Positive definiteness of the blended force-based quasicontinuum method

Xingjie Helen Li, Mitchell Luskin, Christoph Ortner

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


The development of consistent and stable quasicontinuum models for multidimensional crystalline solids remains a challenge. For example, proving the stability of the force-based quasicontinuum (QCF) model [M. Dobson and M. Luskin, M2AN Math. Model. Numer. Anal., 42 (2008), pp. 113-139] remains an open problem. In one and two dimensions, we show that by blending atomistic and Cauchy-Born continuum forces (instead of a sharp transition as in the QCF method) one obtains positive-definite blended force-based quasicontinuum (B-QCF) models. We establish sharp conditions on the required blending width.

Original languageEnglish (US)
Pages (from-to)1023-1045
Number of pages23
JournalMultiscale Modeling and Simulation
Issue number3
StatePublished - 2012


  • Atomistic-to-continuum
  • Blending
  • Quasicontinuum
  • Stability


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