3. Numerical formulation

Knowing the temperature and the proportion of phase \(\beta\) at the previous moment and the temperature at the current instant, the aim is to determine the proportion of phase \(\beta\) at the current instant \({Z}_{\mathrm{\beta }}^{t}\).

At a given moment, we are looking for solution \({Z}_{\mathrm{\beta }}^{t}\) such as \(G({Z}_{\mathrm{\beta }}^{t})=0\), an equation that is solved by a Newton method with controlled limits:

(3.1)\[ {Z} _ {\ mathrm {\ beta}}} ^ {i+1}} = {Z} _ {\ mathrm {\ beta}} ^ {i} -\ frac {G ({Z} _ {\ mathrm {\ beta}}} {\ beta}}} ^ {beta}}} ^ {beta}} ^ {beta}} ^ {beta}} ^ {beta}} ^ {beta}} ^ {beta}} ^ {i})}} {G\ text {i})}}\]

The stopping criterion is given by the following condition:

(3.2)\[ \ text {If} G ({Z} _ {\ mathrm {\ beta}}} ^ {i+1})\ le {10} ^ {-6}\ text {then} {Z} _ {\ mathrm {\ beta} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}} ^ {beta}}\]

3.1. Determining the direction of evolution

To know which model to integrate at a given moment \(t\), all you have to do is make the following observations:

• If \({Z}_{\mathrm{\beta }}^{t-1}<{Z}_{\mathrm{\beta }}^{\mathit{eq}}({T}^{t})\) and if we integrate the model into the cooling, we must have \({Z}_{\mathrm{\beta }}^{t}<{Z}_{\mathrm{\beta }}^{t-1}<{Z}_{\mathrm{\beta }}^{\mathit{eq}}({T}^{t})\). However, this is contrary to the condition for applying the model to cooling, which assumes that \({Z}_{\mathrm{\beta }}^{t}>{Z}_{\mathrm{\beta }}^{\mathit{eq}}({T}^{t})\). It is therefore necessary to choose the heating model.

• If \({Z}_{\mathrm{\beta }}^{t-1}>{Z}_{\mathrm{\beta }}^{\mathit{eq}}({T}^{t})\) and if we integrate the model into the heating, we must have \({Z}_{\mathrm{\beta }}^{t}>{Z}_{\mathrm{\beta }}^{t-1}>{Z}_{\mathrm{\beta }}^{\mathit{eq}}({T}^{t})\). However, this is contrary to the condition for applying the model to heating, which assumes that \({Z}_{\mathrm{\beta }}^{t}<{Z}_{\mathrm{\beta }}^{\mathit{eq}}({T}^{t})\). It is therefore necessary to choose the cooling model.

3.2. Integration of equations

3.2.1. Template for META_LEMA_ANI

In the case of using the law of mechanical behavior META_LEMA_ANI, the metallurgical phase is an internal variable of the law of behavior. It is therefore determined during the integration of the latter. The resolution of these equations is then ensured by the code generator MFront.

3.3. Templates for other cases

In any other case, integration is carried out in the following manner:

• Heated model: the solution is such as \({Z}_{\mathrm{\beta }}^{t-1}\le {Z}_{\mathrm{\beta }}^{t}<{Z}_{\mathrm{\beta }}^{\mathit{eq}}({T}^{t})\). The function \({G}_{c}({Z}_{\mathrm{\beta }})\) and its derivative are given by:

:math:`{G}_{c}({Z}_{mathrm{beta }})={Z}_{mathrm{beta }}-{Z}_{mathrm{beta }}^{t-1}-mathrm{Delta }t{A}_{c}mathrm{exp}left(-frac{E}{RT}right){

T-{T}_{mathit{eq}}

}^{M}`

:math:`G{text{“}}_{c}({Z}_{mathrm{beta }})=1+Mmathrm{Delta }t{A}_{c}mathrm{exp}left(-frac{E}{RT}right){

T-{T}_{mathit{eq}}

}^{M-1}T{text{“}}_{mathit{eq}}`

• Cooling model: the solution is such as \({Z}_{\mathrm{\beta }}^{\mathit{eq}}({T}^{t})<{Z}_{\mathrm{\beta }}^{t}\le {Z}_{\mathrm{\beta }}^{t-1}\). The function \({G}_{r}({Z}_{\mathrm{\beta }})\) and its derivative are given by:

:math:`{G}_{r}({Z}_{mathrm{beta }})={Z}_{mathrm{beta }}-{Z}_{mathrm{beta }}^{t-1}+mathrm{Delta }t

T-{T}_{mathit{eq}}

mathrm{exp}left({A}_{r}+{B}_{r}

T-{T}_{mathit{eq}}

right){Z}_{mathrm{beta }}

1-{Z}_{mathrm{beta }}

`

:math:`begin{array}{c}G{text{“}}_{c}({Z}_{mathrm{beta }})=1+mathrm{Delta }t

T-{T}_{mathit{eq}}

mathrm{exp}left({A}_{r}+{B}_{r}

T-{T}_{mathit{eq}}

right)left(1-2{Z}_{mathrm{beta }}right)\ -mathrm{Delta }tmathit{sig}(T-{T}_{mathit{eq}})T{text{“}}_{mathit{eq}}mathrm{exp}left({A}_{r}+{B}_{r}

T-{T}_{mathit{eq}}

right){Z}_{mathrm{beta }}(1-{Z}_{mathrm{beta }}){1+{B}_{r}

T-{T}_{mathit{eq}}

}end{array}`

With:

(3.3)\[\begin{split} \ {\ begin {array} {ccc}\ text {If} {Z}} _ {\ mathrm {\ beta}} =0&\ text {for} & T {\ text {'}} _ {\ mathit {eq}}} =1000\\ mathit {eq}}} =1000\\ mathit {eq}}} =1000\\ mathit {eq}}} =1000\\ mathit {eq}} =1000\\\ mathit {eq}}} =1000\\\ mathit {eq}} =1000\\\\\ text {eq}} =1000\\\\\ text {eq}} =1000\\\\\ text {eq}} =1000\\\\ text {eq}} =1000\\\\\ text {\ text {'}} _ {\ mathit {eq}} =\ frac {1} {eq}}} =\ frac {1}}} =\ frac {1} {1- {Z} _ {\ mathrm {\ eq}} _ {\ mathrm {\ beta}}}}\ frac {\ beta}}}}\ frac {\ beta}}}}\ frac {\ beta}}}}\ right)\ right)\ right)} ^ {\ frac {1} {n} -1}\ frac {1} {Z} _ {\ mathrm {\ beta}} _ {\ mathrm {\ beta}}}}\ frac {\ beta}}}}\ right)\ right)\ right)} ^ {\ frac {1} {} _ {\ mathrm {\ beta}}}}\\\ text {Si}\\ text {Si}\ mathrm {0.99} < {Z} _ {\ mathrm {\ beta}}\ le 1&\ text {for}} & T {\ text {'}} & T {\ text {'}}} & T {\ text {for}} & T {\ text {for}} & T {\ text {'}} & T {\ text {'}} & T {\ text {'}} & T {\ text {'}} & T {\ text {'}} & T {\ text {'}} & T {\ text {'}} & T {\ text {'}} & T {\ text {'}} & T {\ text {'}} & T {\ {0.99})\ end {array}\end{split}\]