Continuum-based sensitivity analysis for coupled atomistic and continuum simulations for 2-D applications using bridging
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RESEARCH PAPER
Continuum-based sensitivity analysis for coupled atomistic and continuum simulations for 2-D applications using bridging scale decomposition Yunxiang Wang · Kuang-Hua Chang
Received: 18 March 2012 / Revised: 2 October 2012 / Accepted: 14 November 2012 / Published online: 22 December 2012 © Springer-Verlag Berlin Heidelberg 2012
Abstract The concept of linking simulations at multiple length and time scales is found useful for studying local physical phenomena such as crack propagation. Many multi-scale methods, which couple molecular dynamics models with continuum models, have been proposed over the last decade. One of the most advanced methods developed recently is the bridging scale method, in which the total displacement is decomposed into orthogonal coarse and fine scales. This paper presents the continuum-based sensitivity analysis for two-dimensional coupled atomistic and continuum problems using the bridging scale method. A variational formulation for the bridging scale decomposition is developed based on the Hamilton’s principle. The continuum-based variational formulation provides a uniform and generalized system of equations from which the differential equations can be obtained naturally. The sensitivity expressions for both direct differentiation method (DDM) and adjoint variable method (AVM) are derived in a continuum setting. Due to its efficiency for crack problems, the direct differentiation method is chosen to be implemented numerically and applied to two examples, including
This paper has been presented in the 23rd International Congress of Theoretical and Applied Mechanics (ICTAM2012), Beijing, China, 19–24 August, 2012. Y. Wang () School of Aerospace and Mechanical Engineering, The University of Oklahoma, 865, Asp avenue, Room 238, Norman, OK 73019, USA e-mail: [email protected] K.-H. Chang School of Aerospace and Mechanical Engineering, The University of Oklahoma, 865, Asp avenue, Room 201, Norman, OK 73019, USA e-mail: [email protected]
a crack propagation problem. Both material and sizing design variables are included to reveal the impact of design changes at the macroscopic level to the responses at the atomistic level. Also demonstrated is the feasibility of achieving the variation of the time history kernel computed using numerical procedures. The sensitivity coefficients calculated are shown to be accurate compared with overall finite difference method. The physical implications of the sensitivity results are also discussed, which accurately predict the behavior of the structural responses. Keywords Bridging scale method · Multi-scale simulation · Design sensitivity analysis
1 Introduction During the last decade, a number of multi-scale simulation methods, which provide rather simplified treatments of the interface between the atomistic and continuum regions, have been proposed by different research groups. Abraham et al. (1998) developed the macroscopic, atomistic, ab initio dynamics (MAAD) method in which tight binding (TB), molecular dynamics (MD) and finite element
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