A geometric formulation of linear elasticity based on discrete exterior calculus. (1st February 2022) - Boom, Pieter D.; Kosmas, Odysseas; Margetts, Lee; Jivkov, Andrey P.
Abstract: 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. Highlights: 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.