  A computational approach is introduced for the study of the rheological
properties of complex fluids and soft materials. The approach allows for a
consistent treatment of microstructure elastic mechanics, hydrodynamic
coupling, thermal fluctuations, and externally driven shear flows. A mixed
description in terms of Eulerian and Lagrangian reference frames is used for
the physical system. Microstructure configurations are represented in a
Lagrangian reference frame. Conserved quantities, such as momentum of the fluid
and microstructures, are represented in an Eulerian reference frame. The
mathematical formalism couples these different descriptions using general
operators subject to consistency conditions. Thermal fluctuations are taken
into account in the formalism by stochastic driving fields introduced in
accordance with the principles of statistical mechanics. To study the
rheological responses of materials subject to shear, generalized periodic
boundary conditions are developed where periodic images are shifted relative to
the unit cell to induce shear. Stochastic numerical methods are developed for
the formalism. As a demonstration of the methods, results are presented for the
shear responses of a polymeric fluid, lipid vesicle fluid, and a gel-like
material.
