An efficient monolithic solution scheme for FE$^2$ problems

TL;DR

This paper introduces a monolithic FE$^2$ scheme that solves macro and micro displacements simultaneously, significantly reducing computational costs in multiscale simulations.

Contribution

It presents a novel monolithic FE$^2$ approach with a common Newton-Raphson loop, implemented in Abaqus, that decreases micro-scale iteration costs.

Findings

Reduces computational costs by up to 60%.

Requires minimal modifications to existing staggered schemes.

Successfully implemented in commercial FE software Abaqus.

Abstract

The FE$^2$ method is a very flexible but computationally expensive tool for multiscale simulations. In conventional implementations, the microscopic displacements are iteratively solved for within each macroscopic iteration loop, although the macroscopic strains imposed as boundary conditions at the micro-scale only represent estimates. In order to reduce the number of expensive micro-scale iterations, the present contribution presents a monolithic FE$^2$ scheme, for which the displacements at the micro-scale and at the macro-scale are solved for in a common Newton-Raphson loop. In this case, the linear system of equations within each iteration is solved by static condensation, so that only very limited modifications to the conventional, staggered scheme are necessary. The proposed monolithic FE$^2$ algorithm is implemented into the commercial FE code Abaqus. Benchmark examples demonstrate that the monolithic scheme saves up to ~60% of computational costs.