Using EikoTwin DIC to complete mechanical regularization - adding mechanical conditions to field strain measurements.

Author:Lucas AngénieuxEikoSimCompany R&D Engineer

In EikoTwin DIC, the calculation of the displacement field is performed on the simulation grid. However, since this mathematical problem is essentially ill-posed, there is a mechanical regularization parameter that allows the user to add mechanical conditions to the field measurements. Unlike most digital image correlation software, in EikoSim, we choose to use a mechanical 'filter' to maintain mechanical significance through regularization rather than filtering manually. We will explore the implications of this regularization in this article, providing some recommendations for its use.

Why add mechanical regularization?

In digital image correlation (DIC), to calculate the displacement field from a set of images, we use the gray levels of the reference image and the subsequent images to track the displacement between the two (for more information on the basics of DIC and global methods). The gray level conservation assumption allows for the establishment of a minimization problem (1) in matrix form:

The Gaussian-Newton type format used to solve this problem has good convergence properties, but it may fail to converge to satisfactory results if the loss function has many flat parts or local minima: this problem is referred to as ill-posed (see Figure 1). Therefore, we will look for a way to modify this function to allow the algorithm to converge despite these issues.Figure 1 – Mathematically ill-posed problem – Local minima in the red circlesOne solution to solve complex problems is to use regularization. This technique involves adding a penalty term to the loss function to discard solutions that do not meet certain criteria expected in the result field. In the case of mechanical regularization, mechanically inadmissible displacement fields (e.g., too 'high frequency') will be penalized. This solution acts as a mechanical 'filter', which can be close to the filters used in most current DIC software, but here it is based solely on mechanical assumptions.Mechanical regularization in EikoTwin DICTo add mechanical information to the problem, we can base it on solid constitutive equations and simplify assumptions based on the material considered. Except for construction materials or multiphase materials, we can choose Hooke's law in linear, homogeneous, and isotropic elasticity (LHI), which will allow faithful handling of many industrial situations.According to Hooke's law and the static equilibrium of solids (3), we modify scheme (1) to the following form (4) [1]:Thus, the new system is equivalent to minimizing a new loss function, which is a linear combination of the image correlation function and the regularization function (see Figure 2). Both functions must be weighted to avoid one function having too significant an impact on the measurement field. This is the operation of the regularization length chosen by the user in EikoTwin DIC.In EikoTwin DIC, the user selects three parameters when starting the measurement of the displacement field and strain field: maximum number of iterations, convergence criteria, and regularization length. The choice of this length affects the results as well as the convergence speed. Therefore, it is necessary to understand its physical significance to make the best choice.

Below, we will discuss different examples. These examples are based on virtual images generated using EikoTwin Virtual, a software developed by EikoSim (for more details, see the article on using EikoTwin Virtual for virtual testing). Therefore, it is possible to generate correlated images for which we fully understand the geometry and theoretical displacement and strain fields, using them as references.

The regularization length is viewed as the size of the local approximation area of the linear elastic model.

As mentioned earlier, mechanical regularization is based on constitutive laws: Hooke's law in linear, homogeneous, and isotropic elasticity. In the absence of mechanical regularization, the finite element formalism has already imposed continuity of displacements from one node to another during the measurement process. Regularization allows us to add mechanical assumptions locally to a set of nodes. Therefore, the regularization length can be viewed as the size of the local area where the model is approximated by a linear elastic model. Thus, an excessively large regularization length will restrict the motion of parts to rigid body motion, while a regularization length smaller than the element size will have no effect.

In the example of Figure 3, we observe a flat part with a displacement step of 2 mm along the normal of the part (in the right image, the green curve represents the displacement profile of the part used in the virtual image for this example). Here, the applied regularization length (100 mm) can be viewed as the length for measuring the step from 0 to 2 mm. The graph on the right shows the displacement profile according to the normal, where the 'transition zone' is 100 mm. This area is indeed the region where the effects of the previously seen linear elastic constitutive relationship are effectively visible.Figure 3 – Regularization length, viewed as the 'transition' length at discontinuities.

为了给问题添加力学信息，我们可以基于固体本构方程，并根据所考虑的材料简化假设。除建筑材料或多相材料外，我们可以选择线性、均质和各向同性弹性（LHI）中的胡克定律，这将允许忠实地处理许多工业情况。

根据胡克定律和固体的静态平衡（3），我们将方案（1）修改为以下形式（4）[1]：

σ––––(x––)=C––––––––:ε––(x––)(2)

∇––.σ––––+f––=0(3)

([Figure 1 – Mathematically ill-posed problem – Local minima in the red circlesOne solution to solve complex problems is to use regularization. This technique involves adding a penalty term to the loss function to discard solutions that do not meet certain criteria expected in the result field. In the case of mechanical regularization, mechanically inadmissible displacement fields (e.g., too 'high frequency') will be penalized. This solution acts as a mechanical 'filter', which can be close to the filters used in most current DIC software, but here it is based solely on mechanical assumptions.]+[Figure 1 – Mathematically ill-posed problem – Local minima in the red circlesReg])⋅{To add mechanical information to the problem, we can base it on solid constitutive equations and simplify assumptions based on the material considered. Except for construction materials or multiphase materials, we can choose Hooke's law in linear, homogeneous, and isotropic elasticity (LHI), which will allow faithful handling of many industrial situations.According to Hooke's law and the static equilibrium of solids (3), we modify scheme (1) to the following form (4) [1]:Thus, the new system is equivalent to minimizing a new loss function, which is a linear combination of the image correlation function and the regularization function (see Figure 2). Both functions must be weighted to avoid one function having too significant an impact on the measurement field. This is the operation of the regularization length chosen by the user in EikoTwin DIC.}–[Figure 1 – Mathematically ill-posed problem – Local minima in the red circlesRegMechanical regularization in EikoTwin DICu}(4)

因此，新系统等效于最小化新的损失函数，即图像相关函数和正则化函数的线性组合（见图2）。必须对这两个函数进行加权，以避免其中一个函数对测量场产生太重要的影响。这是用户在EikoTwin DIC中选择的正则化长度的操作。

EikoTwin-DIC中的正则化长度

在EikoTwin DIC中，用户在启动位移场和应变场测量时选择3个参数：最大迭代次数、收敛准则和正则化长度。该长度的选择会影响结果以及收敛速度。因此，有必要理解其物理含义，以便做出最佳选择。

在下面，我们将讨论不同的示例。这些示例基于使用EikoTwin Virtual生成的虚拟图像，EikoTwin Virtual是EikoSim开发的软件（有关更多说明，请参阅关于使用EikoTwin Virtual进行虚拟测试的文章）。因此，有可能生成相关图像，我们完全了解这些图像的几何形状和理论位移和应变场，并将其用作参考。

正则化长度视为线弹性模型局部近似区域的大小

如前所述，力学正则化基于本构定律：线性、均质和各向同性弹性中的胡克定律。在没有机械正则化的情况下，有限元形式主义已经在测量过程中施加了从一个节点到另一个节点的位移连续性。正则化允许我们在局部向一组节点添加机械假设。因此，正则化长度可以视为局部区域的大小，其中模型由线弹性模型近似。因此，过大的正则化长度会将零件的运动限制为刚体运动，而小于元素大小的正则化长度则不会产生任何影响。

在图3的示例中，我们观察到一个平面零件，其沿零件法线的位移步长为2 mm（在右侧的图中，绿色曲线表示应用于该示例中使用的虚拟图像中零件的位移轮廓）。这里，应用的正则化长度（100 mm）可以被视为测量0到2 mm步长的长度。右侧的图表显示了根据法线的位移剖面，其中“过渡区”为100 mm。该区域确实是之前所见的线弹性本构关系的影响有效可见的区域。

图3——正则化长度，被视为不连续处的“过渡”长度

In practice, care must be taken not to choose a regularization length that is too significant compared to the magnitude being measured, so as not to lose the information contained in the initial measurement, which may produce noise (without regularization length).

Regularization length as the cutoff frequency of a low-pass filter

One can also view the regularization length as a filter that can cut off high-frequency variations in the displacement field. In this case, the wavelength associated with the cutoff frequency is equal to the regularization length.

Figure 4 below provides relevant illustrations of this vision. Here, we observe the case of a 2000 mm L-shaped model. The virtual image of the model (using EikoTwin Virtual) is generated according to the outline in Figure 4: pseudo-sinusoidal normal displacements along the Y direction, with a fixed amplitude, whose wavelength varies continuously from 100 mm to 1500 mm along the length of the model. The size of these elements is approximately 60 mm.

Figure 4 – Example of a solid model with variable wavelength pseudo-sinusoidal displacement

We then analyze the measured displacement field with different mechanical regularization lengths (from 50 mm at the top to 1000 mm at the bottom) in Figure 5. As expected, regularization has no effect on the minimum length (values smaller than the element size; the signal is noisy, but all displacement fluctuations have been measured). For the maximum regularization length (corresponding to half the plate size), there are no longer very high-frequency fluctuations ("noise") at the scale of the elements (especially in areas close to the vertical plane); the small wavelengths before this 1000 mm length are significantly attenuated. Only waves with wavelengths greater than 1000 mm can be measured correctly. On the right side, the normal displacement profile is plotted along a straight line on the Z-axis of the part. We can then observe different displacement profiles based on the wavelength used, highlighting the low-pass filter-like behavior of the regularization.

Towards a more mathematical view of mechanical regularization

Another way to view the impact and weight of mechanical regularization in problem (4) is shown in Figure 6 below:

Figure 6 – The effect of regularization length on the loss function

In Figure 6, we see that the regularization length has an impact on the mechanical regularization loss function [1]. Therefore, in this case, length L1 (too small) has a minimal effect on the total loss function, and the right side will retain local minima that do not correspond to the solution, which may prevent the algorithm from converging correctly. Length L3, which is too long, gives too much weight to mechanical regularization, thus distinguishing the global minimum solution. We see that length L2 will replace the local minima on the right side and support the minimum on the left side, which is the solution.

How to choose the appropriate mechanical regularization length?

It should be noted that the regularization length depends on the situation being studied (mesh, measurement area, phenomena to be observed, etc.), and there is no "ideal" length in absolute terms. Users can choose the most suitable regularization length for their specific situation by finding the best compromise between computational speed, convergence, and detail level (i.e., richness of the displacement field). This article aims to better understand the impact of mechanical regularization on the solution field and the physical significance of the length to be applied.

To illustrate the content of this section, we will look at the strain measured on the virtual image of a conventional tensile specimen that is 95 mm long with a mesh size of 2 mm in the area of interest. The reference strain field along the tensile axis (used to generate the image) is shown in Figure 7 below.

Figure 7 – The tensile strain field of the specimen used in this example,

In EikoSim, we first recommend conducting a comprehensive measurement without regularization. This provides preliminary results without any mechanical regularization effects. In the second step, a regularization length of 2 or 3 element sizes may be related to improved convergence in the absence of measurement noise. Analyzing the effect of regularization length on specific quantities of interest (displacement sensors, strain gauges, etc.) or fields (displacement, strain, residuals) also helps optimize this choice. This is what we did for the sample in this case. From the erosion and our images, it is more important to focus on the residuals and strains in the middle of the specimen rather than the displacement field.

The residual field is an effective tool for judging the choice of regularization length. Users can initiate different measurements by increasing the length and study its effect on the residual field (using batch mode plugins is particularly relevant here). Starting from a certain value, the regularization length will increase the residuals, especially in areas where the part is most deformed. In fact, the measured displacement field will resemble rigid body motion, in which case the contribution of the regularization term in the global function will be disproportionate. Therefore, we can study the values of the observed local increases in residuals, especially in the areas of interest we are testing. In our example, we can see in Figure 8 that starting from a regularization length of 15 mm, the residuals increase, even exploding after 30 mm, which is a very poor indication of displacement field measurement. Therefore, it is recommended to choose a length between 0 and 10 mm (i.e., 0 to 5 elements).

Figure 8 – Effect of regularization length on the residual field

The tensile strain profile along the specimen is shown in Figure 9 (by plotting the strain along the tensile axis at a fixed time). According to the explanation of the regularization effects given above, the impact of the chosen regularization length on the measurements conducted in EikoTwin DIC can be directly observed here. If the regularization length is too large, the field will be smoothed and crushed, and the strain measured along the profile will decrease. These profiles confirm the previously observed choice of 6 mm (i.e., 3 elements) regularization length versus 15 mm length. This situation illustrates the importance of regularization (filtering high frequencies that interfere with test results) while showing the negative effects that may arise from choosing a length that is too long. Therefore, special attention must be paid to the choice of regularization length to achieve the best possible compromise between computational speed and convergence and the level of detail in the measurements.

Figure 9 – The effect of regularization length on the tensile strain profile of the specimen

Conclusion

In this article, we have seenIn the example of Figure 3, we observe a flat part with a displacement step of 2 mm along the normal of the part (in the right image, the green curve represents the displacement profile of the part used in the virtual image for this example). Here, the applied regularization length (100 mm) can be viewed as the length for measuring the step from 0 to 2 mm. The graph on the right shows the displacement profile according to the normal, where the 'transition zone' is 100 mm. This area is indeed the region where the effects of the previously seen linear elastic constitutive relationship are effectively visible.that mechanical regularization is a mathematical tool that imposes mechanical conditions on the measurement of fields through digital image correlation. Mechanical regularization allows for the correction of certain errors that may occur during testing: the simulation mesh is too fine, the speckle pattern is not suitable for the mesh size, measurement noise is too significant, hindering the correct interpretation of results, etc...

Different physical or mathematical methods of regularization length help to better understand its use in order to achieve excellent results during testing. In practice, an analysis without regularization is first conducted, and then the length is gradually evolved by looking at the quantities of interest specific to the test, allowing for the selection of an appropriate regularization length. Lengths equal to 2 or 3 element sizes are generally suitable for most cases.

Biomechanics Reference

[1] Arturo Mendoza, Jan Neggers, François Hild, Stéphane Roux. Complete Mechanical Regularization
Applied to Digital Image and Volume Correlation. Computer Methods in Applied Mechanics and
Engineering, Elsevier, 2019, 355, pp.27-43. 10.1016/j.cma.2019.06.005 . hal-02148780,Complete Mechanical Regularization Applied to Digital Image and Volume Correlation – Open Archive HAL (archives-ouvertes.fr)