T.
Belytschko and S. P. Xiao
International Journal for Multiscale Computational Engineering, Vol 1, 2003
This article develops coupling methods for continuum models with molecular models. Two methods are studied here: an overlapping domain decomposition method, which has overlapping domain; and an edge-to-edge decomposition method, which has an interface between the two models. These two methods enforce compatibility on the overlapping domain or interface nodes/atoms by the Lagrange multiplier method or the augmented Lagrangian method.
This paper is available in PDF form .
S. P. Xiao and T. Belytschko
Computer Methods in Applied Mechanics and Engineering, Vol
193, 1645-1669, 2004
A multiscale method based on the overlapping domain decomposition called the bridging domain method for coupling continuum models with molecular models is described. In the bridging domain coupling method, the continuum domain and molecular domain are overlaid at the interface. The total Hamiltonian is taken to be a linear combination of the continuum and molecular Hamiltonian. We enforce the compatibility on the overlapping subdomain by Lagrange multipliers or by the augmented Lagrangian method. An explicit algorithm for dynamics solutions is developed. Results show that this multiscale method can avoid spurious wave reflections at the molecular/continuum interface without any additional filtering procedures. A multiple-time-step algorithm is also implemented and it saves considerable computation time.
This paper is available in PDF form .
S. P. Xiao and W. X. Yang
International Journal of Computational Methods, Vol
2(3): 293-313, 2005
Since meshfree particle methods have advantages on
simulating the problems involving extremely large deformations, fractures etc.,
they become attractive options to be used in the hierarchical multiscale
modeling to approximate a large number of atoms. This paper proposes a
nanoscale meshfree particle method with the implementation of the
quasicontinuum technique. The intrinsic properties of the material associated
with each particle will be sought from the atomic level via the Cauchy-Born
rule. The studies of a nano beam and a cracked nano plate will show that such a
hierarchical modeling can be beneficial from the advantages of meshfree particle
methods.
This paper is available in PDF form
.
S. P. Xiao
and W. X. Yang
International Journal of Computational Science and Engineering, 2(3/4):
213-220, 2007
In this paper, a meshfree particle method with the stress
point integration scheme is studied. It has been shown that this meshfree
particle method with Lagrangian kernel can provide a
stable method. A finite element mapping technique is introduced to insert
stress points and to calculate volumes associated with particles/stress points
so that the triangulation and the Voronoi diagram can
be avoided. This meshfree particle method can be used for nanoscale simulations
via the implementation of the Cauchy-Born rule. It can also be coupled with
molecular dynamics based on the bridging domain coupling technique.
This paper is available in PDF form .
S.
P. Xiao, and W. X. Yang
Computational Materials Science, 37: 374-379, 2006
In this study, we
develop a temperature-related Cauchy-Born (TCB) rule for multiscale modeling of
crystalline solids based on the assumptions that deformation is locally
homogeneous and atoms have the same local vibration mode. When employing the
TCB rule in the nanoscale continuum approximation, the first Piola-Kirchhoff
stress can be explicitly computed as the first derivative of the Helmholtz free
energy density to the deformation gradient. Since the Helmholtz free energy is temperature-dependent,
multiscale methods consisting of the TCB rule embedded continuum model can be
used to elucidate temperature-related physical phenomena at the nanoscale.
Stress analyses of canonical ensembles verify the continuum approximation with
the TCB rule by comparing the calculated Cauchy stresses with the outcomes of
molecular dynamics simulations. As an application of the TCB rule in multiscale
modeling, the nanoscale meshfree particle method with the TCB rule demonstrates
the same crack propagation phenomenon in a nanoplate
as molecular dynamics. This example shows that the temperature effects are
significant on the crack propagation speed when the temperature is in a
particular range.
This paper is available in PDF form .
S. P. Xiao, and W.
X. Yang
International Journal for Numerical Methods in Engineering,
69: 2099-2125, 2007
A new homogenization technique, the temperature-related
Cauchy–Born (TCB) rule, is proposed in this paper with the consideration
of the free energy instead of the potential energy. Therefore, temperature
effects at the nanoscale can be investigated using continuum approximation with
the implementation of the TCB rule. The TCB rule is verified via stress
analyses of several crystalline solids. Temperaturerelated
material instability is also studied. In addition, a new hierarchical
multiscale method is developed through implementing the TCB rule into meshfree
particle methods. Quasicontinuum meshfree particle simulations are conducted to
investigate bending of nanobeams, crack propagation
in nanoplates and a three-dimensional nanoindentation problem.
This paper is available in PDF form
.
W. X. Yang, and S. P.
Xiao
Computational Materials Science, 41: 431-439, 2008
In this
paper, we first study the material stability of nanostructured materials via
the continuum linearized stability analysis technique
with the temperature-related Cauchy-Born (TCB) rule. As a temperature-related
homogenization technique, the TCB rule considers the free energy instead of the
potential so that temperature effects on material stability can be
investigated. In addition, we develop a thermo-mechanical coupling model
through implementing the thermal diffusion equation into nanoscale continuum
approximation. Crack propagation at a nanoplate is
studied as an example. Since the nanoscale phenomenon of bond breaking is
involved when crack propagates, temperature increasing around the crack tip due
to the released potential is considered in our thermo-mechanical coupling
model.
This paper is available in PDF form .
S. P. Xiao, J. Ni, and S. W. Wang
Journal of Computational and Theoretical Nanoscience,
2008, in press
This paper presents a study on the feasibility of applying high performance computing (HPC) to the Bridging Domain Multiscale (BDM) method, so that featured scalable multiscale computations can be achieved. Wave propagation in a molecule chain through an entire computational domain is employed as an example to demonstrate its applicability and computing performance when multiscale-based simulations are conducted in a large-scale parallel computing environment. In addition, the conceptual idea and computing framework using Grid computing technologies is proposed to enhance future multiscale computations in nanotechnology.
This paper is available in PDF form .