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A posteriori analysis and adaptive algorithms for blended type atomistic-to-continuum coupling with higher-order finite elements

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COMPUTER PHYSICS COMMUNICATIONS 2025年 310卷

作者： Wang, Yangshuai Natl Univ Singapore Dept Math 10 Lower Kent Ridge Rd Singapore 119076 Singapore

The accurate and efficient simulation of material systems with defects using atomistic-to-continuum (a/c) coupling methods is a significant focus in computational materials science. Achieving a balance between accuracy and computational cost requires the application of a posteriori error analysis and adaptive algorithms. In this paper, we provide a rigorous a posteriori error analysis for three common blended a/c methods: the blended energy-based quasi-continuum (BQCE) method, the blended force-based quasi-continuum (BQCF) method, and the atomistic/continuum blending with ghost force correction (BGFC) method. We discretize the Cauchy-Born model in the continuum region using first- and second-order finite element methods, with the potential for extending to higher-order schemes. The resulting error estimator provides both an upper bound on the true error and a reliable lower bound, subject to a controllable truncation term. Furthermore, we offer an a posteriori analysis of the energy error. We develop and implement an adaptive mesh refinement algorithm applied to two typical defect scenarios: a micro-crack and a Frenkel defect. In both cases, our numerical experiments demonstrate optimal convergence rates with respect to degrees of freedom, in agreement with a priori error estimates.

关键词： atomistic-to-continuum coupling Crystalline defects A posteriori error estimate Adaptivity Higher-order finite element methods

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ANALYSIS OF AN OPTIMIZATION-BASED atomistic-to-continuum coupling METHOD FOR POINT DEFECTS

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ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE 2016年第1期50卷 1-41页

作者： Olson, Derek Shapeev, Alexander V. Bochev, Pavel B. Luskin, Mitchell Univ Minnesota Minneapolis MN 55455 USA Skolkovo Inst Sci & Technol Skolkovo Russia Sandia Natl Labs Computat Math POB 5800MS 1320 Albuquerque NM 87185 USA

We formulate and analyze an optimization-based atomistic-to-continuum (AtC) coupling method for problems with point defects. Application of a potential-based atomistic model near the defect core enables accurate simulation of the defect. Away from the core, where site energies become nearly independent of the lattice position, the method switches to a more efficient continuum model. The two models are merged by minimizing the mismatch of their states on an overlap region, subject to the atomistic and continuum force balance equations acting independently in their domains. We prove that the optimization problem is well-posed and establish error estimates.

关键词： atomistic-to-continuum coupling atomic lattice constrained optimization point defect

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(IN-)STABILITY AND STABILIZATION OF QNL-TYPE atomistic-to-continuum coupling METHODS

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MULTISCALE MODELING & SIMULATION 2014年第3期12卷 1258-1293页

作者： Ortner, Christoph Shapeev, Alexander V. Zhang, Lei Univ Warwick Math Inst Coventry CV4 7AL W Midlands England Univ Minnesota Sch Math Minneapolis MN 55455 USA Shanghai Jiao Tong Univ Inst Nat Sci Dept Math Shanghai 200240 Peoples R China Shanghai Jiao Tong Univ MOE Key Lab Sci & Engn Comp Shanghai 200240 Peoples R China

We study the stability of ghost force-free energy-based atomistic-to-continuum (a/c) coupling methods. In one dimension we essentially complete the theory by introducing a universally stable a/c coupling as well as a stabilization mechanism for unstable coupling schemes. We then present a comprehensive study of a two-dimensional scalar planar interface setting, as a step towards a general two-dimensional/three-dimensional vectorial analysis. Our results point out various new challenges. For example, we find that none of the ghost force-free methods known to us is universally stable (i.e., stable under general interaction and general loads). We then explore to what extent our one-dimensional stabilization mechanism can be extended.

关键词： atomistic-to-continuum coupling quasicontinuum stability

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A Posteriori Error Control for Three Typical Force-Based atomistic-to-continuum coupling Methods for an atomistic Chain

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Numerical Mathematics(Theory,Methods and Applications) 2019年第1期12卷 233-264页

作者： Hao Wang Shaohui Liu Feng Yang School of Mathematics Sichuan UniversityNo.24 South Section OneChengdu610065China Department of Applied Mathematics and Statistics Stony Brook UniversityStony BrookNew York11794-3600U.S College of Mathematical Science Beijing UniversityZhongguancun Road 1Beijing100000China

We consider the problem of a posteriori error estimates and adaptivity for three typical force-based atomistic-to-continuum coupling *** the residual and the stability estimates,we derive computable a posteriori error estima-tors for the three methods in the energy norm and formulate adaptive algorithms using these *** numerical experiments show optimal convergence rates of these *** efficiency of the estimators are also demonstrated numerically.

关键词： atomistic-to-continuum coupling a posteriori error estimate adaptivity force based method

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ANALYSIS OF THE RESIDUAL-TYPE AND THE GRADIENT RECOVERY-TYPE A POSTERIORI ERROR ESTIMATORS FOR A CONSISTENT atomistic-to-continuum coupling METHOD IN ONE-DIMENSION

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MULTISCALE MODELING & SIMULATION 2018年第2期16卷 679-709页

作者： Wang, Hao Yang, Siyao Sichuan Univ Sch Math 24 South Sect One Chengdu 610065 Sichuan Peoples R China Natl Univ Singapore Fac Sci Dept Math Singapore 119077 Singapore

We consider the a posteriori error estimation for an atomistic-to-continuum coupling scheme for a generic one-dimensional many-body next-nearest-neighbor interaction model in one dimension. We derive and rigorously prove the efficiency of the residual type estimator. We prove the equivalence between the residual type and the gradient recovery-type estimator in the continuum region and propose a (novel) hybrid a posteriori error estimator by combining the two types of estimators. Our numerical experiments illustrate the optimal convergence rate of the adaptive algorithms using these estimators, whose effectivity indices are also presented.

关键词： atomistic-to-continuum coupling quasi-continuum method a posteriori error analysis

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An atomistic-to-continuum coupling method for heat transfer in solids

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2008年第41-42期197卷 3351-3365页

作者： Wagner, G. J. Jones, R. E. Templeton, J. A. Parks, M. L. Sandia Natl Labs Fluid Thermal Sci & Engn Dept Livermore CA 94551 USA Sandia Natl Labs Mech Mat Dept Livermore CA 94551 USA Sandia Natl Labs Applied Math & Applicat Dept Albuquerque NM 87123 USA

In this work, we present a seamless, energy-conserving method to couple atomistic and continuum representations of a temperature field in a material. This technique allows a molecular dynamics simulation to be used in localized regions of the computational domain, surrounded and overlaid by a continuum finite element representation. Thermal energy can pass between the two regions in either direction, making larger simulations of nano-scale thermal processes possible. We discuss theoretical developments and numerical implementation details. In addition, we present and analyze a set of representative simulations. Published by Elsevier B.V.

关键词： atomistic-to-continuum coupling heat transfer finite elements multi-scale simulations

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ANALYSIS OF PATCH-TEST CONSISTENT atomistic-to-continuum coupling WITH HIGHER-ORDER FINITE ELEMENTS

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ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE 2017年第6期51卷 2263-2288页

作者： Dedner, Andreas Ortner, Chistoph Wu, Huan Univ Warwick Math Inst Zeeman Bldg Coventry CV4 7AL W Midlands England

We formulate a patch test consistent atomistic-to-continuum coupling (a/c) scheme that employs a second-order (potentially higher-order) finite element method in the material bulk. We prove a sharp error estimate in the energy-norm, which demonstrates that this scheme is (quasi-) optimal amongst energy-based sharp-interface a/c schemes that employ the Cauchy-Born continuum model. Our analysis also shows that employing a higher-order continuum discretisation does not yield qualitative improvements to the rate of convergence.

关键词： atomistic models coarse graining atomistic-to-continuum coupling quasicontinuum method error analysis

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On atomistic-to-continuum coupling by blending

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MULTISCALE MODELING & SIMULATION 2008年第1期7卷 381-406页

作者： Badia, Santiago Parks, Michael Bochev, Pavel Gunzburger, Max Lehoucq, Richard Univ Politecn Cataluna CIMNE ES-08034 Barcelona Spain Sandia Natl Labs Albuquerque NM 87185 USA Florida State Univ Sch Computat Sci Tallahassee FL 32306 USA

A mathematical framework for the coupling of atomistic and continuum models by blending them over a subdomain subject to a constraint is developed. Using the framework, four classes of atomistic-to-continuum (AtC) blending methods are established, their consistency is studied, and their relative merits are discussed. In addition, the framework helps clarify the origin of ghost forces and formalizes the notion of a patch test. Numerical experiments with the AtC methods are used to illustrate the theoretical results.

关键词： atomistic-to-continuum coupling blending methods multiscale simulation coupling methods

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THE ROLE OF THE PATCH TEST IN 2D atomistic-to-continuum coupling METHODS

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ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE 2012年第6期46卷 1275-1319页

作者： Ortner, Christoph Univ Warwick Math Inst Coventry CV4 7AL W Midlands England

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P-1-finite element discretisation of Cauchy-Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the interface correction, and (iv) connectedness of the atomistic region. The extent to which these assumptions are necessary is discussed in detail.

关键词： atomistic models atomistic-to-continuum coupling quasicontinuum method coarse graining ghost forces patch test consistency

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Bridging Methods for atomistic-to-continuum coupling and Their Implementation

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Communications in Computational Physics 2010年第4期7卷 831-876页

作者： Pablo Seleson Max Gunzburger Department of Scientific Computing Florida State UniversityTallahasseeFL 32306-4120USA

Several issues connected with bridging methods for atomistic-to-continuum(AtC)coupling are *** coupling approaches using various energy blending models are studied as well as the influence that model parameters,blending functions,and grids have on simulation *** use the Lagrange multiplier method for enforcing constraints on the atomistic and continuum displacements in the bridge *** also show that continuum models are not appropriate for dealing with problems with singular loads,whereas AtC bridging methods yield correct results,thus justifying the need for a multiscale *** investigate models that involve multiple-neighbor interactions in the atomistic region,particularly focusing on a comparison of several approaches for dealing with Dirichlet boundary conditions.

关键词： Multiscale modeling atomistic-to-continuum coupling boundary treatment

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