A Simple and Efficient Lagrange Multiplier Based Mixed Finite Element for Gradient Damage

TL;DR

This paper introduces a novel Lagrange multiplier-based mixed finite element method for gradient damage models, offering mesh-independent, robust, and efficient simulations without additional degrees of freedom or stabilization parameters.

Contribution

The proposed formulation is simple, efficient, and does not require cross-element information, enabling straightforward implementation and improved convergence over existing methods.

Findings

Mesh-independent damage solutions demonstrated.

Robustness under severe damage and cyclic loading.

Computational cost only slightly higher than elastic calculations.

Abstract

A novel finite element formulation for gradient-regularized damage models is presented which allows for the robust, efficient, and mesh-independent simulation of damage phenomena in engineering and biological materials. The paper presents a Lagrange multiplier based mixed finite element formulation for finite strains. Thereby, no numerical stabilization or penalty parameters are required. On the other hand, no additional degrees of freedom appear for the Lagrange multiplier which is achieved through a suitable FE-interpolation scheme allowing for static condensation. In contrast to competitive approaches from the literature with similar efficiency, the proposed formulation does not require cross-element information and thus, a straightforward implementation using standard element routine interfaces is enabled. Numerical tests show mesh-independent solutions, robustness of the solution procedure for states of severe damage and under cyclic loading conditions. It is demonstrated that the computing time of the gradient damage calculations exceeds the one of purely elastic computations only by an insignificant amount. Furthermore, an improved convergence behavior compared to alternative approaches is shown.