A geometric formulation of linear elasticity based on discrete exterior calculus

A direct formulation of linear elasticity of cell complexes based on discrete exterior calculus is presented. The primary unknowns are displacements, represented by a primal vector-valued 0-cochain. Displacement differences and internal forces are represented by a primal vector-valued 1-cochain and a 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 Poisson’s 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. Excellent 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. •Formulation of linear elasticity using discrete exterior calculus (DEC).•Derivation of new discrete sharp musical isomorphism.•Derivation of non-local and non-diagonal discrete Hodge star for linear elasticity.•Numerical validation using standard mechanical problems with analytic solutions.