An analysis of the quasicontinuum method

By Jaroslaw Knap, Magdalena Ortiz. 2001

425 citations recorded by the source corpus.

The aim of this paper is to present a streamlined and fully three-dimensional version of the quasicontinuum (QC) theory of Tadmor et al. (Philos. Mag. A 73 (1996) 1529; Langmuir 12 (1996) 4529) and to analyze its accuracy and convergence characteristics. Specifically, we assess the effect of the summation rules on accuracy; we determine the rate of convergence of the method in the presence of strong singularities, such as point loads; and we assess the effect of the refinement tolerance, which controls the rate at which new nodes are inserted in the model, on the development of dislocation microstructures.

Subfields: Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/5da8bf18a8d7dddf99ee7d95ce41da449b36c3b5

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In Materials Chemistry

Citations in the corpus

On the mechanical modeling of the extreme softening/stiffening response of axially loaded tensegrity prisms (2014, methodology): “, 2012); and the numerical modeling of large-scale tensegrity lattices under general loading conditions, via the quasicontinuum method and local maximum-entropy schemes (Knap and Ortiz, 2001; Bompadre et al., 2012; Fraternali et al., 2012).”
A Unified Interpretation of Stress in Molecular Systems (2010, methodology): “Some common atomistic/continuum coupling methods are quasicontinuum [53, 59], coupling of lengthscales [51], cluster quasicontinuum [ 31 ], bridging domain [66], coupled atomistics and discrete dislocations [54], and heterogeneous multiscale methods [15], to name just a few.”
Analysis of the heterogeneous multiscale method for elliptic homogenization problems (2004, methodology): “Other numerical methods that use local microscale solvers to help extract macroscale behavior are found in [26, 27].”
A force-based coupling scheme for peridynamics and classical elasticity (2013, methodology): “Methods attempting to couple nonlocal and local models, particularly, atomistic and continuum ones, inlcuding the Quasicontinuum method [21, 36, 41] and the Arlequin aproach [5, 6, 8, 9], among others, are provided in the literature; for a benchmark study of different atomistic-to-continuum coupling”
An introduction to computational nanomechanics and materials (2004, background): “Because of the homogeneous deformation restriction that the Cauchy–Born rules enforces upon the underlying atomistic system, a non-local QC formulation [118] was developed, which allows atomistic defects such as dislocations to occur.”
A multiscale quasicontinuum method for dissipative lattice models and discrete networks (2014, methodology): “The QC formulations that depart from a force-equilibrium however (Knap and Ortiz, 2001; Kulkarni et al., 2008), also appear to be energetically inconsistent for conservative systems.”
A multiscale projection method for the analysis of carbon nanotubes (2004, background): “Remarks: Coarse scale formulation that does not use continuum constitutive update scheme has also been proposed in the quasi-continuum method [17].”
Structural solution using molecular dynamics: Fundamentals and a case study of epoxy-silica interface (2011, background): “…one of the earliest methods to deal with this problem, initially applied to crystalline materials such as metals (Tadmor et al., 1996; Shenoy et al., 1998; Shenoy et al., 1999; Knap and Ortiz, 2001; Miller and Tadmor, 2002; Curtin and Miller, 2003; Shimokawa et al., 2004; Miller and Tadmor, 2007).”
Atomistic/continuum coupling in computational materials science (2003, methodology): “The original implementation was for two-dimensional static equilibrium problems, but the method has been naturally extended to three-dimensional in both [41,54] and to ﬁnite temperatures by [58].”
A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods (2009, methodology): “The value of n can be determined in a number of ways [24], but one physically sensible approach is to assign all atoms within the Voronoi cell of node J to n as discussed earlier.”
Atomistic to continuum limits for computational materials science (2007, methodology): “Just to name one method among others, let us mention the quasicontinuum method, from [88, 106, 107, 125, 126,132–134], that more or less involves energy functionals of the form (1.”