A method to reduce the degrees freedom in molecular mechanics simulation is presented. Although the approach is formulated for amorphous materials in mind, it is equally applicable to crystalline materials. The method can be selectively applied to regions where molecular displacements are expected to be small while simultaneously using classical molecular mechanics (MM) for regions undergoing large deformation. The accuracy and computational efficiency of the approach is demonstrated through the simulation of a polymer-like substrate being indented by a rigid hemispherical indentor. The region directly below the indentor is modelled by classical molecular mechanics while the region further away has the degrees of freedom (DOFs) reduced by about 50 times. The results of automatically reverting regions of reduced DOFs back to classical MM also demonstrate the capability of performing adaptive simulations

Deng, M.

Lim, K. M.

Tan, V. B. C.

Tay, T. E.

University of Queensland eSpace

5th Australasian Congress on Applied Mechanics, ACAM 2007  
10-12 December 2007, Brisbane, Australia 
Adaptive and coupled continuum-molecular mechanics simulations of 
amorphous materials 
 
V.B.C. Tan, M. Deng, K.M. Lim and T.E. Tay 
 
Department of Mechanical Engineering, National University of Singapore, Singapore 117576 
Abstract: A method to reduce the degrees freedom in molecular mechanics simulation is presented. 
Although the approach is formulated for amorphous materials in mind, it is equally applicable to 
crystalline materials. The method can be selectively applied to regions where molecular displacements 
are expected to be small while simultaneously using classical molecular mechanics (MM) for regions 
undergoing large deformation. The accuracy and computational efficiency of the approach is 
demonstrated through the simulation of a polymer-like substrate being indented by a rigid 
hemispherical indentor. The region directly below the indentor is modelled by classical molecular 
mechanics while the region further away has the degrees of freedom (DOFs) reduced by about 50 
times. The results of automatically reverting regions of reduced DOFs back to classical MM also 
demonstrate the capability of performing adaptive simulations. 
Keywords: Pseudo amorphous cell, multiscale simulation, molecular simulation, adaptive simulations, 
amorphous materials. 
1 Introduction 
Many recent technological advances have been made possible through an understanding of materials 
and fabrication of structures at the molecular level. At this length scale, where molecular interactions 
dominate, mechanical analyses are almost always performed using molecular mechanics (MM) 
simulations. While they have provided important insights into the mechanical properties of material 
systems at the nanometer scale, MM simulations are limited to systems of at most micrometer 
dimensions because of the large number of molecules involved. In situations where events at the 
molecular level are strongly coupled to events at a much larger scale, multiscale simulation techniques 
to bridge molecular and continuum levels are needed to accurately describe these multi-physics 
problems. A multiscale simulation technique, formulated to accommodate the amorphous nature of 
polymer systems, is presented in this paper.  
 
2 Multiscale methods 
Literature on multiscale models with regards to coupled atomistics and continua can be categorized as 
either (i) formulation of new multiscale approaches, (ii) improvements to existing models or (iii) 
application of multiscale models. Of the various multiscale approaches that have been proposed, 
those that have seen significant applications can be broadly classified into (i) Coarsed Grained 
Molecular Dynamics, (ii) handshake models, (iii) Quasicontinuum models and (iv) projection models. 
 
In Coarsed Grained Molecular Dynamics (CGMD), not all molecules are represented independently in 
the computational model. Instead, clusters of molecules are grouped together to form a bead or a 
grain. Each bead is then treated as a large molecule in what is then essentially classical Molecular 
Dynamics simulation [1-3]. 
 
Another common approach in multiscale modelling, is through the use of a handshake region in the 
computational model. The handshake region bridges the atomistic domain with the continuum domain 
of the model [4,5]. The main idea in handshaking is to allow the atomistic domain and the continuum 
domain to overlap. At the overlap, a field variable (normally the potential energy) is assumed to take 
on a weighted combination of the magnitudes of the same variable in the continuum and atomistic 
domains. The weight is normally a function that decreases monotonically from unity to zero through 
the overlap. 
  
In the Quasicontinuum (QC) method, there is essentially no distinction between atomistic and 
continuum description of the problem domain. This is achieved by partitioning the domain into non-
overlapping cells akin to the finite element method. The cells overlay the molecules of the material 
with the vertices of the cell coinciding with some (not all) pre-selected representative molecules. The 
degrees of freedom are the position (or displacement) of the representative molecules. The QC 
method seeks the configuration of representative molecules that gives the minimum potential energy 
of the computational domain. The two main simplifications in the QC method are that (i) the position of 
each molecule in a cell is taken to be a function of the representative molecules of the cell and (ii) the 
average energy of a cell can be approximated from a group of molecules within the cell [6-9].  
 
The last group of multiscale methods is the projection method. By projection, it means that the field 
variable, usually the displacement of the molecules, is projected onto a reduced solution space. The 
idea stems from classical works in decomposing a complete solution into fine and coarse scales and 
solving for the fine scale only in regions that require it [10]. The fine scale solution subspace is 
orthogonal to the coarse scale solution subspace so that the total solution is simply the addition of the 
fine and coarse scale solutions. The method starts by defining the solution space for the coarse scale, 
which again is normally similar to the finite element solution space. The fine scale solution is then 
defined to be the difference between the complete and coarse scale solution. The coarse scale 
solution is usually obtained from minimizing some L2 error norm, e.g. minimizing the sum of the 
squares of error in molecular displacement. 
 
3 The Pseudo Amorphous Cell 
While the abovementioned methods have been formulated with general applications in mind, none 
have been applied to model the mechanical behavior of polymers. The mechanical modeling of 
polymers even at solely atomistic scale is by itself an area that has not been firmly established. 
Different methods of modeling polymers are still being proposed. Modeling polymers is complex 
because they are generally amorphous and molecular interaction comprises pair-wise bonds, long 
range non-bond interact ion, bond bending and bond torsion. Moreover polymer chains can comprise 
more than 10,000 monomers per chain. Unlike lattices of metal atoms, the equilibration of polymer 
chains in bulk normally does not end up with the same equilibrated configurations. One way of 
determining the mechanical properties of polymers in bulk is through the use of the Amorphous Cell 
[11]. In this method, the bulk polymer model comprises a periodic assembly of repeating parent 
polymer chains. The repeat intervals in the three spatial directions are conveniently represented by a 
cuboid or cell. The parent chains are obtained by equilibrating chains of specified number of 
monomers with the constraints that the chains replicate themselves according to the dimensions of the 
cell. 
 
The Pseudo Amorphous Cell (PAC) method for multiscale simulation in this paper makes use of the 
Amorphous Cell as building blocks of polymer domains to be investigated. Rather than solving for the 
position of all the molecules of the constituent Amorphous Cell, displacement of the vertices of the 
cells (a.k.a. nodes) are the unknowns. The molecular displacements are then recovered from the 
nodal displacement through a mapping function, T. The PAC method allows regions where atomistic 
details are important to be modeled by classical molecular mechanics (MM). The connection between 
the PAC cell regions and the MM region is seamlessly effected by augmenting the T transformation 
with unit matrices where the molecular displacements are to be solved directly from MM. The 
computational effort to solve the coupled system of equations can be shown to be dependent on the 
size of the MM region if the PAC cell regions do not undergo large deformation. This is in 
correspondence with the way the polymer domain is expected to be partitioned  PAC cell and 
continuum methods at regions of small strain gradients with computationally costly MM domains at 
regions of high deformation. In the PAC method, the T matrix is derived to extract the molecular 
displacements according to the modes of deformation of the cells (i.e. stretch and shear) which turn 
out to be very different from linear interpolation. 
 
  
4 Results and discussion 
To demonstrate the PAC cell method, simulations of nanoindentation of a polymer-like substrate were 
performed. The polymer domain is 5 cells across by 16 cells deep. The PAC cell method is applied to 
the bottom half of the substrate while molecular mechanics was used for the top half. The parent chain 
of the amorphous cell consists of 50 monomers. Hence, a 50:1 reduction in degrees of freedom is 
achieved for the PAC cell domain. The indentation process was also simulated using a fully molecular 
mechanics simulation for the entire domain for comparison. 
 
Figures 1 and 2 are the strain contours of the substrate obtained from the multiscale and fully 
molecular mechanics simulations respectively at the same indentation depth. The agreement of both 
plots show the accuracy of the PAC cell method.  
 
 
 
 
Fig. 1.  Strain contours from multiscale simulation 
 
 
 
Fig. 2.  Strain contours from molecular mechanics simulation 
 
Although the displacements of atoms within PAC cells are not, and need not, be solved during the 
simulation, individual atom displacement can be easily obtained through the mapping function T at any 
stage of the simulation. This implies that whenever and wherever higher resolution in displacement is 
required, PAC cells can selectively be reverted to atomistic domains.  
 
Four sets of simulations are performed  a full molecular mechanics simulation, a coupled MM-PAC 
simulation without adaptivity as described previously and two adaptive MM-PAC simulations. For the 
two adaptive simulations, the substrate initially comprises an MM domain in the top four rows of 
  
amorphous cells with the remaining substrate represented by PAC cells. The PAC cells are reverted to 
atomistic domains when they experience a strain (y) of 2% fo r one of the adaptive simulation and a 
strain of 3% for the other.  
 
The interatomic potential of atoms which are bonded to each other is given by,  
 
 
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and the Lennard-Jones pairwise potential for nonbonded atoms is given by,
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with a cutoff of 5.2=r . 
 
The parameters of the Lennard-Jones potential are set to unity while the parameters for the bond 
potential are 30=k  and 5.10 =R . 
 
The indentation force versus depth curves from all four sets of simulations are shown in Figure 3. It 
shows that the indentation force predicted from the multiscale simulation and fully molecular 
mechanics simulations as the indentor pushes into the substrate are almost identical. 
 
-20
-10
0
10
20
30
40
50
60
-0.5 0.5 1.5 2.5 3.5 4.5
d
F
Adaptive (2%)
Adaptive (3%)
Non adaptive
MM (full)
 
Fig. 3.  Comparison of force vs indentation depth graphs from multiscale and molecular mechanics 
  
 
It should be noted that a fourth fully molecular mechanics simulation was performed but with a 
substrate only half as thick. The results were very different. This shows that the accuracy achieved by 
the multiscale simulation is not because deformation is localized and completely resides within the 
molecular mechanics domain of all simulations.   
 
References 
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networks: a molecular dynamics study of epoxies. Macromolecules, Vol. 37, No. 2, pp 630-637, 2004. 
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Murat M., Paul W., Santos S., Suter U.W. and Tries V. Bridging the gap between atomistic and coarse-
grained models of polymers: status and perspectives. In Advances in polymer science 152: 
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[4] Abraham F.F., Broughton J.Q., Bernsten N. and Kaxiras E. Spanning the continuum to quantum length 
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[5] Broughton J.Q., Abraham F.F., Bernsten N. and Kaxiras E. Concurrent coupling of length scales: 
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[6] Tadmor E.B., Ortiz M. and Phillips R. Quasicontinuum analysis of defects in solids. Philosophical 
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