A gap element for treating non-matching discrete interfaces

Song, Yeo-Ul; Youn, Sung‐Kie; Park, K. C.

doi:10.1007/s00466-015-1186-6

Cited by 17 publications

(8 citation statements)

References 48 publications

(76 reference statements)

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“…In the literature, formulation (26) is sometimes called the localized Lagrange multipliers based formulation (we abbreviate this as LLM), where the term 'localized' is used to associate the multipliers λ C , λ L and u with the corresponding sub-domains, see e.g. [55][56][57]. Table 2 briefly summarizes the considered formulations.…”

Section: We Now Introduce Two Functions Onmentioning

confidence: 99%

A non-intrusive global/local approach applied to phase-field modeling of brittle fracture

Gerasimov

Noii

Allix

et al. 2018

Adv. Model. and Simul. in Eng. Sci.

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This paper aims at investigating the adoption of non-intrusive global/local approaches while modeling fracture by means of the phase-field framework. A successful extension of the non-intrusive global/local approach to this setting would pave the way for a wide adoption of phase-field modeling of fracture, already well established in the research community, within legacy codes for industrial applications. Due to the extreme difference in stiffness between the global counterpart of the zone to be analized locally and its actual response when undergoing extensive cracking, the main foreseen issues are robustness, accuracy and efficiency of the fixed point iterative algorithm which is at the core of the method. These issues are tackled in this paper. We investigate the convergence performance when using the native global/local algorithm and show that the obtained results are identical to the reference phase-field solution. We also equip the global/local solution update procedure with relaxation/acceleration techniques such as Aitken's 2 -method, the Symmetric Rank One and Broyden's methods and show that the iterative convergence can be improved significantly. Results indicate that Aitken's 2 -method is probably the most convenient choice for the implementation of the approach within legacy codes, as this method needs only tools already available for the so-called sub-modeling approach, a strategy routinely used in industrial contexts.

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Section: We Now Introduce Two Functions Onmentioning

confidence: 99%

A non-intrusive global/local approach applied to phase-field modeling of brittle fracture

Gerasimov

Noii

Allix

et al. 2018

Adv. Model. and Simul. in Eng. Sci.

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“…A simple nonmatching interface problem between two substructures is described in Figure 2. Through the interface compatibility conditions and total potential energy equation in the nonmatching problem, 13,14 the equations of motion can be written as

[\frac{d^{2}}{d t^{2}} M + K] x = f,

\frac{d^{2}}{d t^{2}} M + K = [\begin{array}{ccc} \frac{d^{2}}{d t^{2}} {boldM}_{s} + {boldK}_{s} & boldB & bold0 \\ {boldB}^{T} & bold0 & prefix- {boldL}_{f} \\ bold0 & prefix- {boldL}_{f}^{T} & bold0 \end{array}], x = [\begin{array}{ccc} {boldu}_{s} \\ {bold-italicλ}_{ℓ} \\ {boldu}_{b} \end{array}], f = [\begin{array}{ccc} {boldf}_{s} \\ bold0 \\ bold0 \end{array}],

where L f is the assembly Boolean matrix for the nonmatching interface. To obtain the assembly Boolean matrix L f , we first consider the interface compatibility conditions.…”

Section: A General Formulation Of the F‐cms (New) Methodsmentioning

confidence: 99%

“…Unlike the dual CB method, the localized Lagrange multipliers were utilized in the F‐CMS method. It provides unique independent interface constraints to prevent rank deficiency including the corner edge assemble case 10‐12 and easy interface treatments of non‐matching mesh problems may be available to satisfy the displacement compatibility and the force equilibrium conditions 13,14 . The F‐CMS method is also suitable for parallel algorithm such as the FE tearing and interconnect (FETI) method like the dual CB method 15,16 .…”

Section: Introductionmentioning

confidence: 99%

An iterative scheme of flexibility‐based component mode synthesis with higher‐order residual modal compensation

Chung

Kim²,

Chae

et al. 2021

Numerical Meth Engineering

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An Iterative Flexibility‐based Component Mode Synthesis (IF‐CMS) is presented for the model reduction of partitioned structural dynamic systems via localized Lagrange multipliers. A distinct IF‐CMS feature is the inclusion of hierarchical residual flexibility at each iteration, resulting in considerably improved accuracy compared with the classical Craig–Bampton (CB) CMS method. The present IF‐CMS method does not increase the number of substructural modes during the hierarchical iteration process. In particular, to alleviate ill‐conditioning during the iteration steps, the proposed IF‐CMS method adopts the so‐called (qd, α, ub)‐formulation by condensing Lagrange multipliers from the original F‐CMS method. The performance of the proposed method is evaluated through numerical examples, which illustrates improved accuracy over existing CMS methods.

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“…To formulate the coupling of different levels within Global-Local scheme, a single Lagrange multiplier method leads to redundant interface conditions in the case of many local domains (i.e., over-constrained condition for more than two domains, thus leads to the linear dependency of the imposed constraints), see for instance [70] and references therein. Therefore, inconsistency conditions due to the over-constrained interface displacement continuity appears and this leads to the non-unique solutions [71,70,29,80]. But this is not the case for the localized Lagrange multiplier (LLM) approach which provides no redundancy for the interface conditions and leads to unique and stable solutions [70].…”

Section: Introductionmentioning

confidence: 99%

“…This issue has been extensively studied in the context of the mortar methods [75,38,78]. In contrast, LLM through the introduction of an intermediate surface on which both displacements and forces are introduced as added variables thus offering a regularization of stiffness mismatch issues [80]. This is achieved by enforcing an additional weak from to our system of equations that are designed to satisfy both the displacement compatibility condition and force equilibrium conditions [80].…”

Section: Introductionmentioning

confidence: 99%

Multilevel Global-Local techniques for adaptive ductile phase-field fracture

Aldakheel,

Noii,

Wick

et al. 2021

Preprint

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This paper outlines a rigorous variational-based multilevel Global-Local formulation for ductile fracture. Here, a phase-field formulation is used to resolve failure mechanisms by regularizing the sharp crack topology on the local state. The coupling of plasticity to the crack phase-field is realized by a constitutive work density function, which is characterized through a degraded stored elastic energy and the accumulated dissipated energy due to plasticity and damage. Two different Global-Local approaches based on the idea of multiplicative Schwarz' alternating method are proposed: (i) A global constitutive model with an elastic-plastic behavior is first proposed, while it is enhanced with a single local domain, which, in turn, describes an elastic-plastic fracturing response. (ii) The main objective of the second model is to introduce an adoption of the Global-Local approach toward the multilevel local setting. To this end, an elastic-plastic global constitutive model is augmented with two distinct local domains; in which, the first local domain behaves as an elastic-plastic material and the next local domain is modeled due to the fracture state. To further reduce the computational cost, predictor-corrector adaptivity within Global-Local concept is introduced. An adaptive scheme is devised through the evolution of the effective global plastic flow (for only elastic-plastic adaptivity), and through the evolution of the local crack phase-field state (for only fracture adaptivity). Thus, two local domains are dynamically updated during the computation, resulting with two-way adaptivity procedure. The overall response of the Global-Local approach in terms of accuracy/robustness and efficiency is verified using single-scale problems. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

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A gap element for treating non-matching discrete interfaces

Cited by 17 publications

References 48 publications

A non-intrusive global/local approach applied to phase-field modeling of brittle fracture

A non-intrusive global/local approach applied to phase-field modeling of brittle fracture

An iterative scheme of flexibility‐based component mode synthesis with higher‐order residual modal compensation

Multilevel Global-Local techniques for adaptive ductile phase-field fracture

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