[Paper Review] An Optimization-Based Atomistic-to-Continuum Coupling Method

Derek Olson, Pavel Bochev+2 more|arXiv (Cornell University)|Apr 17, 2013

Block Copolymer Self-Assembly15 references4 citations

TL;DR

This paper introduces an optimization-based atomistic-to-continuum (AtC) coupling method that formulates the coupling as a constrained optimization problem minimizing the mismatch between atomistic and continuum displacements in the overlap region. By treating the atomistic and continuum models independently with virtual Dirichlet controls on interfaces, the method avoids blending and achieves optimal convergence rates in a one-dimensional linearized next-nearest-neighbor model.

ABSTRACT

We present a new optimization-based method for atomistic-to-continuum (AtC) coupling. The main idea is to cast the coupling of the atomistic and continuum models as a constrained optimization problem with virtual Dirichlet controls on the interfaces between the atomistic and continuum subdomains. The optimization objective is to minimize the error between the atomistic and continuum solutions on the overlap between the two subdomains, while the atomistic and continuum force balance equations provide the constraints. Splitting of the atomistic and continuum problems instead of blending them and their subsequent use as constraints in the optimization problem distinguishes our approach from the existing AtC formulations. We present and analyze the method in the context of a one-dimensional chain of atoms modeled using a linearized two-body next-nearest neighbor interactions.

Motivation & Objective

• To develop a novel AtC coupling method that avoids energy or force blending by treating atomistic and continuum models independently.
• To minimize the discrepancy between atomistic and continuum solutions in the overlap region via constrained optimization.
• To establish theoretical convergence and stability using virtual Dirichlet controls on artificial interfaces.
• To provide a framework where key properties of the method can be inferred from the individual atomistic and continuum constituents.
• To demonstrate optimal convergence rates in a one-dimensional linearized two-body interaction model.

Proposed method

• The method formulates AtC coupling as a constrained optimization problem with virtual Dirichlet controls on the interfaces between atomistic and continuum subdomains.
• The objective function minimizes the ℓ²-norm of the difference between atomistic and continuum displacements over the overlap region Ωₒ.
• The constraints enforce force balance in the atomistic and continuum subdomains Ωₐ and Ωᶜ, respectively.
• The approach separates the atomistic and continuum problems rather than blending them, distinguishing it from existing energy- or force-based blending methods.
• The method uses a variational formulation based on displacement fields and applies boundary conditions via virtual controls to ensure well-posedness.
• Theoretical analysis relies on solving linear systems for displacement modes and estimating norms using matrix inversion bounds and geometric series.

Experimental results

Research questions

• RQ1 Can a non-blending AtC method be formulated using optimization to minimize displacement mismatch in the overlap region?
• RQ2 How does the use of virtual Dirichlet controls on artificial interfaces affect the stability and convergence of the coupled system?
• RQ3 What is the convergence rate of the optimization-based AtC method in comparison to existing blending-based formulations?
• RQ4 Can key properties of the coupled method be derived directly from the properties of the individual atomistic and continuum models?
• RQ5 How do the coefficients in the displacement solution behave under asymptotic scaling, and what does this imply for error bounds?

Key findings

• The method achieves optimal convergence rates in the ℓ²-norm of the displacement error, with the error bounded by a constant times the mesh size.
• The norm of the displacement solution in the atomistic region is bounded by the inverse of the matrix T_L, which converges to a fixed matrix T as L→∞.
• The error estimate for the overlap region depends on the eigenvalues of the quadratic form, with the smallest eigenvalue bounded below by (1/24)γ² for small γ.
• The convergence rate is independent of the overlap size when properly scaled, indicating robustness to domain decomposition.
• Theoretical analysis confirms that the method’s stability and convergence can be deduced from the properties of the atomistic and continuum models alone.
• The method avoids the consistency and stability issues common in blended AtC methods by minimizing mismatch rather than blending energies or forces.