Prediction of Distribution of Microstructural Parameters Inmetallic Materials Described by Differential Equations With Recrystallization Term

Approximation of solutions of DDEs under nonstandard assumptions via Euler scheme

“…The assumptions above, especially (F3),(F4), are inspired by the real-life model describing evolution of dislocation density, see [8,11,18,19,20,22] for detailed description and discussion of the DDE involved. In Lemma 3.1 below we prove that under the assumptions (F1)-(F3) the equation (1.1) has unique solution z = z(t) on the whole interval [−τ, (n + 1)τ ].…”

“…Metal phase change model. For another numerical experiments we consider a modified model describing a phase change of metallic materials from [20,Chapter 3.3], see also [8,11,18,19,22]. It can be described by delay differential equation due to delay in the response to the change in processing conditions.…”

“…Meanwhile, it turns out that real world applications need nonstandard assumptions. For example, a phase change of metallic materials can be described by a delay differential equation due to delay in the response to the change in processing conditions [8,11,18,19,20,22]. In [8,19] authors considered a case when a right-hand side function is one-dimensional, locally Hölder continuous and monotone.…”

“…For example, a phase change of metallic materials can be described by a delay differential equation due to delay in the response to the change in processing conditions [8,11,18,19,20,22]. In [8,19] authors considered a case when a right-hand side function is one-dimensional, locally Hölder continuous and monotone. In this paper we generalize the techniques used in [8] and study the error of the classical Euler scheme for a multidimensional case, when a right-hand side function is locally Hölder continuous and satisfies one-side Lipschitz conditions.…”

Mean field and full field modelling of microstructure evolution and phase transformations during hot forming and cooling of low carbon steels

“…Full coupling was applied, what means that flow stress calculated by the IVM model was returned to the constitutive law in the FE model. The results of these simulations are presented by Morkisz et al (2019) and are not repeated here. Explicit finite difference solution of equation ( 4) is performed in each time step of the FE method and, in consequence, this approach does not influence the computing costs of the FE code.…”