Coarse-grained descriptions of multiple scale processes in solid systems

Several extant hybrid atomistic-continuum computational schemes designed to handle coupled processes on vastly separated spatial scales are based on a "dynamic" coarse graining of the continuum by means of finite elements. Such an exact coarse-graining treatment of the one-dimensional harmonic chain of identical atoms was carried out as a test. It is shown that the error in thermomechanical properties (e.g., the tension) engendered by "dynamic" finite-element coarse graining can be substantial, depending on the thermodynamic state. An alternative "static" finite-element coarse-graining description, which is an extension to nonzero temperature of the "quasicontinuum" procedure of Tadmor, Ortiz, and Phillips, is proposed in an attempt to correct this error. The extended quasicontinuum technique applied to the pure one-dimensional harmonic chain yields the exact solution, thus indicating its promise for more general applications. Problems anticipated in the extension of the technique to realistic three-dimensional models of solids are discussed.