Stability Of A Force-Based Hybrid Method With Planar Sharp Interface

Abstract

Abstract. We study a force-based hybrid method that couples atomistic model with CauchyBorn elasticity model with sharp transition interface. We identify stability conditions that guarantee the convergence of the hybrid scheme to the solution of the atomistic model with second order accuracy, as the ratio between lattice parameter and the characteristic length scale of the deformation tends to zero. Convergence is established for hybrid schemes with planar sharp interface for system without defects, with general finite range atomistic potential and simple lattice structure. The key ingredient of the proof is regularity and stability analysis of elliptic systems of difference equations. We apply the results to atomistic-to-continuum scheme for a 2D triangular lattice with planar interface.Key words. Multiscale method, atomistic-to-continuum, stability analysis, force-based coupling AMS subject classifications. 65N12; 74S301. Introduction. Multiscale methods couple together atomistic and continuum models have received intense investigations in recent years; see, e.g., [1,11,23,30,31]. Generally speaking, there are two main categories of methods coupling atomistic and continuum models: energy-based methods and force-based methods. The energybased methods employ an energy that is a mixture of atomistic energy and continuum elastic energy. The energy functional is then minimized subject to suitable boundary conditions to obtain the deformed state of the system. The force-based methods work instead at the level of force balance equations. The forces derived from atomistic and continuum models are coupled together. The force balance equations supplemented with suitable boundary conditions are solved to obtain the deformed state of the system.From a numerical analysis point of view, the key issue for these multiscale methods is the consistency and stability analysis of the coupled schemes [11, Chapter 7]. In this paper, we study force-based atomistic-to-continuum hybrid methods in two and three dimension with sharp transition between the atomistic and continuum regions. In our previous work [22], we developed the stability analysis in general dimension for a force-based atomistic-to-continuum method with smooth transition between the two regions. The main focus of the current paper is to extend the stability analysis of [22] to hybrid schemes with sharp interface between atomistic and continuum models.Comprehensive reviews for force-based hybrid methods can be found in [25, Section 5 and Section 6] and [30, Section 12.5]. A class of force-based methods uses a handshake region (transition region) between the atomistic and continuum regions.

Analysis of blended atomistic/continuum hybrid methods

“…Both in [24] and [21] the main motivation of force-blending was that stability of the scheme can be proben, while the stability of sharp-interface force-based a/c couplings is entirely open at this point [8,9,10,23] 3.2. Approximation Error Estimates.…”

“…In particular, the technique of Lemma 4.10, which is the main new technical ingredient to prove stability of B-QCE and B-QCF, is unlikely to generalise to sharp-interface couplings. To that end the ideas present in [23] and [33] are more promising starting points.…”

A Concurrent Global–Local Numerical Method for Multiscale PDEs

“…This implies that the proposed hybrid method actually mixes the flux/stress in a weak sense, which is different to the approach in [30,31] that mixes the forces in a strong sense. This is more appropriate because Problem (1.1) is in divergence form.…”

“…In this contribution, we propose a new concurrent global-local method to capture both the average information and the local microscale information simultaneously, as we shall explain in more details below. The current approach is mainly inspired by the recent work [30,31] by two of the authors, in which a hybrid method that couples force balance equations from the atomistic model and the Cauchy-Born elasticity is proposed and analyzed. Such method is proven to have sharp stability and optimal convergence rate.…”

Force-based atomistic/continuum blending for multilattices

“…The primary advantage of force-based methods is that the forces can easily be defined in a way to avoid spurious interface effects (ghost forces); that is, the defect-free perfect crystal is a bona fide equilibrium configuration of the AtC force operator. The cost of defining the BQCF method and other forcebased methods to be free of ghost forces is that these force fields are no longer conservative, which creates significant challenges in their numerical analysis [15,27]. The blended force-based methods, originally studied in [23,7,5,26], seek to overcome this problem by a smooth blending between atomistic and continuum forces over a region called the blending, overlap, or handshake region.…”