Analysis of an optimization-based atomistic-to-continuum coupling method for point defects

Olson, Derek; Shapeev, Alexander V.; Bochev, Pavel B.; Luskin, Mitchell

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 1-41.

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.

Classification : 65N99, 65G99, 73S10
Mots clés : Atomistic-to-continuum coupling, atomic lattice, constrained optimization, point defect

P. Bauman, H. Ben Dhia, N. Elkhodja, J. Oden and S. Prudhomme, On the application of the Arlequin method to the coupling of particle and continuum models. Comput. Mech. 42 (2008) 511–530 | DOI | MR | Zbl

X. Blanc, C. Le Bris and P.L. Lions, Atomistic to continuum limits for computational materials science. ESAIM: M2AN 41 (2007) 391–426. | DOI | Numdam | MR | Zbl

M. Born and K. Huang, Dynamical Theory of Crystal Lattices, 1st edition. Clarendon Press (1954). | MR | Zbl

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods. In vol. 15. Springer (2008). | MR | Zbl

V.I. Burenkov, On extension of functions with preservation of seminorm. Trudy Mat. Inst. Steklov. 172 (1985) 71–85. | MR | Zbl

M. Discacciati, P. Gervasio and A. Quarteroni, The interface control domain decomposition (icdd) method for elliptic problems. SIAM J. Control Optim. 51 (2013) 3434–3458. | DOI | MR | Zbl

M. Discacciati, P. Gervasio and A. Quarteroni, Interface control domain decomposition methods for heterogeneous problems. Int. J. Numer. Methods Fluids 76 (2014) 471–496. | DOI | MR

M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN 42 (2008) 113–139. | DOI | Numdam | MR | Zbl

E. Weinan, J. Lu and J.Z. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74 (2006) 214115. | DOI

V. Ehrlacher, C. Ortner and A.V. Shapeev, Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations (2013). Preprint . | arXiv | MR

L.C. Evans, Partial Differential Equations. Grad. Stud. Math., 2nd edition. American Mathematical Society (2010). | MR

P. Gervasio, J.L. Lions and A. Quarteroni, Heterogeneous coupling by virtual control methods. Numer. Math. 90 (2001) 241–264. | DOI | MR | Zbl

M. Giaquinta. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press (1983). | MR | Zbl

J. Hubbard and B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 4th edition. Matrix Editions (2009). | Zbl

B. Van Koten and M. Luskin, Analysis of energy-based blended quasi-continuum approximations. SIAM J. Numer. Anal. 49 (2011) 2182–2209. | DOI | MR | Zbl

X. Li, L. Luskin, C. Ortner and A. Shapeev, Theory-based benchmarking of the blended force-based quasicontinuum method. Comput. Methods Appl. Mech. Eng. 268 (2014) 763–781. | DOI | MR | Zbl

X. Li, M. Luskin and C. Ortner, Positive definiteness of the blended force-based quasicontinuum method. Multiscale Model. Simul. 10 (2012) 1023–1045. | DOI | MR | Zbl

X. Li, C. Ortner, A.V. Shapeev and B. Van Koten, Analysis of Blended Atomistic/Continuum Hybrid Methods (2014). Preprint . | arXiv | MR

J.L. Lions, Virtual and effective control for distributed systems and decomposition of everything. J. Anal. Math. 80 (2000) 257–297. | DOI | MR | Zbl

J.L. Lions and O. Pironneau, Algorithmes paralleles pour la solution de problemes aux limites. C. R. Acad. Sci.-Series I-Math. 327 (1998) 947–952. | MR | Zbl

J. Lu and P. Ming, Convergence of a force-based hybrid method in three dimensions. Commun. Pure Appl. Math. 66 (2013) 83–108. | DOI | MR | Zbl

M. Luskin and C. Ortner, Atomistic-to-continuum coupling. Acta Numerica 22 (2013) 397–508. | DOI | MR | Zbl

M. Luskin, C. Ortner and B. Van Koten, Formulation and optimization of the energy-based blended quasicontinuum method. Comput. Methods Appl. Mech. Eng. 253 (2013) 160–168. | DOI | MR | Zbl

R. Miller and E. Tadmor, A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17 (2009) 053001. | DOI

D. Olson, P. Bochev, M. Luskin and A. Shapeev, An Optimization-Based Atomistic-to-Continuum Coupling Method. SIAM J. Numer. Anal. 52 (2014) 2183–2204. | DOI | MR | Zbl

D. Olson, P. Bochev, M. Luskin and A. Shapeev, Development of an optimization-based atomistic-to-continuum coupling method. In Proc. of LSSC 2013, edited by I. Lirkov, S. Margenov and J. Wasniewski. Springer Lect. Notes Comput. Sci. Springer-Verlag, Berlin, Heidelberg (2014). | MR

M. Ortiz, R. Phillips and E.B. Tadmor, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529–1563. | DOI

C. Ortner, A posteriori existence in numerical computations. SIAM J. Numer. Anal. 47 (2009) 2550–2577. | DOI | MR | Zbl

C. Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods. ESAIM: M2AN 46 (2012) 1275–1319. | DOI | Numdam | MR | Zbl

C. Ortner and A Shapeev, Analysis of an energy-based atomistic/continuum approximation of a vacancy in the 2d triangular lattice. Math. Comput. 82 (2013) 2191–2236. | MR | Zbl

C. Ortner and A.V. Shapeev, Interpolants of Lattice Functions for the Analysis of Atomistic/Continuum Multiscale Methods (2012). Preprint . | arXiv

C. Ortner, A.V. Shapeev and L. Zhang, (In-)stability and stabilisation of QNL-type atomistic-to-continuum coupling methods (2013). Preprint . | arXiv | MR

C. Ortner and E. Süli. A note on linear elliptic systems on $R^d$ (2012). Preprint . | arXiv

C. Ortner and F. Theil, Justification of the Cauchy–Born approximation of elastodynamics. Arch. Ration. Mech. Anal. 207 (2013) 1025–1073. | DOI | MR | Zbl

C. Ortner and L. Zhang, Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces: A two-dimensional model problem. SIAM J. Numer. Anal. 50 (2012) 2940–2965. | DOI | MR | Zbl

C. Ortner and L. Zhang, Atomistic/Continuum Blending with Ghost Force Correction (2014). Preprint . | arXiv | MR

R. Phillips, Crystals, defects and microstructures: modeling across scales. Cambridge University Press (2001).

R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl

A.V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions. SIAM J. Multiscale Model. Simul. 9 (2011) 905–932. | DOI | MR | Zbl

T. Shimokawa, J.J. Mortensen, J. Schiotz and K.W. Jacobsen, Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region. Phys. Rev. B 69 (2004) 214104. | DOI

E. Stein, Singular integrals and differentiability properties of functions. In vol. 2. Princeton University Press (1970). | MR | Zbl

E. Tadmor and R. Miller, Modeling Materials Continuum, Atomistic and Multiscale Techniques, 1st edition. Cambridge University Press (2011). | Zbl

E Weinan and P. Ming, Cauchy–Born rule and the stability of crystalline solids: static problems. Arch. Rat. Mech. Anal. 183 (2007) 241–297. | DOI | MR | Zbl

S.P. Xiao and T. Belytschko, A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193 (2004) 1645–1669. | DOI | MR | Zbl