72 J. Yang and K. Bhattacharya (a) (b) (c) 200 300 400 500 600 700 x-Disp U -5 -4 -3 -2 -1 0 1 2 3 4 5 200 300 400 500 600 700 x-Disp U -6 -4 -2 0 2 4 Number of nodes Total element errors 106 105 104 101 102 103 Adaptive ALDIC Uniform refined global DIC Fig. 7.1 The superior efficiency of the newly proposed ALDIC method. Mode-I fracture displacement obtained using different DIC algorithms. (a) Global DIC method. (b) Adaptive mesh ALDIC method. (c) Comparison of total a posterior error between Global DIC method and our adaptive mesh ALDIC method Table 7.1 Comparison of computation time cost of Adaptive ALDIC and Global DIC algorithms for synthetic mode-I fracture experiment Adaptive ALDIC Global DIC Adaptive level Min subset Solve Estimate Mark Refine Element size Computation cost Subproblem 1 Subproblem 2 1 128 ×128 12.85 s 0.29 s 45.03 s 0.003 s 0.113 s 128 ×128 1634.5 s 2 64 ×64 27.40 s 0.28 s 51.41 s 0.162 s 0.229 s 64 ×64 3345.3 s 3 32 ×32 131.14 s 0.54 s 46.83 s 0.003 s 0.139 s 32 ×32 1651.2 s Total 316.4 s image as the reference image, and synthesize deformed images using mode-I fracture deformation using Max Williams’ exact solution ⎧ ⎨ ⎩ u = KI 2E r 2π (1+ν) (2k −1)cos θ 2 −cos 3θ 2 v = KI 2E r 2π (1+ν) (2k −1)sin θ 2 −sin 3θ 2 (7.11) where E=70GPa, ν =0.3, KI =15GPa √pixel are setting properties of the fracture deformation. We start from uniform finite element mesh (element size: 128 ×128 pixels) and adaptively refine elements where a posteriori error is large. All the simulations are done using MATLAB on a personal laptop with 2.6 GHz Inter Core i5 Processor, 8GB 1600 MHz DDR3 Memory. Figure 7.1a plots the displacement field with the uniform finite element of size 32 ×32 pixels, shows the Fig. 7.1b shows the solution where the smallest finite element size is also 32 ×32 pixels. The displacement fields agree, but the computation time of Adaptive ALDIC is only 20% of the Global DIC. Figure 7.1c shows both in the Adaptive ALDIC and Global DIC, with more number of nodes, the results total error will decrease and the error of Adaptive ALDIC decreases faster than Global DIC. When we have the same number of nodes, Adaptive ALDIC has less total error compared with Global DIC. We compare the computation time of the new proposed adaptive mesh ALDIC method and uniform mesh Global DIC in Table 7.1, where we can find this new adaptive ALDIC method decreases computation time significantly compared with current Global DIC algorithm. 7.5 Conclusions In this paper, we report a new Global DIC algorithm that uses an adaptive mesh. It builds on our recent work on the augmented Lagrangian digital image correlation (ALDIC) [12]. We solve the resulting problem using the alternating direction method of multipliers (ADMM). Compared with current Global DIC algorithm, this new adaptive Global DIC algorithm decreases computation time to 20% or even one order of magnitude with no loss (and some gain) in accuracy. Besides 2D-DIC, the early results of the application of Adaptive ALDIC to 3D-DVC are encouraging [13]. Acknowledgement We gratefully acknowledge the support of the US Air Force Office of Scientific Research through the MURI grant ‘Managing the Mosaic of Microstructure’ (FA9550-12-1-0458).
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