmentioning

confidence: 99%

“…In this note, we will review our latest results on conforming stabilised finite element methods for the plate obstacle problem, cf. [19]. Our method is based on a saddle point formulation with the contact force appearing as an additional unknown (Lagrange multiplier).…”

Section: Introductionmentioning

confidence: 99%

“…Numerical approximation of fourth-order variational inequalities has been previously studied, e.g., in [24,4,3,16,2] but to our knowledge Nitsche's or stabilised methods using conforming C 1 -continuous elements were for the first time proposed and rigorously analysed in [19].…”

Section: Introductionmentioning

confidence: 99%

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Nitsche’s Method for the Obstacle Problem of Clamped Kirchhoff Plates

Gustafsson

Stenberg

Videman

2019

Lecture Notes in Computational Science and Engineering

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“…The stabilisation parameter is chosen as α = 10 −5 , and the marking and adaptive refinement strategies are as in [19]. For this problem the contact set is one-dimensional.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…0,E , with V n (u h ) and M nn (u h ) denoting the jumps over interior edges of the Kirchhoff shear force and the normal moment, see [18,19] for more details.…”

Section: Nitsche's Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nitsche’s Method for the Obstacle Problem of Clamped Kirchhoff Plates

Gustafsson

Stenberg

Videman

2019

Lecture Notes in Computational Science and Engineering

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“…With regard to unilateral contact of thin structures, a Nitsche approximation of the unilateral contact between a Kirchhoff-Love plate and a rigid obstacle is presented in [24,23]. In these references, the origin of the proposed method is the Kirchhoff-Love plate model subject to a unilateral contact condition.…”

Section: Introduction and Kinematics Of Plate Modelsmentioning

confidence: 99%

Nitsche-based models for the unilateral contact of plates

2021

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This paper aims to present different Nitsche-based models for the unilateral contact of plate structures. Our analysis is based on the consideration of Nitsche’s method on a 3D structure with kinematic assumptions of thin or thick plate theories. This approach is compared to that of Gustafsson, Stenberg and Videman which consists of Nitsche’s method applied directly on a 2D plate model. To simplify the presentation, we focus on the contact of an elastic plate with a rigid obstacle. The different approaches are compared numerically in terms of reliability compared to the 3D elastic model. The aim of this paper is to present different Nitsche based models for the unilateral contact of plate structures. Our analysis is based on the consideration of Nitsche’s method on a 3D structure with kinematic assumptions of thin or thick plate theories. This approach is compared to the one of Gustafsson, Stenberg and Videman which consists of Nitsche's method directly applied on a 2D plate model. To simplify the presentation, we focus on the contact of an elastic plate with a rigid obstacle. The different approaches are numerically compared in terms of reliability compared to the 3D elastic model.

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First-order least-squares method for the obstacle problem

Führer

2019

Numer. Math.

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We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error estimates including the case of non-conforming convex sets are given and optimal convergence rates are shown for the lowest-order case. We provide also a posteriori bounds that can be be used as error indicators in an adaptive algorithm. Numerical studies are presented.Date: October 14, 2018. 2010 Mathematics Subject Classification. 65N30, 65N12, 49J40. Key words and phrases. First-order system, least-squares method, variational inequality, obstacle problem, a priori analysis, a posteriori analysis.Acknowledgment. This work was supported by CONICYT through FONDECYT project "Least-squares methods for obstacle problems" under grant 11170050. 1 arXiv:1801.09622v1 [math.NA] 29 Jan 2018on. For simplicity assume a zero obstacle (the remainder of the paper deals with general non-zero obstacles). Then, the problem reads −∆u ≥ f, u ≥ 0, (−∆u − f )u = 0 in some domain Ω and u| ∂Ω = 0. Introducing the Lagrange multiplier (or reaction force) λ = −∆u − f and σ = ∇u, we rewrite the problem as a first-order system, see also [2,3,9,16],Note that f ∈ L 2 (Ω) does not imply more regularity for u so that λ ∈ H −1 (Ω) is only in the dual space in general. However, observe that div σ + λ = −f ∈ L 2 (Ω) and therefore the functionalwhere · , · denotes a duality pairing, is well-defined for div σ + λ ∈ L 2 (Ω). We will show that minimizing J over a convex set with the additional linear constraints u ≥ 0, λ ≥ 0 is equivalent to solving the obstacle problem. We will consider the variational inequality associated to this problem with corresponding bilinear form a(·, ·). An issue that arises is that a(·, ·) is not necessarily coercive. However, as it turns out, a simple scaling of the first term in the functional ensures coercivity on the whole space. In view of the aforementioned properties, this means that our method is unconstrained stable. The recent work [16] based on a Lagrange formulation (without reformulation to a first-order system) considers augmenting the trial spaces with bubble functions (mixed method) resp. adding residual terms (stabilized method) to obtain stability.Furthermore, we will see that the functional J evaluated at some discrete approximation (u h , σ h , λ h ) with u h , λ h ≥ 0 is an upper bound for the error. Note that for λ h ∈ L 2 (Ω) the duality λ h , u h reduces to the L 2 inner product. Thus, all the terms in the functional can be localized and used as error indicators.Additionally, we will derive and analyse other variational inequalities that are also based on the first-order reformulation. The resulting methods are quite similar to the least-squares scheme since they share the same residual terms. The only difference is that the compatibility condition λu = 0 is incorporated in a different, non-symmetric, way. We will present a uniform analysis that covers the least-squa...

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A stabilised finite element method for the plate obstacle problem

Cited by 7 publications

References 41 publications

Nitsche’s Method for the Obstacle Problem of Clamped Kirchhoff Plates

Nitsche’s Method for the Obstacle Problem of Clamped Kirchhoff Plates

Nitsche-based models for the unilateral contact of plates

First-order least-squares method for the obstacle problem

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