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Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics. (English) Zbl 1330.74066

Summary: In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.

MSC:

74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
65K10 Numerical optimization and variational techniques
74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
74G65 Energy minimization in equilibrium problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
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