A Geometric Formulation of Linear Elasticity Based on Discrete Exterior Calculus

TL;DR

This paper introduces a novel geometric formulation of linear elasticity using discrete exterior calculus, enabling accurate simulations of elastic behavior in complex lattice structures.

Contribution

It presents a new discrete exterior calculus-based approach to model linear elasticity, linking displacements, forces, and constitutive relations on cell complexes.

Findings

Numerical simulations match analytical solutions accurately.

The method effectively models elastic behavior in various lattice structures.

Provides a foundation for future dissipative process formulations.

Abstract

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.