*3.1. Training of ML Yield Function*

A reference material with Hill-type anisotropy is defined with the material parameters as given in Table 1. The yield locus of this reference material for plane-stress conditions with *σ*<sup>3</sup> = 0 is plotted in Figure 1, in which the yield locus of the reference material with Hill-like anisotropy is compared with that of an isotropic material with the same yield strength *σy*.

**Figure 1.** Yield locus for plane-stress conditions (*σ*<sup>3</sup> = 0) and Hill-like anisotropy with parameters given in Table 1 (red line) and for an isotropic material with the same yield strength, but *H*<sup>1</sup> = *H*<sup>2</sup> = *H*<sup>3</sup> = 1 (blue line). The values of the principal stresses are normalized by the yield strength *σy*.

**Table 1.** Elastic and plastic material parameters defining the reference material with Hill-like anisotropy in its plastic flow behavior. For simplicity, isotropic elastic behavior and ideal plasticity without work hardening are assumed in this work.


The thus-defined reference material is used to produce training data—and later also test data—for the machine learning algorithm. To accomplish this, a set of stress values in form of principal deviatoric stresses is produced in a way to cover the complete space of polar angles and also sufficiently many equivalent stresses in the elastic and plastic regimes. This is conveniently achieved by creating a set of *n*ang equally distributed polar angles *θ* (*k*) in the range of [−*π*, *π*] and a set of *n<sup>s</sup>* equivalent stresses *σ* J2 *eq* (*l*) in the range [0.1*σy*, 5*σy*]. Note that for the entire procedure, the yield strength *σ<sup>y</sup>* of the material is assumed to be known. This is not a restriction, because the yield strength of an unknown material can be easily determined from the input data in a pre-analysis step.

The transformation into principal stresses is performed as

$$
\sigma\_{\text{train}}^{(j)} = \sqrt{2/3} \,\sigma\_{\epsilon q}^{\text{l}2(l)} \left( \mathfrak{a} \cos \theta^{(k)} + \mathfrak{b} \sin \theta^{(k)} \right) \tag{22}
$$

with

$$j = k + (l - 1)n\_{\text{ang}} \qquad (k = 1, \dots, n\_{\text{ang}}; l = 1, \dots, n\_{\text{s}}) \tag{23}$$

and the unit angles *a* and *b* spanning the deviatoric stress plane as given above. This produces a set of *n<sup>t</sup>* = *n*ang*n<sup>s</sup>* principal stresses with which, finally, the set of result vectors

$$y^{(j)} = \text{sgn}\left(f(\boldsymbol{\sigma}\_{\text{train}}^{(j)})\right) \qquad \text{with} \qquad j = (1, \ldots, n\_t) \,,\tag{24}$$

is generated by evaluating the yield function *f* of the reference material, as defined in Equations (2) and (5), with the material parameters given in Table 1.

In the numerical example given here, the full training data set comprises *n*ang = 36 values for the polar angle and *n<sup>s</sup>* = 28 values for the equivalent stress for each angle, resulting in a total of *n<sup>t</sup>* = 1008 training data sets. Concerning the effort to create this data, it is noted here that only the number of angles *n*ang is relevant for the number of experiments or micromechanical simulations necessary to generate the training data, because each angle defines a load case from which several stresses in the elastic and plastic regime will result, and other training data points can be easily constructed from this raw data by linear scaling, as described above. The implications of the number of load cases required to achieve an accurate representation of the ML yield function will be further discussed in Section 4. A graphical representation of the training data is shown in Figure 2, where also the different ways of representing the anisotropy of the yield function with Hill-type equivalent stresses and von Mises (J2) equivalent stresses are demonstrated. The actual training data comprise four additional sets of polar angles associated with larger equivalent stresses scaled to values of up to *σ*eq = 5*σ<sup>y</sup>* to prevent the decision function from falling back to zero in this regime, which might cause erroneous results in FEA. To enforce the periodicity of the training ML yield function and its gradient, the training data is periodically repeated within the training algorithm, such that the polar angle covers a range <sup>−</sup>1.3*<sup>π</sup>* <sup>&</sup>lt; *<sup>θ</sup>* <sup>&</sup>lt; 1.3*π*.

**Figure 2.** Polar plots of a subset of the training data produced from the anisotropic yield function of the reference material: (**a**) Von Mises (J2) equivalent stresses according to Equation (3) are used, such that the yield strength, rather than the equivalent stress, is a function of the polar angle *θ*. (**b**) Equivalent stresses are calculated according to the Hill definition in Equation (5) to achieve a constant yield strength by mapping the equivalent stresses accordingly. In both figures, the yield locus is indicated by a solid black line, data points in the elastic regime are plotted in blue color and data in the plastic regime in red color. Both figures represent the same stress data, only mapped in a different way; all stresses are normalized with the reference yield strength *σy*.

With this data set, the training of the SVC algorithm is performed. Using the training parameters *C* = 10 and *γ* = 4 results in a very good training score of above 99%. However, to evaluate the true quality of the training procedure and to judge whether overfitting has occurred, it is necessary to verify the results with an independent set of test data, which has not been used for training purposes. The error produced on such test data sets with 480 random deviatoric stresses as data points is below 1%, and the *R* 2 -correlation coefficient between test data and training data is above 98%, which leads to the conclusion that the trained ML yield function has a very high accuracy and robustness. A variation of the training parameters revealed that the results are rather insensitive to the parameter *C*, which can be varied between 2 < *C* < 20 without having a pronounced influence on the results, whereas changing the parameter *γ* by more then 20% causes a significant deterioration of the training results. The resulting SVC decision function, defined in Equation (18), is plotted together with the training data in Figure 3 in the deviatoric stress space.

**Figure 3.** Field plot of the trained SVM decision function defined in Equation (18), where areas in purple color shades represent negative values and brown shades represent positive values. The numerical value of the decision function is not relevant because only its sign is taken into account in the flow rule. The isoline for *f*SVC = 0 is represented as a black line. Training data are plotted in light blue color for data with negative values (elastic) and in brown color for positive values (plastic).

Finally, to demonstrate the quality of the ML yield function in the full principal stress space, the predicted categories are plotted in three different slices corresponding to different plane-stress conditions, together with the yield function of the reference material in Figure 4. It is seen that the training data in the deviatoric space covers in fact only a single line in each slice, which demonstrates the power of reducing the dimensionality of the data-oriented yield function by exploiting basic physical principles.

**Figure 4.** Color map of the trained SVC prediction of the yield function in slices through the principal stress space defined by plane-stress conditions: (**a**) *σ*<sup>3</sup> = 0; (**b**) *σ*<sup>1</sup> = 0; (**c**) *σ*<sup>2</sup> = 0. Brown regions indicate values of "+1" (plasticity) and purple regions values of "-1" (elasticity). The ML yield locus, corresponding to the isoline for *f*SVC = 0, is represented as a black line; the yield locus of the Hill-like anisotropic reference material is indicated as a red line. The training data points are plotted with the same color code as in Figure 3.
