Multiscale Coupling Method And Algorithm Of Atomistic System For Crystal Defect

Many fundamental problems in material science and engineering can be modeled by partial differential equations that reflect the macroscopic or continuum properties of materials.In spite of the tremendous successes,continuum models also have limitations,accuracy being one of them.When a material has defects or atomistic features,a continuum model may not be able to capture them.For example,in material fracture,the atomic bond has to be considered.Because of this,one might be tempted to switch to a fully atomistic model that is accurate and takes the full atomistic structure into account.However,this is not an efficient strategy,because atomistic systems involved with extremely fine scales are too large to handle even using most powerful computers available.This is where multi-scale modeling is particularly useful.By coupling continuum and atomistic models,it takes advantage of both the simplicity and efficiency of the continuum(or coarse-grained)models,as well as the accuracy of the atomistic models.In this study,through carrying out a rigorous a posteriori analysis of the residual,we firstly design an adaptive mesh refinement algorithm for atomistic/continuum coupling in two dimensions;we also construct a residual based a posteriori error indicator for QM(quantum mechanics)and MM(molecular mechanics)coupling methods,the corresponding adaptive QM/MM algorithm is proposed;we also consider the consistency of a coarse-grained approximation to the dynamical atomistic chain.The major innovations are as follows:Atomistic/continuum coupling methods aim to achieve optimal balance between accuracy and efficiency.Adaptivity is the key for the efficient implementation of such methods.In this study,we carry out a rigorous a posteriori analysis of the residual,the stability constant,and the error bound,for a consistent atomistic/continuum coupling method in 2D.We design and implement the corresponding adaptive mesh refinement algorithm,and the convergence rate with respect to degrees of freedom is quasi-optimal compare with a priori error estimates.QM(quantum mechanics)and MM(molecular mechanics)coupling methods are widely used in simulations of crystalline defects.In this study,we construct a residual based a posteriori error indicator for QM/MM coupling approximations.We show that this error indicator is reliable and can be computed efficiently by certain sampling techniques.Based on the error indicator and Dorfler marking strategy,we design an adaptive QM/MM algorithm for crystalline defects and demonstrate the efficiency of this algorithm with some numerical experiments.We mainly consider the consistency of the coarse-grained model with respect to the grain(mesh)size to provide a justification to the goodness of such an approximation.We either add a viscous term to the coarse-grained MD model or apply a space average to the coarse-grained MD solutions to reduce the characteristic oscillations with different frequencies.The coarse-grained solution is also compared with the solution of the Riemann problem of the(macroscopic)continuum model(a nonlinear wave equation of mixed type)to show how well the coarse-grained model can approximate the macroscopic behavior of the material.