@@ -17,7 +17,7 @@ The different steps of this methods are:
## 2. Inverse method procedure
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.
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.
### 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:
Of course for the summation, the stress tensors have to be expressed in the same coordinates system.
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.
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.
### 2.2 Guessing initial forces value
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).
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.
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.
__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]
