Applications of conic programming in non-smooth mechanics

TL;DR

This paper reviews how conic programming techniques can be applied to solve complex non-smooth problems in nonlinear mechanics, improving robustness and efficiency of numerical solutions.

Contribution

It introduces new formulations of non-smooth mechanics problems as conic programs, leveraging advanced interior-point algorithms for better computational performance.

Findings

Conic programming effectively handles non-smooth mechanics problems.

Formulations improve robustness and computational efficiency.

Applicable to plasticity, membranes, cracks, and fluid flows.

Abstract

In the field of nonlinear mechanics, many challenging problems (e.g. plasticity, contact, masonry structures, nonlinear membranes) turn out to be expressible as conic programs. In general, such problems are non-smooth in nature (plasticity condition, unilateral condition, etc.), which makes their numerical resolution through standard Newton methods quite difficult. Their formulation as conic programs alleviates this difficulty since large-scale conic optimization problems can now be solved in a very robust and efficient manner, thanks to the development of dedicated interior-point algorithms. In this contribution, we review old and novel formulations of various non-smooth mechanics problems including associated plasticity with nonlinear hardening, nonlinear membranes, minimal crack surfaces and visco-plastic fluid flows.