Yang, Qingcheng
(2016)
MULTIRESOLUTION MOLECULAR MECHANICS: THEORY AND APPLICATIONS.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
A general mathematical framework, Multiresolution Molecular Mechanics (MMM), is proposed to consistently coarse-grain molecular mechanics at different resolutions in order to extend the length scale of nanoscale modeling of crystalline materials. MMM is consistent with molecular mechanics in the sense that the constitutive description such as energy and force calculations is exactly the same as molecular mechanics and no empirical and phenomenological constitutive relationships in continuum mechanics are employed. As such, MMM can converge to full molecular mechanics naturally.
As many coarse-graining approaches, MMM is based on approximating the total potential energy of a full atomistic model. Analogous to quadrature rules employed to evaluate energy integrals in finite element method (FEM), a summation rule is required to evaluate finite energy summations. Most existing summation rules are specifically designed for the linear interpolation shape function and their extensions to high order shape functions are currently not clear. What distinguishes MMM from existing works is that MMM proposes a novel summation rule framework SRMMM that is valid and consistent for general shape functions. The key idea is to analytically derive the energy distribution of the coarse-grained atomistic model and then choose some quadrature-type (sampling) atoms to accurately represent the derived energy distribution for a given shape function. The optimal number, weight and position of sampling atoms are also determined accordingly, similar to the Gauss quadrature in FEM. The governing equations are then derived following the variational principle.
The proposed SRMMM is verified and validated numerically and compared against many other summation rules such as Gauss-quadrature-like rule. And SRMMM demonstrates better performance in terms of accuracy and computational cost. The convergence property of MMM is also studied numerically and MMM shows FEM-like behavior under certain circumstance. In addition, MMM has been applied to solve problems such as crack propagation, atomic sheet shear, beam bending and surface relaxations by employing high order interpolation shape functions in one (1D), two (2D) and three dimensions (3D) .
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
20 September 2016 |
| Date Type: |
Publication |
| Defense Date: |
17 March 2016 |
| Approval Date: |
20 September 2016 |
| Submission Date: |
15 July 2016 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
188 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Swanson School of Engineering > Mechanical Engineering |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
MULTIRESOLUTION MOLECULAR MECHANICS; SUMMATION RULE; GENERALIZED QUASICONTINUUM METHOD; COARSE-GRAINED ATOMISTICS; ATOMISTIC-TO-CONTINUUM COUPLING; MULTISCALE MODELING |
| Date Deposited: |
20 Sep 2016 19:30 |
| Last Modified: |
15 Nov 2016 14:34 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/28638 |
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