  We consider an immersed elastic body that is actively driven through a
structured fluid by a motor or an external force. The behavior of such a system
generally cannot be solved analytically, necessitating the use of numerical
methods. However, current numerical methods omit important details of the
microscopic structure and dynamics of the fluid, which can modulate the
magnitudes and directions of viscoelastic restoring forces. To address this
issue, we develop a simulation platform for modeling viscoelastic media with
tensorial elasticity. We build on the lattice Boltzmann algorithm and
incorporate viscoelastic forces, elastic immersed objects, a microscopic
orientation field, and coupling between viscoelasticity and the orientation
field. We demonstrate our method by characterizing how the viscoelastic
restoring force on a driven immersed object depends on various key parameters
as well as the tensorial character of the elastic response. We find that the
restoring force depends non-monotonically on the rate of diffusion of the
stress and the size of the object. We further show how the restoring force
depends on the relative orientation of the microscopic structure and the
pulling direction. These results imply that accounting for previously neglected
physical features, such as stress diffusion and the microscopic orientation
field, can improve the realism of viscoelastic simulations. We discuss possible
applications and extensions to the method.
