Inspection of ratcheting models for pathological error sensitivity and overparametrization

Abstract

Accurate analysis of plastic strain accumulation under stress-controlled cyclic loading is vital for numerous engineering applications. Typically, models of plastic ratcheting are calibrated against available experimental data. Since actual experiments are not exactly accurate, one should check the identification protocols for pathological dependencies on experimental errors. In this paper, a step-by-step algorithm is presented to estimate the sensitivities of identified material parameters. As a part of the sensitivity analysis method, a new mechanics-based metric in the space of material parameters is proposed especially for ratcheting-related applications. The sensitivity of material parameters to experimental errors is estimated, based on this metric. Moreover, a relation between pathological error sensitivity and overparametrization is established. This relation gives rise to a new criterion of overparametrization. The advantages of the new overparametrization criterion are exposed and its plausibility is checked by alternative criteria, like the consideration of correlation matrices and validation of identified parameters on “unseen” data. For demonstration purposes, the accumulation of irreversible strain in the titanium alloy VT6 (Russian analog of Ti-6Al-4V) is analysed. Three types of phenomenological models of plastic ratcheting are considered. They are the Armstrong-Frederick model as well as the first and the second Ohno-Wang models. Based on real data, a new rule of isotropic hardening is proposed for greater accuracy of simulation. The ability of the sensitivity analysis to determine reliable and unreliable parameters is demonstrated.

Appendices

Appendix A: Fast computation of $\vec {p}^{\ (j)}$

We discuss a quick computation of the parameter vectors $\vec {p}^{\ (j)} \in \mathbb {R}^{n}$, corresponding to the jth draw of noisy data. The procedure is the same as in [38]. Recall that $\overrightarrow{Exp} \in \mathbb {R}^{N_{\text {exp}}}$ is the vector of available experimental data, $\overrightarrow{Mod}(\vec {p}) \in \mathbb {R}^{N_{\text {exp}}}$ is the corresponding modelling response, $\vec {p} = (\vec {p}_c, \vec {p}_K) \in \mathbb {R}^{n}$ is the vector of unknown material parameters. Within the sensitivity analysis, the actual experimental data are replaced by the noisy data $\overrightarrow{Exp} + \overrightarrow{Noise}$. The optimal set of parameters corresponding to noise-free data is denoted as $\vec {p}^{\ *}$. The Jacobian of the model response at $\vec {p}^{\ *}$ is the operator

$$\begin{aligned} \mathbf {J} := \frac{\partial \overrightarrow{Mod}(\vec {p})}{\partial \vec {p}}|_{\vec {p}^{\ *}} \in \mathbb {R}^{N_{\text {exp}} \times n}. \end{aligned}$$

(44)

Assuming only small changes in parameters, we linearize the model response near $\vec {p}^{\ *}$:

$$\begin{aligned} \overrightarrow{Mod}^{lin}(\vec {p}) := \overrightarrow{Mod}(\vec {p}^{\ *}) + \mathbf {J}(\vec {p} - \vec {p}^{\ *}). \end{aligned}$$

(45)

Now, the parameter set $\vec {p}^{\ (j)}$, $j = 1, 2, ..., N_{\text {noise}}$ is the minimizer of the error function for noisy data

$$\begin{aligned}&\Phi ^{\text {noisy}}(\vec {p}) := \overrightarrow{Resid}^{\text {T}} \ \overrightarrow{Resid}, \end{aligned}$$

(46)

$$\begin{aligned}&\overrightarrow{Resid} := \overrightarrow{Exp} + \overrightarrow{Noise} - \overrightarrow{Mod}^{lin} \nonumber \\&\quad = \overrightarrow{Exp} + \overrightarrow{Noise} - \overrightarrow{Mod}(\vec {p}^{\ *}) - \mathbf {J}(\vec {p} - \vec {p}^{\ *}). \end{aligned}$$

(47)

Abbreviate by $\overrightarrow{A}$ the following vector:

$$\begin{aligned} \overrightarrow{A} := \overrightarrow{\text {Exp}} + \overrightarrow{\text {Noise}} - \overrightarrow{\text {Mod}(\vec {p}^{\ *})} - \mathbf {J} \ \vec {p}^{\ *} . \end{aligned}$$

(48)

Then the error function is a quadratic form of $\vec {p}$, given by

$$\begin{aligned} \Phi ^{\text {noisy}}(\vec {p}) = \big (\overrightarrow{A} - \mathbf {J} \vec {p}\big )^{\text {T}} \big (\overrightarrow{A} - \mathbf {J} \vec {p}\big ). \end{aligned}$$

(49)

Its derivative with respect to $\vec {p}$ is a linear function of the unknown parameter vector $\vec {p}$

$$\begin{aligned} \frac{\displaystyle \partial \Phi ^{\text {noisy}}(\vec {p})}{\displaystyle \partial \vec {p} } = - 2 \big ( \overrightarrow{A} - \mathbf {J} \vec {p} \big )^{\text {T}} \ \mathbf {J}. \end{aligned}$$

(50)

The stationarity condition $\frac{\displaystyle \partial \Phi ^{\text {noisy}}(\vec {p})}{\displaystyle \partial \vec {p} } = 0$ yields a system of linear algebraic equations with respect to $\vec {p}$. Then the analytical solution is

$$\begin{aligned} \vec {p}^{\ (j)} = \big ( \mathbf {J}^{\text {T}} \mathbf {J} \big )^{-1} \big ( \mathbf {J} \big )^{\text {T}} \ \overrightarrow{A}. \end{aligned}$$

(51)

In fact, this semi-analytical solution represents a single iteration of the Gauss-Newton method [49].

Unfortunately, due to matrix multiplication, the condition number of $\mathbf {J}^{\text {T}} \mathbf {J}$ can be very large. This effect may falsify the results of (51). To resolve this problem, $\mathbf {Q} \mathbf {R}$ decomposition of $\mathbf {J}$ should be implemented:

$$\begin{aligned}&\mathbf {J} = \mathbf {Q} \ \mathbf {R} \in \mathbb {R}^{N_{\text {exp}} \times n}, \quad \mathbf {Q} \in \mathbb {R}^{N_{\text {exp}} \times n},\nonumber \\&\quad \mathbf {Q}^{\text {T}} \mathbf {Q} = id_{\mathbb {R}^{n}} \in \mathbb {R}^{n \times n}, \quad \mathbf {R} \in \mathbb {R}^{n \times n}. \end{aligned}$$

(52)

Here, $\mathbf {R}$ is upper triangular. Substituting this into (51), we have after some computations

$$\begin{aligned} \vec {p}^{\ (j)} = \mathbf {R}^{-1} \mathbf {Q}^{\text {T}} \ \overrightarrow{A}. \end{aligned}$$

(53)

Since the matrix $\mathbf {R}^{-1} \mathbf {Q}^{\text {T}}$ is pre-computed, the parameter cloud is evaluated extremely efficiently even for a large number of draws ($N_{\text {draws}} \ge 10,000$). The condition number of the matrix $\mathbf {R}$ behaves like the square root of the condition number of $\mathbf {J}^{\text {T}} \mathbf {J}$. Thus, (53) gives more robust solution than (51).

Appendix B: Correlation matrices

Let $\mathbf {J}$ be the Jacobian, defined in (44). The correlation matrix $\mathbf {Corr} \in \mathbb {R}^{n \times n}$ is defined as follows [7]:

$$\begin{aligned} \mathbf {Corr}_{i j} = \mathbf {P}_{i j}/ \sqrt{ \mathbf {P}_{i i} \mathbf {P}_{j j} }, \quad \text {where} \quad \mathbf {P} = \big ( \mathbf {J}^{\text {T}} \ \mathbf {J} \big )^{-1} \in \mathbb {R}^{n \times n}. \end{aligned}$$

(54)

We say that there is a strong correlation between parameters $p_i$ and $p_j$, if $\mathbf {Corr}_{i j} \approx \pm 1$. In that case, a slight change in $p_i$ can be counteracted by a change in $p_j$, still leaving the model response $\overrightarrow{Mod}$ virtually the same. In such situations the minimum of the error functional $\Phi$ lies in a “horizontal ravine” (cf. Fig. 10 in [32]). In the sense of Euclidean metric, a large correlation between parameters is characteristic for ill-defined optimization problems.

Table 7 Correlation matrix for AF model with $N_{\text {branches}} = 2$

Table 8 Correlation matrix for AF model with $N_{\text {branches}} = 3$

Table 9 Correlation matrix for AF model with $N_{\text {branches}} = 4$

Table 10 Correlation matrix for OW− I model with $N_{\text {branches}} = 2$

Table 11 Correlation matrix for OW-I model with $N_{\text {branches}} = 3$

Table 12 Correlation matrix for OW-I model with $N_{\text {branches}} = 4$

Table 13 Correlation matrix for OW-II model with $N_{\text {branches}} = 2$

Table 14 Correlation matrix for OW-II model with $N_{\text {branches}} = 3$

Table 15 Correlation matrix for OW-II model with $N_{\text {branches}} = 4$

Appendix C: Implementation of Sobol’s sequence

For each draw of the Monte Carlo method, the stochastic model (41) requires 40 independent random numbers $\varepsilon _k \in \mathcal {N}(0,\sigma ^2)$ (20 numbers to obtain noisy data for each test). Within the quasi Monte Carlo method, they are obtained in the following way. First, we set the properties of the Sobol sequence. Dimensions is the number of terms of the Sobol sequence in each draw; we use $Dimensions =40$. Skip is the number of initial points to omit from Sobol’s sequence, we put $Skip = 10^3$. Leap is the interval between points of the sequence; $Leap = 3 \cdot 10^2$ in our case.

Recall that $N_{\text {noise}}$ is the number of draws. Calling Sobol’ generator [8] we obtain a matrix $S \in \mathbb {R}^{N_{\text {noise}} \times Dimensions}$ of quasi-random numbers uniformly distributed over the interval [0, 1]. Then for the jth draw, the corresponding quasi-random variables with the normal distribution are:

$$\begin{aligned}&\varepsilon _{j,2i+1} = \cos ({2\pi } S_{j,2i+1}) \cdot \sqrt{-2 \ln {S_{j,2i+1}}}, \nonumber \\&\quad \varepsilon _{j,2i+2} = \sin ({2\pi } S_{j,2i+1}) \cdot \sqrt{-2 \ln {S_{j,2i+1}}}, \end{aligned}$$

(55)

for $j = 1, 2, ..., N_{\text {noise}}$, $i = 1, 2, ..., Dimensions$.

Appendix D: Fast computation of the distance

The fast computation of the distance between two sets of parameters relies on the linearization of the strain response function $\varepsilon _{11}(t)$ with respect to the material parameters. For the fixed stress-controlled loading history (Fig. 15) we evaluate the derivative

$$\begin{aligned} d \mathbf {\varepsilon }/d \vec {p} (t) = \frac{\partial {\varepsilon _{1 1}(t,\vec {p})} }{\partial {\vec {p}}}|_{\vec {p} = \vec {p}^{\ *}}, \quad t \in [0,T_{\text {metric}}]. \end{aligned}$$

(56)

For the parameter set $\vec {p}$ close to the center of the cloud $\vec {p}^{\ *}$, the axial strain $\varepsilon _{1 1}(t,\vec {p})$ is approximated as

$$\begin{aligned} \varepsilon _{1 1}(t,\vec {p}) = \varepsilon _{1 1}(t,\vec {p}^{\ *}) + d \mathbf {\varepsilon }/d \vec {p} (t) \cdot (\vec {p} - \vec {p}^{\ *}), \end{aligned}$$

(57)

where $\varepsilon _{1 1}(t,\vec {p}^{\ *})$ is the strain history related to the center of the parameter cloud. Then the mechanics-based distance between $\vec {p}^{\ *}$ and $\vec {p}$ is

$$\begin{aligned} \text {dist}(\vec {p}, \vec {p}^{\ *}) = \max \limits _{t \in [0,T_{\text {metric}}]} | d \mathbf {\varepsilon }/d \vec {p} (t) \cdot (\vec {p} - \vec {p}^{\ *}) |. \end{aligned}$$

(58)

Appendix E: AF versus OW-II regarding error sensitivities

Table 5 says that the material parameters of the AF models are more sensitive to experimental errors than the parameters of the OW-II models. To obtain an intuitive insight into this effect, recall that for each instance of noisy data the set $\vec {p}^{\ (j)}$ is computed through (48) and (53). Thus, the scatter of parameters $\vec {p}^{\ (j)}$ depends on the matrix $\mathbf {R} \in \mathbb {R}^{n \times n}$. Recall that $\mathbf {R}$ is obtained by the $\mathbf {Q} \mathbf {R}$ decomposition of the Jacobian $\mathbf {J} \in \mathbb {R}^{N_{exp} \times n}$ (Appendix A). The condition numbers of $\mathbf {R}$ are listed in Table 16 for both models; they are evaluated with respect to the $l_2$ norm. Interestingly, the dependence of the condition number on $N_{\text {branches}}$ is not monotonic. However, the AF models exhibit much larger condition number than the OW-II models in all the cases. Therefore, in the Euclidean $l_2$ metric, the parameter vector of the AF models is more sensitive to the noise than for the OW-II models. This high sensitivity corresponds to the results shown in Table 5.

Table 16 Condition numbers of $\mathbf {R}$

As noted in Appendix A, the condition number of the matrix $\mathbf {R}$ behaves like the square root of the condition number of $\mathbf {J}^{\text {T}} \mathbf {J}$. The corresponding values are summarized in Table 17. Recall that the columns of $\mathbf {J}$ are derivatives of the model response with respect to individual parameters. The large condition number of $\mathbf {J}^{\text {T}} \mathbf {J}$ means that the columns of the Jacobian $\mathbf {J}$ are “close” to being linearly dependent. Geometrically, this means that the columns “almost” lie in an $(n-1)$-dimensional subspace of $\mathbb {R}^{N_{exp}}$. Table 17 suggests that this is likely the case for the AF model.

Table 17 Condition numbers of $\mathbf {J}^{\text {T}} \mathbf {J}$

For the AF models, the minimum eigenvalues $\lambda _{\text {min}}$ of the matrix $\mathbf {R}$ are shown in Table 18. Since the eigenvalues are close to zero, the corresponding eigenvectors represent certain directions in the space of parameters, such that the model response $\vec {Mod}(\vec {p})$ is almost the same along those directions (Table 18). Recalling (35), we conclude that a small increase in $c_1$ can be compensated by an increase in $\gamma$ and $\beta$.

Table 18 Minimum eigenvalues and eigenvectors of $\mathbf {R}$ for the AF models