The Nitsche method applied to a class of mixed-dimensional coupling problems

Rabinovich, Daniel; Ofir, Yoav; Givoli, Dan

doi:10.1016/j.cma.2014.02.006

Cited by 9 publications

(4 citation statements)

References 43 publications

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“…By coupling this shell model to a fully 3D phase‐field description, the reduced computational effort of a structural element is preserved while refraining from any assumptions or simplifications in the phase‐field. Although the idea of combining solid and structural elements to describe various parts of a structure, such as done for instance in References 59‐63, or even different parts within each other for reinforced structures as for example, in References 64‐66 is not new, the idea to combine solid and structural elements for different fields of a coupled BVP has not been presented before.…”

Section: Introductionmentioning

confidence: 99%

Phase‐field modeling of brittle fracture along the thickness direction of plates and shells

Ambati

Heinzmann

Seiler

et al. 2022

Numerical Meth Engineering

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The prediction of fracture in thin-walled structures is decisive for a wide range of applications. Modeling methods such as the phase-field method usually consider cracks to be constant over the thickness which, especially in load cases involving bending, is an imperfect approximation. In this contribution, fracture phenomena along the thickness direction of structural elements (plates or shells) are addressed with a phase-field modeling approach. For this purpose, a new, so called "mixed-dimensional" model is introduced, which combines structural elements representing the displacement field in the two-dimensional shell midsurface with continuum elements describing a crack phase-field in the three-dimensional solid space. The proposed model uses two separate finite element discretizations, where the transfer of variables between the coupled twoand three-dimensional fields is performed at the integration points which in turn need to have corresponding geometric locations. The governing equations of the proposed mixed-dimensional model are deduced in a consistent manner from a total energy functional with them also being compared to existing standard models. The resulting model has the advantage of a reduced computational effort due to the structural elements while still being able to accurately model arbitrary through-thickness crack evolutions as well as partly along the thickness broken shells due to the continuum elements. Amongst others, the higher accuracy as well as the numerical efficiency of the proposed model are tested and validated by comparing simulation results of the new model to those obtained by standard models using numerous representative examples.

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Section: Introductionmentioning

confidence: 99%

Phase‐field modeling of brittle fracture along the thickness direction of plates and shells

Ambati

Heinzmann

Seiler

et al. 2022

Numerical Meth Engineering

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“…The most typical methods include the asymptotic homogenization method [5,6], the multilevel finite element method (FE2) [7,8], the representative volume element method (RVE) [9], the heterogeneous multiscale method (HMM) [10,11], multiscale methods based on Nitsche method [12,13], multiscale methods based on boundary conditions transfer [14,15], domain decomposition type methods [16,17] and other homogenization type methods [18,19] etc. Otherwise, many recent researchers focused on the cases of beam and plate structures based on homogenization concepts.…”

Section: Introductionmentioning

confidence: 99%

A two-level method for static and dynamic analysis of multilayered composite beam and plate

Zhang

Yin

Zhang

et al. 2016

Finite Elements in Analysis and Design

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“…The critical role played by the wave number in Helmholtz equation needs to be investigated in the framework of Nitsche's formulation. Indeed, the authors in [32] reported large error for certain choices of angular frequencies which they characterized as resonances. Such a phenomenon can actually be connected to the inf-sup constant.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Hanso and Hermansson adopted a Nitsche's method to couple non‐matching meshes in fluid‐structure vibration problems; they reported promising results for the eigenstate analysis of a fluid domain enclosed inside an elastic container. Rabinovich et al, applied Nitsche's method to couple mixed‐dimensional problems considering time‐harmonic response in elastic structures. A fundamental component missing in the current literature is a detailed analysis taking into consideration the non‐elliptic nature of the problem.…”

Section: Introductionmentioning

confidence: 99%

Nitsche's method for Helmholtz problems with embedded interfaces

Zou

Aquino

Harari

2016

Numerical Meth Engineering

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SUMMARY In this work, we use Nitsche’s formulation to weakly enforce kinematic constraints at an embedded interface in Helmholtz problems. Allowing embedded interfaces in a mesh provides significant ease for discretization, especially when material interfaces have complex geometries. We provide analytical results that establish the well-posedness of Helmholtz variational problems and convergence of the corresponding finite element discretizations when Nitsche’s method is used to enforce kinematic constraints. As in the analysis of conventional Helmholtz problems, we show that the inf-sup constant remains positive provided that the Nitsche’s stabilization parameter is judiciously chosen. We then apply our formulation to several 2D plane-wave examples that confirm our analytical findings. Doing so, we demonstrate the asymptotic convergence of the proposed method and show that numerical results are in accordance with the theoretical analysis.

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The Nitsche method applied to a class of mixed-dimensional coupling problems

Cited by 9 publications

References 43 publications

Phase‐field modeling of brittle fracture along the thickness direction of plates and shells

Phase‐field modeling of brittle fracture along the thickness direction of plates and shells

A two-level method for static and dynamic analysis of multilayered composite beam and plate

Nitsche's method for Helmholtz problems with embedded interfaces

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