Development of an Optimization-Based Atomistic-to-Continuum Coupling Method

Atomistic-to-Continuum (AtC) coupling methods are a novel means of computing the properties of a discrete crystal structure, such as those containing defects, that combine the accuracy of an atomistic (fully discrete) model with the efficiency of a continuum model. In this note we extend the optimization-based AtC, formulated in arXiv:1304.4976 for linear, one-dimensional problems to multi-dimensional settings and arbitrary interatomic potentials. We conjecture optimal error estimates for the multidimensional AtC, outline an implementation procedure, and provide numerical results to corroborate the conjecture for a 1D Lennard-Jones system with next-nearest neighbor interactions.Comment: 12 pages, 3 figure

Symmetries of 2-lattices and second order accuracy of the Cauchy-Born model

We show that the Cauchy-Born model of a single-species 2-lattice is second order if the atomistic and continuum kinematics are connected in a novel way. Our proof uses a generalization to 2-lattices of the point symmetry of Bravais lattices. Moreover, by identifying similar symmetries in multispecies pair interaction models, we construct a new stored energy density, using shift gradients but not strain gradients, that is also second order accurate. These results can be used to develop highly accurate continuum models and atomistic/continuum coupling methods for materials such as graphene, hcp metals, and shape memory alloys

A three-layer-mesh bridging domain for coupled atomistic-continuum simulations at finite temperature: formulation and testing

pre-printAlthough concurrent multiscale methods have been well developed for zero-temperature simulations, improvements are needed to meet challenges pertaining to finite-temperature simulations. Bridging domain method (BDM) is one of the most efficient and widely-used multiscale atomistic-continuum techniques. It is recently revealed that the BDM coupling algorithm has a cooling effect on the atoms in the bridging domain (BD), and application of thermostats to rectify the cooling effect in the original BDM formulation is unstable. We propose improvement of the BDM formulation for finite-temperature simulations by employing a three-layer mesh structure in the BD, consisting of coarse, meso, and atomic meshes. The proposed method uses a mesh-independent physics-based discrimination between thermal and mechanical waves to define and introduce a meso mesh that is independent of the finite-element (FE) mesh. Temperature stability in the BD is achieved by constraining only the mechanical part of atomic motion to the FE displacements while unconstrained thermal vibrations are thermostatted using local thermostats in the BD. The new architecture of three-layer-mesh BD effectively mitigates the temperature cooling effect encountered by the conventional BDM as well as suppresses the spurious mechanical wave reflection. Numerical simulations have shown the robustness and accuracy of the proposed multiscale method at finite temperature

Analysis of blended atomistic/continuum hybrid methods

We present a comprehensive error analysis of two prototypical atomistic-to-continuum coupling methods of blending type: the energy-based and the force-based quasicontinuum methods. Our results are valid in two and three dimensions, for finite range many-body interactions (e.g., EAM type), and in the presence of lattice defects (we consider point defects and dislocations). The two key ingredients in the analysis are (1) new force and energy consistency error estimates; and (2) a new technique for proving energy norm stability of a/c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium

MULTIRESOLUTION MOLECULAR MECHANICS: THEORY AND APPLICATIONS

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)