  A direct formulation of linear elasticity of cell complexes based on discrete
exterior calculus is presented. The primary unknown are displacements,
represented by primal vector-valued 0-cochain. Displacement differences and
internal forces are represented by primal vector-valued 1-cochain and dual
vector-valued 2-cochain, respectively. The macroscopic constitutive relation is
enforced at primal 0-cells with the help of musical isomorphisms mapping
cochains to smooth fields and vice versa. The balance of linear momentum is
established at primal 0-cells. The governing equations are solved as a Laplace
equation with a non-local and non-diagonal material Hodge star. Numerical
simulations of several classical problems with analytic solutions are presented
to validate the formulation. Good agreement with known solutions is obtained.
The formulation provides a method to calculate the relations between
displacement differences and internal forces for any lattice structure, when
the structure is required to follow a prescribed macroscopic elastic behaviour.
This is also the first and critical step in developing formulations for
dissipative processes in cell complexes.
