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> UNDER CONSTRUCTION
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Photoelatic image: inverse method analysis
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==========================================
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## 1. Overview
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The inverse method analysis provide a quantitative estimation of the force network inside a granular media made in a photoelastic material. With such analysis the contact forces magnitude and orientation can be determined under some assumptions. The main idea of inverse method is to generate a numerical photoelastic picture that matches the experimental one, like illustrated below (from [1]).
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>JK: if we want to use this in a new paper we might need to either get permission from RSI to reprint or just do a new version of the image
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![Example](uploads/c4fd38c19a5c2c6263c786a28a3ff9bd/Example.png)
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The inverse method analysis provide a quantitative estimation of the force network inside a granular media made in a photoelastic material. With such analysis the contact forces magnitude and orientation can be determined under some assumptions. The main idea of inverse method is to generate a numerical photoelastic picture that matches the experimental one, like illustrated below (the experimental picture comes from [the matlab implementation of the method](https://github.com/jekollmer/PEGS)).
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<img src="uploads/7e9d15a3fc844774cb987ea3fb530f0e/Force_measurement.svg" width="800">
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To achieve a good matching an optimization procedure is used, this procedure required a numerical computation of the photoelastic signal in the granular media. For cylindrical particles such an analytical expression of this signal can be obtained using the theory of elasticity and the theory of photoelasticity.
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... | ... | @@ -64,17 +62,30 @@ The optimization algorithm must be able to handle non-linear problem and possibl |
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In the case of low quality pictures, the initial guess of the forces distribution may be not precise enough to converge toward a correct results. A correct result is identified by a threshold value to reach by the MSE or SSIM value.
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To improve the initial guess of the forces provided to the optimizer, the forces identified on some disks are applied on the adjacent disk using the Newton's third law. Then the optimization procedure is called with all possible combination of active contact (the propagated contact force being always active). If an optimization procedure succeed, the newly identified forces are propagated to they neighbors.
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> I have to work on the following images to explain propagation steps with particle 139 and/or 480.
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> JK: Maybe a smaller crop just showing the affected particle and its neighbour would give a clearer view on what is happening ?
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<img src="uploads/cadedc89a2659bbcb5d0d927d5767e85/Picture.png" width="250">
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<img src="uploads/093ba577f1853595e148c32d324fb0da/synthetic0.png" width="250">
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<img src="uploads/7ec3e96162cda9ad086b560426877142/synthetic1.png" width="250">
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<img src="uploads/5cf649d69e2a1180a8a59511b339ceb9/synthetic2.png" width="250">
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<img src="uploads/7e5f9fa31da2d739f34f372157221460/synthetic3.png" width="250">
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<img src="uploads/a0ea651b60494eb790cef8ef30a2dac9/synthetic4.png" width="250">
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To improve the initial guess of the forces provided to the optimizer, the forces identified on some disks are applied on the adjacent disk using the Newton's third law. Then the optimization procedure is called with all possible combination of active contact (the propagated contact force being always active). If an optimization procedure succeed, the newly identified forces are propagated to they neighbors. This procedure is illustrated below.
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> The forces must be identified in order to reproduce the photoelastic signal of the Fig.2. The Fig.3 shows the gradient value at each point of the particles, the G2 value on each particle and the sum of the contact force magnitude.
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>
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> <img src="uploads/1bc4999fead9eee0109faeb2ed086c96/Propa_1_2.svg" width="800">
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>
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> The optimizer is call for all particle with all combination of active contact. The sum of the contact force magnitude is evenly distributed on all active contact. The Fig.4 give all possible combination for the particle number 1 (top left), the photoelastic signal obtained after optimization and the SSIM value. The Fig.5 shows the result of this first run of optimization, the solved particles are the green ones.
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>
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> <img src="uploads/8725ddd3b0c2adab6d6e51b2a9e719c0/algoForceJeuxHautGauche_TXT.svg" width="800">
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>
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> The next iteration of optimization will be done on particles number 5 and 7 as thy both have a solved contact from the neighbors (particles 3 and 4). All possible combination for the particle number 5 are illustrated on Fig.6. note that the contact force value with particle 3 and 4 are the same for all combination. After this step of iteration the particle 5 is solved but not the 7 (Fig.7).
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>
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> <img src="uploads/c67491baa350db50f7252e0499c9b7b4/second_it.svg" width="800">
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>
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> The next iteration of optimization will only be carried out on the particles 6 and 7. The only unknowns are the contact forces with the walls. This step allows us to retrieve the photoelastic response of the Fig.3.
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>
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> The following pictures shows the same algorithm on a more complex example. The first picture is the experimental photoelastic response. The next pictures illustrate the iterations 1 to 5, after that the algorithm does not reach the specified SSIM value on any new particle and the optimization process ends.
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>
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> <img src="uploads/cadedc89a2659bbcb5d0d927d5767e85/Picture.png" width="250">
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> <img src="uploads/093ba577f1853595e148c32d324fb0da/synthetic0.png" width="250">
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> <img src="uploads/7ec3e96162cda9ad086b560426877142/synthetic1.png" width="250">
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> <img src="uploads/5cf649d69e2a1180a8a59511b339ceb9/synthetic2.png" width="250">
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> <img src="uploads/7e5f9fa31da2d739f34f372157221460/synthetic3.png" width="250">
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> <img src="uploads/a0ea651b60494eb790cef8ef30a2dac9/synthetic4.png" width="250">
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## 3. Examples
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