Abstract
The paper investigates model reduction techniques that are based on a nonlocal quasi-continuum-like approach. These techniques reduce a large optimization problem to either a system of nonlinear equations or another optimization problem that are expressed in a smaller number of degrees of freedom. The reduction is based on the observation that many of the components of the solution of the original optimization problem are well approximated by certain interpolation operators with respect to a restricted set of representative components. Under certain assumptions, the “optimize and interpolate” and the “interpolate and optimize” approaches result in a regular nonlinear equation and an optimization problem whose solutions are close to the solution of the original problem, respectively. The validity of these assumptions is investigated by using examples from potential-based and electronic structure-based calculations in Materials Science models. A methodology is presented for using quasi-continuum-like model reduction for real-space DFT computations in the absence of periodic boundary conditions. The methodology is illustrated using a basic Thomas–Fermi–Dirac case study.
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References
Allen M.P. and Tildesley D.J. (1987). Computer Simulation of Liquids. Clarendon Press, Oxford
Atkinson K.E. (1989). An Introduction to Numerical Analysis. Wiley, New York
Bertsekas D.P. (1982). Constrained Optimization and Lagrange Multiplier Methods. Academic, New York
Blanc, X., LeBris, C., Lions, P.-L.: Atomistic to continuum limits for computational materials science. Math. Model. Numer. Anal. (to appear) (2007)
Lu W.E.J. and Yang J.Z. (2006). Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74: 214115
Fago M., Hayes R., Carter E., Ortiz M. (2004) Density-functional-theory-based local quasicontinuum method: prediction of dislocation nucleation. Phys. Rev. B 70: 100102
Fiacco A.V. (1983). Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic, New York
Fletcher R. (1987). Practical Methods of Optimization. Wiley, Chichester
Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: a modeling language for mathematical programming, chap. 1, 2nd edn. Thomson, Toronto, Canada. Software, other material available at http://www.ampl.com (2003)
Gill, P.E., Murray, W., Saunders, M.A.: User’s guide for SNOPT 5.3: a fortran package for large-scale nonlinear programming. Report NA 97-5, Department of Mathematics, University of California, San Diego (1997)
Kevrekidis, Y., Gear, C.W., Li, J.: The gaptooth method in particle simulations. Phys. Lett. A 190 (2003)
Knap J. and Ortiz M. (2001). An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49: 1899–1923
Knap J. and Ortiz M. (2003). Effect of indenter-radius size on Au(001) nanoindentation. Phys. Rev. Lett. 90(22): 226102-1–226102-4
Koch W. and Holthausen M.C. (2001). A Chemist’s Guide to Density Functional Theory, 2nd edn. John Wiley & Sons Inc., New York
Kunin I. (1982). Elastic media with microstructure, I. One-dimensional models. Springer, Heidelberg
Lin P. (2002). Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comput. 72: 657–675
Miller R.E. and Tadmor E.B. (2002). The quasicontinuum method: overview, applications and current directions. J. Comput. Aided Materials Des. 9: 203–239
Negrut, D., Anitescu, M., Munson, T., Zapol, P.: Simulating nanoscale processes in solids using DFT and the quasicontinuum method (IMECE2005-81755). In: Proceedings of IMECE 2005, ASME International Mechanical Engineering Congress and Exposition (2005)
Ortega J. and Rheinboldt W. (1972). Iterative Solutions of Nonlinear Equations in Several Variables. Academic, New York
Ortner, C., Suli, E.: A-priori analysis of the quasicontinuum method in one dimension. In: Technical Report NA-06/12, Oxford University, Computing Laboratory, Oxford
Rodney, D.: Mixed atomistic/continuum methods: static and dynamic quasi continuum methods. In: Finel, A., Maziere, D., Veron, M. (eds.) Proceedings of the NATO Conference in Thermodynamics, Microstructures and Plasticity. Kluwer, Dordrecht (2003)
Szabo, A., Ostlund, N.: Modern Quantum Chemistry. Dover, New York (1989)
Tadmor E., Ortiz M. and Phillips R.A. (1996). Quasicontinuum analysis of defects in solids. Philos Mag A 73: 1529–1563
Wang Y., Govind N. and Carter E. (1999). Orbital-free kinetic-energy density functionals with a density-dependent kernel. Phys. Rev. B 60: 16350–16358
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Anitescu, M., Negrut, D., Zapol, P. et al. A note on the regularity of reduced models obtained by nonlocal quasi-continuum-like approaches. Math. Program. 118, 207–236 (2009). https://doi.org/10.1007/s10107-007-0188-3
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DOI: https://doi.org/10.1007/s10107-007-0188-3