... | ... | @@ -17,7 +17,7 @@ The different steps of this methods are: |
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## 2. Inverse method procedure
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About the first step every details are given in the dedicated page and nothing has to be added. For the initial guess of the contact force, the $G^2$ value defined in the [gradient analysis](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/gradient-analysis) is used. Depending on the quality of the pictures, some manipulation based on this $`G^2`$ can be done in order to distribute the force over the contacts. A good initial force guessing increases the optimization procedure in speed and [results quality](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/gradient-analysis#1-overview). The forces determination occurs with an optimization procedure. This procedure is described hereafter as it is specific to the inverse method analysis.
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About the first step every details are given in the dedicated page and nothing has to be added. For the initial guess of the contact force, the $G^2$ value defined in the [gradient analysis](/gradient-analysis) is used. Depending on the quality of the pictures, some manipulation based on this $`G^2`$ can be done in order to distribute the force over the contacts. A good initial force guessing increases the optimization procedure in speed and [results quality](/gradient-analysis#1-overview). The forces determination occurs with an optimization procedure. This procedure is described hereafter as it is specific to the inverse method analysis.
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### 2.1 Analytical expression of the photoelastic signal
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... | ... | @@ -36,13 +36,13 @@ The stress field inside the disk is obtained by the superposition of: |
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Of course for the summation, the stress tensors have to be expressed in the same coordinates system.
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In a [circular polariscope](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/reflection-photoelasticity#21-photoelasticity-in-the-reflective-polariscope), the light intensity is a function of the difference of the two principal stress. This difference is equal to $`\sigma_{1}-\sigma_{2} = \sqrt{\left(\left(\sigma_{xx}-\sigma_{yy}\right)^2+\left(2\sigma_{xy}\right)^2\right)}`$ which is easely derivate from the [Mohr's circle](https://en.wikipedia.org/wiki/Mohr%27s_circle) for 2D stress.
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In a [circular polariscope](/reflection-photoelasticity#21-photoelasticity-in-the-reflective-polariscope), the light intensity is a function of the difference of the two principal stress. This difference is equal to $`\sigma_{1}-\sigma_{2} = \sqrt{\left(\left(\sigma_{xx}-\sigma_{yy}\right)^2+\left(2\sigma_{xy}\right)^2\right)}`$ which is easely derivate from the [Mohr's circle](https://en.wikipedia.org/wiki/Mohr%27s_circle) for 2D stress.
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### 2.2 Guessing initial forces value
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From the position and dimensions of the disk, it is easy to construct a possible contact network based on geometrical consideration. A contact occurs if the distance between the two disk centers is lower than the sum of the two radius increased by a small tolerance to overcome imprecision from detection step. This provide the location of all possible contact forces acting on each disk, but it includes some false-contacts (due to the tolerance) or _non-force-bearing_ (which does not transmit any load).
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A correct result of optimization procedure required a good initial force distribution. The [gradient analysis](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/gradient-analysis) provide the $`G^2`$ value which is proportional to the sum of the contact force magnitude $`\sum_i|F_i|`$ acting on the disk. To provide the initial guess of force distribution, two methods are available depending on the quality of the pictures.
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A correct result of optimization procedure required a good initial force distribution. The [gradient analysis](/gradient-analysis) provide the $`G^2`$ value which is proportional to the sum of the contact force magnitude $`\sum_i|F_i|`$ acting on the disk. To provide the initial guess of force distribution, two methods are available depending on the quality of the pictures.
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__In the case of high quality pictures__ (resolution and contrast), a $`G^2_i`$ at each contact $`i`$ can be computed using a small region of interest located near the contact point. The false and _non-force-bearing_ contacts are eliminated if their $`G^2_i`$ value is lower than a threshold value on both contacting disks. The sum of the contact force magnitude is then distributed on the remaining contacts in proportion to the value of $`G^2_i`$. [1]
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... | ... | @@ -131,4 +131,4 @@ Photo-elastic Grain Solver (PEGS): |
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[<go back to home](https://git-xen.lmgc.univ-montp2.fr/PhotoElasticity/Main/wikis/home) |
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\ No newline at end of file |
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[<go back to home](/home) |
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\ No newline at end of file |