Experimental Mechanics DOI 10.1007/s11340-014-9925-8
Mesh-Based Digital Image Correlation Method Using Non-Uniform Elements for Measuring Displacement Fields with High Gradient X. Wang & S. P. Ma
Received: 2 February 2013 / Accepted: 26 June 2014 # Society for Experimental Mechanics 2014
Abstract A new mesh-based digital image correlation (meshDIC) scheme using non-uniform 8-node quadrilateral (Q8) elements is proposed to process displacement fields with regions of both high and low gradient. The effect of element size on Q8-DIC is extensively analyzed, and the results show that when the shape function of an element matches or overmatches the gradient of the displacement field, a larger element size can achieve better results. Conversely, when the shape function under-matches the gradient of the displacement field, a suitable element size should be chosen. Furthermore, a practical and effective method based on the residual error of the DIC calculation is proposed to automatically refine mesh with non-uniform elements. The effectiveness of this method is verified using simulated speckle images that contain one- and two-dimensional non-uniform displacements. The proposed method is also used to improve the accuracy of two experiments that measure displacement fields with high gradient. Keywords Digital image correlation . Mesh refinement . 8-node quadrilateral . Non-uniform element . High gradient . Residual error
Introduction As a convenient and powerful photomechanical method, digital image correlation (DIC) [1–3] has been widely used in various fields. Using DIC, an overall surface deformation field can be obtained by analyzing speckle images from a specimen X. Wang : S. Ma (*) Key Laboratory of Dynamics and Control of Flight Vehicle, Ministry of Education, School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China e-mail:
[email protected]
surface that is generally captured by a non-contact method using various kinds of imaging systems. Therefore, DIC is a very successful technique in two scenarios. The first is deformation measurement at special scales (for example, very large [4] or small [5] scales) or for special materials (for example, soft materials [6]) where traditional gauges cannot be used. The other is the evaluation of damage or failure processes in complicated environments where a heterogeneous deformation field needs to be measured, but where interferometric photomechanical methods are impractical to implement [7]. Traditional DIC resolves the displacement field by matching a number of subsets on a pair of speckle images captured before and after deformation; these two images are often known as the reference and deformed image, respectively. Because of this subset matching, we refer to this kind of DIC as subset-based DIC (subset-DIC). After obtaining the displacement field, the strain field is generally obtained by numerically differentiating the displacement results [1]. To match one specific subset, the displacement field within the subset is assumed to satisfy a simple shape function [2], for example a first order polynomial, which can be evaluated by several deformation parameters. Next, the subset in the reference image is compared with potential subsets in the deformed image and the deformation parameters are searched. As opposed to subset-DIC, mesh-based DIC [8–11] (abbreviated here as mesh-DIC, but also referred to as FEM-DIC or element-based DIC) matches the speckle images on a whole mesh instead of on a single subset, such that displacement continuity between elements is imposed [8–11]. Therefore, an inherent smooth displacement field and better noise suppression are naturally obtained from mesh-DIC [8–10]. In meshDIC, the element could be a 4-node quadrilateral (Q4) [8, 10, 11] or an 8-node quadrilateral (Q8) [9] that represents the displacement within the element with a first-order or secondorder polynomial, respectively. The corresponding mesh-DIC schemes are named Q4-DIC and Q8-DIC, respectively.
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For mesh-DIC, it is reasonable expected that larger elements would be more appropriate for displacement fields with lower gradient, because they are accurate enough to represent such fields and the fewer Degrees of Freedom (DOFs) can reduce optimization error. On the other hand, smaller elements are appropriate for displacement fields with high gradient, because only they can represent high gradient deformation fields accurately. However, it is important to note that the gradient of a deformation field is generally different in different areas. For example, on a specimen with a crack, the deformation in the area close to the crack tip has a high gradient whereas the deformation gradient in other areas would be lower. Considering the analysis above, a mesh with uniform sized elements, as is used currently, is clearly not appropriate. A mesh with non-uniform sized elements should be implemented in DIC analysis in order to obtain better results, i.e., smaller elements in high gradient regions and larger elements in low gradient regions. In other words, the element size should be relative to the gradient of the deformation field. Though some previous studies [12, 13] have tried to use non-uniform mesh in DIC similarly to those used in FEM, the mesh is generated artificially by observation and experience, and is neither accurate nor optimized. Furthermore, in mesh-DIC, the element size also depends on both the amount of information and the level of noise in the speckle image, factors that have never been considered in FEM. A general and systematic method to generate the optimized non-uniform mesh has not yet been reported. In this paper, a new scheme for measuring the displacement fields with regions of both high and low gradient is developed based on Q8-DIC, considering the inherent ability of the Q8 element to represent more heterogeneous displacement fields. The effect of element size on Q8-DIC is systematically studied, and qualitative rules are determined. A Q8-DIC method with non-uniform elements is then proposed to calculate deformation fields with high gradient. An iterative method to refine the mesh based on the residual error is developed to obtain the optimal element size. Numerical and real experiments show that the proposed method can generate better results than a Q8-DIC that uses uniform sized elements. This paper is organized as follows. In Section 2, the effect of element size on deformation field representation and noise suppression of Q8-DIC is studied. In Section 3, a Q8-DIC method with non-uniform sized elements is proposed based on the concept of residual error and applied in simulations. In Section 4, the proposed method is verified by experiments. Section 5 concludes the paper.
Effect of Element Size on Mesh-DIC If the element type of mesh-DIC is chosen, the element size is the most important parameter that affects the performance of
the method, which in turn affects the displacement field representation, the noise suppression, and finally the measurement accuracy. Using Q8-DIC as an example, the effect of the element size on the measurement accuracy of mesh-DIC is analyzed in this section using simulated speckle images. Speckle images were generated to simulate four different types of displacement fields with different homogeneities, i.e., translation (0th order), linear deformation (1st order), 2nd order deformation, and 3rd order deformation. The four displacement fields are expressed as u ¼ 0:25
ð1Þ
u¼
x 400
ð2Þ
u¼
1 ðx−210Þðx−90Þ − 12 14641
ð3Þ
u¼
3 ðx−210Þðx−150Þðx−90Þ þ 10 421875
ð4Þ
The speckle image pairs, i.e., the reference image and deformed images, were created correspondingly. The reference image, cropped from a real speckle image, is shown in Fig. 1(a). The four deformed images, corresponding to the displacement fields represented by Equations (1–4), were generated using the simulation algorithm in Ref. [9]. In order to consider the effect of noise, Gaussian noise with zero mean and various standard variations σg (gray level with respect to an 8-bit image) was added to the simulated speckle image pairs. The displacement fields within the area of interest (AOI) (121 pixels × 121 pixels) were calculated using Q8-DIC with a Q8 mesh (Fig. 1(b)) and various element sizes. By varying the size of the element, represented as s, the error for each different mesh was analyzed as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 2 u 1 X e σ¼t ð5Þ ui −ui N −1 i¼1 where N is the number of pixels in AOI, and ui and e ui are the measured and theoretical displacement of pixel i (i=1, 2, …, N), respectively. The calculation errors of Q8-DIC for different displacement fields with different noise level versus element size are shown in Fig. 1(c)–(f). Figure 1(c)–(f) show the following. i) When the gradient of the displacement field is low, i.e., the shape function of the element matches or over-matches the displacement field, larger elements (fewer DOFs) are better. ii) When the gradient of the displacement field is high, i.e., the shape function of the
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Fig. 1 Q8-DIC calculation errors for various displacement fields and noise levels: a reference speckle image, b Q8 mesh when s=61 pixels, c translation (0th order), d linear deformation (1st order), e 2nd order deformation, and f 3rd order deformation
element under-matches the displacement field, larger elements will produce a larger error and there exists an optimal element size. iii) For a specific deformation field, the optimal element size increases with increasing noise because large elements help suppress the influence of noise.
Q8-DIC with Non-Uniform Elements Results in Section 2 indicate that for a given speckle image pair with specific noise level, the choice of element size in Q8DIC depends on the gradient of the displacement field. A larger element should be used for a low gradient field, and a relatively smaller element should be used for a high gradient field. To measure practical deformation fields that generally contain regions of both high and low gradient, an ideal solution for DIC calculation would be to use non-uniform sized elements, i.e., larger elements in regions of low gradient and smaller elements in regions of high gradient. That is to say, the element size in different areas should be determined according to the gradient of the displacement field in that area. Mesh Refinement in Q8-DIC The size of elements used in Q8-DIC can be determined according to the quality of the DIC calculation in the area covered by these elements. In this paper, the indicator used to evaluate the quality of the DIC calculation for an area A is defined as the residual error of speckle image in this area, i.e., EA ¼
1 X ð f A ðx; yÞ−gA ðx þ u; y þ vÞÞ2 ; NA
ð6Þ
where fA = fA(x, y) is the gray level matrix of area A in the reference image and gA = gA(x + u, y + v) is the gray level matrix of the corresponding area in the deformed image. Displacements u and v are calculated by DIC, and NA is the number of pixels in area A. As expressed by Equation (6), a smaller EA indicates a better DIC calculation, while a larger EA indicates a worse calculation. (For the special case when A
is the overall AOI (represented by Ω), EΩ indicates the global accuracy of the DIC calculation.) Based on this residual error, the optimization of non-uniform mesh in Q8-DIC can be achieved according to the flow diagram in Fig. 2. First, the initial value of global residual error E(0) Ω is calculated using the initial values of the displacements (generally u=0 and v=0) and is set to be the upper boundary of the error to start up the refinement loop. A preliminary mesh with uniform larger elements (generally set at 51 pixels) is used in the DIC calculation and the displacement fields are obtained. Then, the gradient of the calculated displacement fields are observed and analyzed, and the need for any element refinement is determined. If there are high gradient areas, the method proceeds with the mesh refinement step. In each mesh refinement step, the residual errors of the overall (i) AOI (E(i) Ω ) and every area under analysis (EA ) are calculated. Then, the “bad” areas, in which the residual error of area E(i) A is larger than the residual error of AOI E(i) , are found. The elements Ω in the “bad” areas are then refined (each element is divided into several small elements) to obtain an improved mesh and the DIC is recalculated using the new mesh. According to this process, a non-uniform mesh can be optimized step by step. Element refinement stops when one of three termination conditions is met: i) the refinement gets a worse result than the previous one; ii) the refinement is sufficiently good to generate an accurate result; or iii) the element is too small to be refined. Conditions i) and ii) can be determined by the comparison between the EΩ of (i − 1) two adjacent steps, that is, E(i) (i=1, 2, 3, …) for Ω >EΩ (i) (i − 1) (i) condition i) and EΩ ≤EΩ but (E(iΩ− 1) −E(i) Ω )/EΩ ≤δ for condition ii), where δ is a preset threshold, such as 5 % as in the case studies in this paper. For condition iii), the limit of the element size is generally empirically set to 8 pixels. In the following sections, the method described above is used to generate a non-uniform mesh and its improvement on the displacement calculation accuracy is verified. One-Dimensional Non-Uniform Deformation A displacement field with high gradient in one-dimension (shown in Fig. 3(a)) was chosen to verify Q8-DIC with non-
Exp Mech Fig. 2 Flow diagram of optimized non-uniform mesh generation
uniform elements and is expressed by the following equation. ðx−150Þ2 10 1 uðx; yÞ ¼ pffiffiffiffiffiffi e− 162 þ 4 3 2π
ð7Þ
The speckle image pairs containing the displacement fields were simulated using the same algorithm as in Section 2. The size of the reference and deformed speckle images was 301 pixels × 301 pixels, and the AOI (201 pixels × 201 pixels) was located in the center of this image. Following the refinement procedure, after the initial value of global residual error E(0) Ω was calculated, a uniform mesh was used for the first Q8-DIC calculation and the displacement field was obtained (as shown in Fig. 3(b)). This displacement field shows the existence of a high gradient area in the middle of the mesh, indicated by shading in Fig. 3(b). The mesh refinement was then started. For the one-dimensional heterogeneous displacement field, the mesh was refined only along the horizontal direction, i.e., the direction of high displacement gradient. The meshes in the subsequent mesh refinement steps are shown in Fig. 3(c)–(e), where the elements that still need to be refined are shaded. The mesh refinement step was stopped after Step 4 since the elements to be refined were too small (6.25 pixels). As a comparison, a fine mesh (shown in Fig. 3(f)) with uniform small elements was also used to obtain a displacement result. The displacement field errors of Q8-DIC (σ, as defined in equation (5)) at each step are shown in Fig. 3(g). It can be seen that the mesh refinement decreases the Q8-DIC calculation
error. The variation of EA at each refinement step is shown in Fig. 3(h). It can be seen from this figure that the mesh refinement decreases the residual error for both the area of high gradient (AOH) and the area of low gradient (AOL). Correspondingly, the residual error of the overall AOI also decreases. The residual error shows that the refined nonuniform elements produce better DIC results than uniform elements with respect to both the uniform large elements (preliminary mesh) and uniform small elements (fine mesh). The calculated displacement and error fields of the different refinement steps and of the comparison fine mesh are shown in Fig. 3(b)–(f). A cross-section of the displacement at y= 151 pixel shown in Fig. 3(i) further verifies the mesh refinement procedure. Two-Dimensional Non-Uniform Deformation In order to verify the method in the two-dimensional case, a two-dimensional non-uniform displacement field with both high and low gradient areas was simulated. A simulated thermal shock was acted on the center of a plate and a twodimensional heterogeneous deformation field, shown in Fig. 4(a), was obtained using FEM. Next, the speckle image pairs containing the displacement fields were simulated using the same algorithm as in Section 2. The size of the reference and deformed speckle images was 201 pixels × 201 pixels and the AOI was the entire speckle image. Similar to the one-dimensional case, after the initial value of global residual error E(0) Ω was calculated, a uniform mesh
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Fig. 3 One-dimensional non-uniform deformation field, the mesh optimization process used for the Q8-DIC calculation, and results: a displacement field (left) and a cross-section of the displacement field and its second derivative (right); b preliminary mesh and results; c Step 2 mesh and results; d Step 3 mesh and results; e Step 4 (final step) mesh and results; f a fine mesh with uniform small elements and results; g variation of σ in the refinement procedure; h variation of E(i) A in the refinement procedure; and i cross-section of the displacement at y=151 pixel
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Fig. 4 Two-dimensional heterogeneous displacement field, the mesh optimization process used for the Q8-DIC calculation, and results: a displacements u (left) and v (right); b preliminary mesh and results; c Step 2 mesh and results; d Step 3 (final step) mesh and results; e a fine mesh with uniform small elements and results; f variation of σ in the refinement procedure; g variation of E(i) A in the refinement procedure; and h cross-section of the displacement at y=51 pixel
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was used for the first Q8-DIC calculation and the displacement field was obtained (as shown in Fig. 4(b)). The displacement field shows the existence of a high gradient area in the center, indicated by shading in the mesh of Fig. 4(b). The mesh refinement was then started. For the two-dimensional heterogeneous displacement field, the mesh was refined along both the horizontal and vertical directions. The meshes in the different mesh refinement steps are shown in Fig. 4(c)–(d), where the elements that still need to be refined are shaded. The mesh refinement step is stopped (3) because E(4) Ω >EΩ at Step 4, and the results of Step 3 are the final results. For comparison, a fine mesh (shown in Fig. 4(e)) with uniform small elements was also used to obtain a displacement result. The displacement field errors of the Q8-DIC (σ) calculation at each step are shown in Fig. 4(f). It can be seen that the mesh refinement decreases the error of the Q8-DIC calculation. The variations of EA in each refinement step are shown in Fig. 4(g). It can be seen from this figure that the mesh refinement decreases the residual error for both the AOH and the AOL. Correspondingly, the residual error of the overall AOI also decreases. The residual error shows that the refined non-uniform elements produce better DIC results than the uniform elements, both with respect to the uniform large elements (preliminary mesh) and the uniform small elements (fine mesh). The calculated displacement and error fields of the different refinement steps and the comparison fine mesh are shown in Fig. 4(b)–(e). A cross-section of the displacement at y=51 pixel shown in Fig. 4(h) further verifies the mesh refinement procedure.
Application In order to evaluate the proposed method in practice, the proposed method was used to process the speckle images from two real experiments in which displacement fields with high gradient were measured.
During the DIC calculation, the mesh was optimized step by step, as shown in Fig. 5(b)–(d). The mesh refinement step (2) (2) (3) (3) was stopped at Step 3 because E(3) Ω
