23 research outputs found
Goal-Oriented Adaptive Mesh Refinement for the Quasicontinuum Approximation of a Frenkel-Kontorova Model
The quasicontinuum approximation is a method to reduce the atomistic degrees
of freedom of a crystalline solid by piecewise linear interpolation from
representative atoms that are nodes for a finite element triangulation. In
regions of the crystal with a highly nonuniform deformation such as around
defects, every atom must be a representative atom to obtain sufficient
accuracy, but the mesh can be coarsened away from such regions to remove
atomistic degrees of freedom while retaining sufficient accuracy. We present an
error estimator and a related adaptive mesh refinement algorithm for the
quasicontinuum approximation of a generalized Frenkel-Kontorova model that
enables a quantity of interest to be efficiently computed to a predetermined
accuracy.Comment: 15 pages, 6 figures, 3 table
An Analysis of the Effect of Ghost Force Oscillation on Quasicontinuum Error
The atomistic to continuum interface for quasicontinuum energies exhibits
nonzero forces under uniform strain that have been called ghost forces. In this
paper, we prove for a linearization of a one-dimensional quasicontinuum energy
around a uniform strain that the effect of the ghost forces on the displacement
nearly cancels and has a small effect on the error away from the interface. We
give optimal order error estimates that show that the quasicontinuum
displacement converges to the atomistic displacement at the optimal rate O()
in the discrete norm and O() in the norm for
where is the interatomic spacing. We also give a proof
that the error in the displacement gradient decays away from the interface to
O() at distance O() in the atomistic region and distance O()
in the continuum region. E, Ming, and Yang previously gave a counterexample to
convergence in the norm for a harmonic interatomic potential.
Our work gives an explicit and simplified form for the decay of the effect of
the atomistic to continuum coupling error in terms of a general underlying
interatomic potential and gives the estimates described above in the discrete
and norms.Comment: 14 pages, 1 figur
A Dynamic Atomistic-Continuum Method for the Simulation of Crystalline Materials
We present a coupled atomistic-continuum method for the modeling of defects
and interface dynamics of crystalline materials. The method uses atomistic
models such as molecular dynamics near defects and interfaces, and continuum
models away from defects and interfaces. We propose a new class of matching
conditions between the atomistic and continuum regions. These conditions ensure
the accurate passage of large scale information between the atomistic and
continuum regions and at the same time minimize the reflection of phonons at
the atomistic-continuum interface. They can be made adaptive if we choose
appropriate weight functions. We present applications to dislocation dynamics,
friction between two-dimensional crystal surfaces and fracture dynamics. We
compare results of the coupled method and the detailed atomistic model.Comment: 48 pages, 20 figure
Efficient a Posteriori Error Control of a Consistent Atomistic/Continuum Coupling Method for Two Dimensional Crystalline Defects
Adaptive atomistic/continuum (a/c) coupling method is an important method for
the simulation of material and atomistic systems with defects to achieve the
balance of accuracy and efficiency. Residual based a posteriori error estimator
is often employed in the adaptive algorithm to provide an estimate of the error
of the strain committed by applying the continuum approximation for the
atomistic system and the finite element discretization in the continuum region.
In this work, we propose a theory based approximation for the residual based a
posteriori error estimator which greatly improves the efficiency of the
adaptivity. In particular, the numerically expensive modeling residual is only
computed exactly in a small region around the coupling interface but replaced
by a theoretically justified approximation by the coarsening residual outside
that region. We present a range of adaptive computations based on our modified
a posteriori error estimator and its variants for different types of
crystalline defects some of which are not considered in previous related
literature of the adaptive a/c methods. The numerical results show that,
compared with the original residual based error estimator, the adaptive
algorithm using the modified error estimator with properly chosen parameters
leads to the same optimal convergence rate of the error but reduces the
computational cost by one order with respect to the number of degrees of
freedom
A Posteriori Error Estimates for Energy-Based Quasicontinuum Approximations of a Periodic Chain
We present a posteriori error estimates for a recently developed
atomistic/continuum coupling method, the Consistent Energy-Based QC Coupling
method. The error estimate of the deformation gradient combines a residual
estimate and an a posteriori stability analysis. The residual is decomposed
into the residual due to the approximation of the stored energy and that due to
the approximation of the external force, and are bounded in negative Sobolev
norms. In addition, the error estimate of the total energy using the error
estimate of the deformation gradient is also presented. Finally, numerical
experiments are provided to illustrate our analysis