Similarity measures for identifying material parameters from hysteresis loops using inverse analysis

Abstract

Sum-of-square based error formulations may be difficult to implement on an inverse analysis consisting of multiple tension-compression hysteresis loops. Five alternative measures of similarity between curves are investigated as useful tools to help identify parameters from hysteresis loops with inverse analyses. A new algorithm is presented to calculate the area between curves. Four additional methods are presented from literature, which include the Partial Curve Mapping value, discrete Fréchet distance, Dynamic Time Warping, and Curve Length approach. These similarity measures are compared by solving a non-linear regression problem resembling a single load-unload cycle. The measures are then used to solve more complicated inverse analysis, where material parameters are identified for a kinematic hardening transversely anisotropic material model. The inverse analysis finds material parameters such that a non-linear FE model reproduces the behavior from five experimental hysteresis loops. Each method was shown to find useful parameters for these problems, and should be considered a viable alternative when sum-of-square based methods may be difficult to implement. It is important to consider multiple similarity measures in cases when it is impossible to obtain a perfect match.

Appendices

Appendix A: algorithm to calculate area between two curves

The area of any simple (non self intersecting) quadrilateral can be expressed by Gauss’s area formula (also known as the shoelace formula). Gauss’s area formula for a simple quadrilateral is

$$ A = \frac{1}{2}\big | x_{1} y_{2} + x_{2} y_{3} + x_{3}y_{4} + x_{4}y_{1} - x_{2}y_{1} - x_{3}y_{2} - x_{4}y_{3} -x_{1} y_{4} \big | $$

(3)

where A is the area and (x_i, y_i) represents the vertices of the quadrilateral. It is worthwhile to note that any complex quadrilateral can become a simple quadrilateral by rearranging the order of the vertices.

The interior angles of a quadrilateral can be used to detect whether a quadrilateral is simple or complex. Any simple quadrilateral will have a sum of interior angles that add up to 360°. If all interior angles are less than 180°, the simple quadrilateral is said to be convex. However if one interior angle is greater than 180°, the simple quadrilateral is said to be concave. The interior angles of complex quadrilaterals will add up to 720°. An example of a complex, concave, and convex quadrilaterals are shown in Fig. 16.

Fig. 16

Examples of complex and simple quadrilaterals. A simple quadrilateral can be either concave or convex

The change of sign of cross products can be used to detect if a quadrilateral is complex (as an interpretation of the interior angles). Let’s consider an arbitrary quadrilateral represented by the following vectors:

$$\begin{array}{@{}rcl@{}} \boldsymbol{AB} &=& <x_{2} - x_{1}, y_{2} - y_{1}>\end{array} $$

(4)

$$\begin{array}{@{}rcl@{}} \boldsymbol{BC} &=& <x_{3} - x_{2}, y_{3} - y_{2}>\end{array} $$

(5)

$$\begin{array}{@{}rcl@{}} \boldsymbol{CD} &=& <x_{4}- x_{3}, y_{4}, - y_{3}>\end{array} $$

(6)

$$\begin{array}{@{}rcl@{}} \boldsymbol{DA} &=& <x_{1} - x_{4}, y_{1} - y_{4}>\end{array} $$

(7)

The sign of the following cross products dictates whether a quadrilateral is self intersecting or not.

$$\begin{array}{@{}rcl@{}} \boldsymbol{AB} \times \boldsymbol{BC} \end{array} $$

(8)

$$\begin{array}{@{}rcl@{}} \boldsymbol{BC} \times \boldsymbol{CD} \end{array} $$

(9)

$$\begin{array}{@{}rcl@{}} \boldsymbol{CD} \times \boldsymbol{DA} \end{array} $$

(10)

$$\begin{array}{@{}rcl@{}} \boldsymbol{DA} \times \boldsymbol{AB} \end{array} $$

(11)

A complex quadrilateral exists if and only if two of the above cross products are negative and the other two are positive. A simple quadrilateral will have at least three of the same sign cross products. The vertices of a complex quadrilateral can be used to create a simple quadrilateral simply by rearranging the order as shown in Fig. 17.

Fig. 17

By swapping the vertices of (x₁, y₁) with (x₂, y₂), the complex quadrilateral of (a) becomes the simple quadrilateral in (b)

A pseudocode algorithm to compute the effective area between two curves is presented as Algorithm 1. The algorithm first ensures that the two curves have the same number of data points. If not, points are added to the curve with fewer points. The total number of quadrilaterals created will be one less than the number of data points. Two consecutive points are taken from each curve, acting as the vertices of the quadrilateral. Note that the order of the data points which represent the curve is important. The first quadrilateral uses the first and second data point from each curve, the second quadrilateral uses the second and third data point from each curve and so forth. Each quadrilateral is then determined to be either simple or complex. If the quadrilateral is complex, the vertices are reordered until the quadrilateral becomes simple. The area of each simple quadrilateral is calculated using the Gauss area formula, and all quadrilateral areas are summed to give an effective area between curves.

Appendix B: line plot for kinematic hardening problem

line plots were performed in between data points of the kinematic hardening problem to visualize the design space with different similarity measures. All objective function values were normalized using

$$ z_{i} = \frac{x_{i} - \min (\boldsymbol{x})}{\max (\boldsymbol{x}) - \min (\boldsymbol{x})} $$

(12)

such that zero is the best found objective function, and one is the worst. The line plots without noise are presented in Fig. 18, and the line plots with noise are presented in Fig. 19. The line plots help to illustrate the state of the design space (for each measure of similarity) between the optima found. Additionally it appears that the Area and Curve Length methods produced smoother design spaces than the PCM and Discrete Fréchet methods. The line searches were calculated from the results of the first SDRM optimization result, and sometimes display a local optimum that the SDRM failed to find.

Fig. 18

Without noise: Line search with normalized objective values from one objective optimum to another objective optimum

Fig. 19

With noise: Line plot with normalized objective values from one objective optimum to another objective optimum

Appendix C: kinematic hardening results

It is difficult to recognize the differences between similarity mea- sures for the kinematic hardening parameter identification from Section “Kinematic hardening parameter identification from five cycles”. This appendix section shows the results from Section “Kinematic hardening parameter identification from five cycles” for the individual hysteresis loops. The results of the hysteresis curve from the parameter identification can be seen in Figs. 20, 21, 22, 23 and 24, and the results with noise in Figs. 25, 26, 27, 28 and 29.

Fig. 20

Area results for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 21

PCM results for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 22

Discrete Fréchet results for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 23

DTW results for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 24

Curve Length results for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 25

Area results with noise for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 26

PCM results with noise for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 27

Discrete Fréchet results with noise for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 28

DTW results with noise for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle

Fig. 29

Curve Length results with noise for kinematic hardening parameter identification a) first cycle b) second cycle c) third cycle d) fourth cycle e) fifth cycle