# A Post-Processing Method Based on Radial Basis Functions for the Fast Retrieval of the Strain Field in Digital Image Correlation Methods

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## Abstract

**:**

## 1. Introduction

## 2. Radial Basis Functions

**x**(source points) in the m-dimensional domain. The RBF interpolator approximates the spotted distribution of information with a continuous analytical function, whose value s(

_{i}**x**) at a target point

**x**is given as:

_{i}are the weights deriving from the above-mentioned retrieval condition and φ are the radial bases. The behaviour of the interpolant between the points depends upon the chosen kernel, i.e., the particular form of the selected radial function. Table 1 reports typical RBF kernels, where $r=\Vert x-{x}_{i}\Vert $ and $\u03f5$ is a shape parameter [20], which should cope with the average grid spacing. As regards the Generalized MultiQuadratic (GMQ) kernel, R and q are parameters subject to the user’s choice. It is evident that the GMQ kernel can assume all the forms reported in Table 1, with the exception of the thin plate spline and the Gaussian. It is worth noticing that a radial basis is a function of the Euclidean distance $\Vert x-{x}_{i}\Vert $ of two points in a multidimensional space that gives in output a simple scalar, reducing the dimension of the problem.

_{i}the input values associated to the source points and with

**g**the vector collecting them in a row, the vector

**γ**of weights has to satisfy the condition:

**M**containing the radial bases computed at each source point. Upon the inversion of the square matrix

**M**, the weights γ

_{i}are known.

**x**) is added at the end of the summation in Equation (1) to enhance the potency of the interpolant, which acquires the capability to reproduce exactly those functions of the same form of h(

**x**). In this case also the polynomial coefficients are unknown, and the system assumes a different form from that in Equation (2). A condition of orthogonality [21] between the weights γ

_{i}of the radial bases and the monomial terms constituting the polynomial supplies additional constraints to solve the enlarged system. In the case of a linear polynomial in a 2D space:

**P**is the constraint matrix deriving from the orthogonality condition:

## 3. The Method

**F**for the points on the plane assumed the form:

_{y}obtained both with FEM and after differentiation of the RBF field for each value of the discretization level, represented with the number of nodes.

_{y}retrieved with RBF remained practically unchanged over the number of nodes, with FEM results approaching from below. With a relatively small number of nodes, however, in the order of 10

^{4}, the results obtained with both methods could be considered very good. In must be noted that the value of deformation studied was very high, in the order of 60%.

_{DC}), as expected, depended on the number of points employed for tracking, demonstrating however a better convergence to finer results with regard to FEM even when employing the method on coarser meshes. It must be noted that the proposed approach was able to retrieve the large displacement strains associated with a multi-axial in-plane deformation. For reasons of space, however, we addressed only cases in which a prevalent deformation along the y axis was applied, similarly to what happens when dealing with a traction machine. For this reason, being the strain along the y direction prominent, the ε

_{x}was not plotted. In the next section, however, the comparison between two literature cases and the proposed method was carried out using, as a term of comparison, the equivalent strain, that took into account the strains along both the x and y axes.

## 4. Experimental Testing

_{y}, equal to 0.063. It is important to stress that in some specimens in which large deformation and stretching occurs, it is difficult to achieve high-quality tracking, especially near large deformation areas. In these cases, RBF are able to fill the gap, interpolating smoothly the missing information from point to point thanks to the analytic representation they provide. In Figure 12, the vertical displacements for the drilled polyethylene specimen are reconstructed, changing the number of tracked points. It can be appreciated how the contour maps and values remain substantially unchanged, even if the density of points near the most distorted area on the side of the hole is far smaller with respect to other parts of the specimen.

## 5. Conclusions

- retrieves the values of strain from large displacements in a fast and accurate manner;
- is easy to apply because it does not require a connectivity organizing the nodes into elements;
- features a faster convergence with respect to FEM.

- is sensitive to local oscillations introduced by noise because differentiation is point-wise and not averaged on an element basis;
- it is suitable to be used in planar cases.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The central square of the plate featuring the coarsest (

**left**) and finest (

**right**) levels of discretization on the biased mesh.

**Figure 4.**Convergence trends for the RBF post-processing method (blue) and FEM (red) on the mesh refined near the hole.

**Figure 5.**The central square of the plate featuring the coarsest (

**left**) and finest (

**right**) levels of discretization on the unbiased mesh.

**Figure 7.**Coarse (

**top**) and fine (

**bottom**) meshes in undeformed (

**left**) and deformed (

**right**) configuration.

**Figure 8.**Strain field computed by means of proposed method on the fine grid increasing the number of tracked points: (

**a**) 299, (

**b**) 407, (

**c**) 582, (

**d**) 794, (

**e**) 1299 and comparison to FEM results with 7104 nodes (

**f**).

**Figure 9.**Maximum strain for the coarse (

**left**) and fine (

**right**) meshes computed using the proposed method while increasing the number of tracked points (red) and reference values on the same grid obtained with FEM (green) and RBF (dashed) computation.

**Figure 10.**Horizontal (

**a**) and vertical displacements (

**b**) for the drilled experimental sample under traction. In (

**c**) the equivalent strain values retrieved by means of the proposed method are compared with those in the literature (

**d**).

**Figure 11.**FEM contour map of the strain along the y direction for the drilled case (

**left**) and its difference with respect to the proposed method (

**right**).

**Figure 12.**Vertical displacements computed by RBF varying the number of tracked points. (

**a**) 10217 points, (

**b**) 4149 points, (

**c**) 2343 points, (

**d**) 1528 points.

**Figure 13.**Unloaded and loaded aluminum specimen (

**a**). Vertical and horizontal displacements due to loading (

**c**) and equivalent strain maps for the aluminum alloy sample using RBF and tracking (

**b**) and using modal tracking (

**d**).

**Figure 14.**FEM contour map of the strain along the y direction for the aluminum specimen (

**left**) and its difference with respect to the proposed method (

**right**).

RBF | φ(r) |
---|---|

Spline type (Rn) | ${r}^{n},nodd$ |

Thin plate spline (TPSn) | ${r}^{n}\mathrm{log}\left(r\right),neven$ |

Multiquadratic (MQ) | $\sqrt{1+{\u03f5}^{2}{r}^{2}}$ |

Inverse multiquadratic (IMQ) | $\frac{1}{\sqrt{1+{\u03f5}^{2}{r}^{2}}}$ |

Inverse quadratic (IQ) | $\frac{1}{1+{\u03f5}^{2}{r}^{2}}$ |

Gaussian (GS) | ${e}^{-{\u03f5}^{2}{r}^{2}}$ |

Generalized multiquadratic (GMQ) | ${({\u03f5}^{2}{r}^{2}+{R}^{2})}^{q}$ |

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**MDPI and ACS Style**

Groth, C.; Chiappa, A.; Porziani, S.; Biancolini, M.E.; Marotta, E.; Salvini, P.
A Post-Processing Method Based on Radial Basis Functions for the Fast Retrieval of the Strain Field in Digital Image Correlation Methods. *Materials* **2022**, *15*, 7936.
https://doi.org/10.3390/ma15227936

**AMA Style**

Groth C, Chiappa A, Porziani S, Biancolini ME, Marotta E, Salvini P.
A Post-Processing Method Based on Radial Basis Functions for the Fast Retrieval of the Strain Field in Digital Image Correlation Methods. *Materials*. 2022; 15(22):7936.
https://doi.org/10.3390/ma15227936

**Chicago/Turabian Style**

Groth, Corrado, Andrea Chiappa, Stefano Porziani, Marco Evangelos Biancolini, Emanuele Marotta, and Pietro Salvini.
2022. "A Post-Processing Method Based on Radial Basis Functions for the Fast Retrieval of the Strain Field in Digital Image Correlation Methods" *Materials* 15, no. 22: 7936.
https://doi.org/10.3390/ma15227936