#### Description

The development of a set of strain tensor functionals that are suitable for characterizing arbitrarily ordered atomistic structures is described. The approach starts by transforming the discrete atomic coordinates to a continuous and differentiable density function using Gaussian kernels. The local geometries can then characterized in terms of a Taylor series expansion about each atomic center, where the nth order derivatives can be determined from the nth order moments of the neighboring atom positions. This is similar to an approach used by Zimmerman et al. [1], except that the neighborhoods here are smooth rather than discrete. The Cartesian moments can be transformed to solid harmonic functions (also called 3D Zernike functions), which retain both radial and angular information. Those functions can be further recast in terms of Rotationally Invariant Functions (RIF) that cleanly separate different types of shape distortions (strains) and orientation factors. Similar RIF descriptions have been earlier used for pattern recognition and image processing [2, 3]. Examples of using these RIF basis functions to classify the deformation geometries observed in Molecular Dynamics simulations of Cu and Ta under strong compression will be shown. The expansions are carried out to fourth order, which is what is required to distinguish between crystal structures. The resulting functionals allow different types of defect structures and deformations to be readily identified, along with the pathways of the deformation processes. The analysis can then be extended to vector quantities (velocities, forces) so that the analogous momentum and stress functions functionals can be defined, leading to a thermodynamically consistent coarse-graining procedure [4]. It is proposed that these RIF bases would be an optimally compact method for defining and comparing atomic potential functions. REFERENCES [1] Zimmerman, J.A., Bammann, D.J., Gao, H. Int. J. Sol. Struct. 2009, 46, 238. [2] Lo, C.-H., Don, H.-S. IEEE Trans. Patt. Analys. Mach. Intel. 1989, 11, 1053. [3] Kindlmann, G., Ennis, D.B., Whitaker, R.T. IEEE Trans. Med. Imag. 2007, 26, 1483. [4] Webb, E.B., Zimmerman, J.A., Seel, S.C. Math. Mech. Solids. 2008, 13, 221.

#### Recommended Citation

Kober, E.,
&
Welch, P.
(2014).
Strain functionals for characterizing atomistic geometries.
In A. Bajaj, P. Zavattieri, M. Koslowski, & T. Siegmund (Eds.).
*
Proceedings of the Society of Engineering Science 51st Annual Technical Meeting, October 1-3, 2014
*,
West Lafayette: Purdue University Libraries Scholarly Publishing Services, 2014.
https://docs.lib.purdue.edu/ses2014/mms/imms/13

Strain functionals for characterizing atomistic geometries

The development of a set of strain tensor functionals that are suitable for characterizing arbitrarily ordered atomistic structures is described. The approach starts by transforming the discrete atomic coordinates to a continuous and differentiable density function using Gaussian kernels. The local geometries can then characterized in terms of a Taylor series expansion about each atomic center, where the nth order derivatives can be determined from the nth order moments of the neighboring atom positions. This is similar to an approach used by Zimmerman et al. [1], except that the neighborhoods here are smooth rather than discrete. The Cartesian moments can be transformed to solid harmonic functions (also called 3D Zernike functions), which retain both radial and angular information. Those functions can be further recast in terms of Rotationally Invariant Functions (RIF) that cleanly separate different types of shape distortions (strains) and orientation factors. Similar RIF descriptions have been earlier used for pattern recognition and image processing [2, 3]. Examples of using these RIF basis functions to classify the deformation geometries observed in Molecular Dynamics simulations of Cu and Ta under strong compression will be shown. The expansions are carried out to fourth order, which is what is required to distinguish between crystal structures. The resulting functionals allow different types of defect structures and deformations to be readily identified, along with the pathways of the deformation processes. The analysis can then be extended to vector quantities (velocities, forces) so that the analogous momentum and stress functions functionals can be defined, leading to a thermodynamically consistent coarse-graining procedure [4]. It is proposed that these RIF bases would be an optimally compact method for defining and comparing atomic potential functions. REFERENCES [1] Zimmerman, J.A., Bammann, D.J., Gao, H. Int. J. Sol. Struct. 2009, 46, 238. [2] Lo, C.-H., Don, H.-S. IEEE Trans. Patt. Analys. Mach. Intel. 1989, 11, 1053. [3] Kindlmann, G., Ennis, D.B., Whitaker, R.T. IEEE Trans. Med. Imag. 2007, 26, 1483. [4] Webb, E.B., Zimmerman, J.A., Seel, S.C. Math. Mech. Solids. 2008, 13, 221.