Hierarchical Rank-One Sequence Convexification for the Relaxation of Variational Problems with Microstructures

TL;DR

This paper introduces an efficient hierarchical rank-one sequence algorithm for approximating the rank-one convex hull, aiding in the relaxation of variational problems with microstructures in nonlinear solid mechanics.

Contribution

It presents a novel hierarchical rank-one sequence method that provides first and second derivatives for energy minimization and stress calculation, applicable to complex microstructures.

Findings

Algorithm efficiently approximates the rank-one convex hull.

Applicable to 2D and 3D nonconvex damage models.

Demonstrates effectiveness on benchmark and real-world problems.

Abstract

This paper presents an efficient algorithm for the approximation of the rank-one convex hull in the context of nonlinear solid mechanics. It is based on hierarchical rank-one sequences and simultaneously provides first and second derivative information essential for the calculation of mechanical stresses and the computational minimization of discretized energies. For materials, whose microstructure can be well approximated in terms of laminates and where each laminate stage achieves energetic optimality with respect to the current stage, the approximate envelope coincides with the rank-one convex envelope. Although the proposed method provides only an upper bound for the rank-one convex hull, a careful examination of the resulting constraints shows a decent applicability in mechanical problems. Various aspects of the algorithm are discussed, including the restoration of rotational invariance, microstructure reconstruction, comparisons with other semi-convexification algorithms, and mesh independency. Overall, this paper demonstrates the efficiency of the algorithm for both, well-established mathematical benchmark problems as well as nonconvex isotropic finite-strain continuum damage models in two and three dimensions. Thereby, for the first time, a feasible concurrent numerical relaxation is established for an incremental, dissipative large-strain model with relevant applications in engineering problems.