An overview of mixed finite elements for the analysis of inelastic bidimensional structures

TL;DR

This paper reviews and analyzes mixed finite element methods for inelastic bidimensional structures, focusing on their theoretical foundations, numerical stability, and performance in elastoplastic media within a unified framework.

Contribution

It provides a comprehensive methodological analysis and comparison of mixed finite element formulations for inelastic structures, highlighting key aspects for high-performance element design.

Findings

Critical review of variational formulations and interpolation spaces

Assessment of numerical stability and solution strategies

Numerical simulations demonstrating performance of various formulations

Abstract

As inelastic structures are ubiquitous in many engineering fields, a central task in computational mechanics is to develop accurate, robust and efficient tools for their analysis. Motivated by the poor performances exhibited by standard displacement-based finite element formulations, attention is here focused on the use of mixed methods as approximation technique, within the small strain framework, for the mechanical problem of inelastic bidimensional structures. Despite a great flexibility characterizes mixed element formulations, several theoretical and numerical aspects have to be carefully taken into account in the design of a high-performance element. The present work aims at providing the basis for methodological analysis and comparison in such aspects, within the unified mathematical setting supplied by generalized standard material model and with special interest towards elastoplastic media. A critical review of the state-of-the-art computational methods is delivered in regard to variational formulations, selection of interpolation spaces, numerical solution strategies and numerical stability. Though those arguments are interrelated, a topic-oriented presentation is resorted to, for the very rich available literature to be properly examined. Finally, the performances of several significant mixed finite element formulations are investigated in numerical simulations.