A variational formulation of dissipative quasicontinuum methods

Rokoš, Ondřej; Beex, Lars; Zeman, Jan; Peerlings, Rhj Ron

doi:10.1016/j.ijsolstr.2016.10.003

Cited by 18 publications

(18 citation statements)

References 43 publications

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“…This may be considered reasonable because right-angled triangles lead to significantly smaller summation errors, cf. [18], and because these triangles also naturally avoid artificial deformations in transition regions that would lead to spurious mesh refinements and non-physical evolution of internal variables. For the refinement, the standard Rivara [45] algorithm conserving non-degeneracy, conformity, and smoothness, is used for each marked element (K ∈ I).…”

Section: : Endmentioning

confidence: 99%

“…where π k red,α denotes the reduced incremental site energy with condensed internal variables in analogy to Π k red defined in (18)…”

Section: Appendix B Explicit Forms Of Gradients and Hessiansmentioning

confidence: 99%

“…The theoretical framework employed in this contribution is closely related to our variational QC formulation for hardening plasticity discussed in [18]. The present work can be viewed as an extension towards lattices with localized damage and with an adaptive refinement strategy.…”

Section: Introductionmentioning

confidence: 99%

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An adaptive variational Quasicontinuum methodology for lattice networks with localized damage

Rokoš

Peerlings

Zeman

et al. 2017

Numerical Meth Engineering

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Lattice networks with dissipative interactions can be used to describe the mechanics of discrete mesostructures of materials such as 3D-printed structures and foams. This contribution deals with the crack initiation and propagation in such materials and focuses on an adaptive multiscale approach that captures the spatially evolving fracture. Lattice networks naturally incorporate non-locality, large deformations and dissipative mechanisms taking place inside fracture zones. Because the physically relevant length scales are significantly larger than those of individual interactions, discrete models are computationally expensive. The Quasicontinuum (QC) method is a multiscale approach specifically constructed for discrete models. This method reduces the computational cost by fully resolving the underlying lattice only in regions of interest, while coarsening elsewhere. In this contribution, the (variational) QC is applied to damageable lattices for engineering-scale predictions. To deal with the spatially evolving fracture zone, an adaptive scheme is proposed. Implications induced by the adaptive procedure are discussed from the energy-consistency point of view, and theoretical considerations are demonstrated on two examples. The first one serves as a proof of concept, illustrates the consistency of the adaptive schemes and presents errors in energies. The second one demonstrates the performance of the adaptive QC scheme for a more complex problem. AN ADAPTIVE VARIATIONAL QUASICONTINUUM METHODOLOGY FOR LATTICE 175 beam (fibre or yarn) versus that of the network. Second, the formulation and implementation of lattice models is generally significantly easier compared with that of alternative continuum models. Large deformations, large yarn reorientations and fracture are easier to formulate and implement (cf. e.g. the continuum model of Peng and Cao [5] that deals with large yarn reorientations). Thanks to the simplicity and versatility of lattice networks, they are furthermore used for the description of heterogeneous cohesive-frictional materials such as concrete. The reason is that discrete models can realistically represent distributed microcracking with gradual softening, implement material structure with inhomogeneities, capture non-locality of damage processes and reflect deterministic or stochastic size effects. Examples of the successful use of lattice models for such materials are given in [6][7][8][9].As lattice models are typically constructed at the meso-scale, micro-scale or nano-scale, they require reduced-model techniques to allow for application-scale simulations. A prominent example is the Quasicontinuum (QC) method, which specifically aims at discrete lattice models. The QC method was originally introduced for conservative atomistic systems by Tadmor et al. [10] and extended in numerous aspects later on, see for example [11][12][13]. Subsequent generalizations for lattices with dissipative interactions (e.g. plasticity and bond sliding) were provided in [14,15]. In principle, the QC is a numerical p...

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Section: : Endmentioning

confidence: 99%

“…where π k red,α denotes the reduced incremental site energy with condensed internal variables in analogy to Π k red defined in (18)…”

Section: Appendix B Explicit Forms Of Gradients and Hessiansmentioning

confidence: 99%

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An adaptive variational Quasicontinuum methodology for lattice networks with localized damage

Rokoš

Peerlings

Zeman

et al. 2017

Numerical Meth Engineering

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“…For the sake of brevity, we limit ourselves to the time-discrete setting; the general theory is discussed in Mielke and Roubíček (2015), whereas applications to continuous systems can be found, e.g., in Mühlhaus and Aifantis (1991); Han and Reddy (1995); Bourdin et al (2000); Mielke et al (2002); Mielke (2003); Bourdin (2007); Bourdin et al (2008); Burke et al (2010); Pham et al (2011);Hofacker and Miehe (2012); Jirásek and Zeman (2015); Mesgarnejad et al (2015). Applications to lattice networks and QC are presented in Rokoš et al (2016Rokoš et al ( , 2017 for isotropic hardening plasticity and localized damage, respectively.…”

Section: Variational Formulation Of Lattice Network With Localized Dmentioning

confidence: 99%

eXtended variational quasicontinuum methodology for lattice networks with damage and crack propagation

Rokoš

Peerlings

Zeman

2017

Computer Methods in Applied Mechanics and Engineering

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Lattice networks with dissipative interactions are often employed to analyze materials with discrete micro-or meso-structures, or for a description of heterogeneous materials which can be modelled discretely. They are, however, computationally prohibitive for engineering-scale applications. The (variational) QuasiContinuum (QC) method is a concurrent multiscale approach that reduces their computational cost by fully resolving the (dissipative) lattice network in small regions of interest while coarsening elsewhere. When applied to damageable lattices, moving crack tips can be captured by adaptive mesh refinement schemes, whereas fully-resolved trails in crack wakes can be removed by mesh coarsening. In order to address crack propagation efficiently and accurately, we develop in this contribution the necessary generalizations of the variational QC methodology. First, a suitable definition of crack paths in discrete systems is introduced, which allows for their geometrical representation in terms of the signed distance function. Second, special function enrichments based on the partition of unity concept are adopted, in order to capture kinematics in the wakes of crack tips. Third, a summation rule that reflects the adopted enrichment functions with sufficient degree of accuracy is developed. Finally, as our standpoint is variational, we discuss implications of the mesh refinement and coarsening from an energy-consistency point of view. All theoretical considerations are demonstrated using two numerical examples for which the resulting reaction forces, energy evolutions, and crack paths are compared to those of the direct numerical simulations.

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“…In last few years, a variational formulation of dissipative QC method has been done by Rokoš at al. [25], a goal-oriented adaptive version of QC algorithm has been introduced in [26] or a meshless QC method has been developed by Kochmann research group [27]. But the application of all mentioned QC methods is still restricted only to systems with regular geometry of particles.…”

Section: Introductionmentioning

confidence: 99%

Quasicontinuum method extended to irregular lattices

Mikeš

Jirásek

2017

Computers & Structures

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The quasicontinuum (QC) method, originally proposed by Tadmor, Ortiz and Phillips in 1996, is a computational technique that can efficiently handle regular atomistic lattices by combining continuum and atomistic approaches. In the present work, the QC method is extended to irregular systems of particles that represent a heterogeneous material. The paper introduces five QC-inspired approaches that approximate a discrete model consisting of particles connected by elastic links with axial interactions. Accuracy is first checked on simple examples in two and three spatial dimensions. Computational efficiency is then assessed by performing three-dimensional simulations of an L-shaped specimen with elastic-brittle links. It is shown that the QC-inspired approaches substantially reduce the computational cost and lead to macroscopic crack trajectories and global load-displacement curves that are very similar to those obtained by a fully resolved particle model.

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A variational formulation of dissipative quasicontinuum methods

Cited by 18 publications

References 43 publications

An adaptive variational Quasicontinuum methodology for lattice networks with localized damage

An adaptive variational Quasicontinuum methodology for lattice networks with localized damage

eXtended variational quasicontinuum methodology for lattice networks with damage and crack propagation

Quasicontinuum method extended to irregular lattices

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