On mathematical aspects of evolution of dislocation density in metallic materials

TL;DR

This paper analyzes the mathematical modeling of dislocation density evolution in metals using delay differential equations, providing error bounds for numerical methods and validating results through industrial process simulations.

Contribution

It introduces a mathematical framework for modeling dislocation evolution with delay differential equations and derives error bounds for explicit Euler and other numerical methods.

Findings

Error bounds for explicit Euler method under H"older continuity.

Comparison of numerical methods like Runge-Kutta.

Validation through simulations of industrial processes.

Abstract

This paper deals with the solution of delay differential equations describing evolution of dislocation density in metallic materials. Hardening, restoration, and recrystallization characterizing the evolution of dislocation populations provide the essential equation of the model. The last term transforms ordinary differential equation (ODE) into delay differential equation (DDE) with strong (in general, H\"older) nonlinearity. We prove upper error bounds for the explicit Euler method, under the assumption that the right-hand side function is H\"older continuous and monotone which allows us to compare accuracy of other numerical methods in our model (e.g. Runge-Kutta), in particular when explicit formulas for solutions are not known. Finally, we test the above results in simulations of real industrial process.