Merge pull request #214 from alxbilger/fixlatex

replace &space; and + by its corresponding character

Author: hugtalbot

30_Components/30_SolidMechanics/20_Spring/80_PolynomialSpringsForceField.md

@@ -30,14 +30,14 @@ $$
 The dedication of Jacobian matrix for PolynomialSpringForceField is given below:
 
 $$
-J_F(u)=\left(S\frac{\partial&space;\sigma}{\partial&space;L}\cdot\frac{1}{l_0}-S\sigma\cdot\frac{1}{|l|}&space;\right)\begin{bmatrix}\frac{\Delta&space;x^2}{l^2}&\frac{\Delta&space;x\Delta&space;y}{l^2}&\frac{\Delta&space;x\Delta&space;z}{l^2}\\\frac{\Delta&space;y\Delta&space;x}{l^2}&\frac{\Delta&space;y^2}{l^2}&\frac{\Delta&space;y\Delta&space;z}{l^2}\\\frac{\Delta&space;z\Delta&space;x}{l^2}&\frac{\Delta&space;z\Delta&space;y}{l^2}&&space;\frac{\Delta&space;z^2}{l^2}\end{bmatrix}+S\sigma\cdot\frac{1}{|l|}\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}
+J_F(u)=\left(S\frac{\partial \sigma}{\partial L}\cdot\frac{1}{l_0}-S\sigma\cdot\frac{1}{|l|} \right)\begin{bmatrix}\frac{\Delta x^2}{l^2}&\frac{\Delta x\Delta y}{l^2}&\frac{\Delta x\Delta z}{l^2}\\\frac{\Delta y\Delta x}{l^2}&\frac{\Delta y^2}{l^2}&\frac{\Delta y\Delta z}{l^2}\\\frac{\Delta z\Delta x}{l^2}&\frac{\Delta z\Delta y}{l^2}& \frac{\Delta z^2}{l^2}\end{bmatrix} + S\sigma\cdot\frac{1}{|l|}\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}
 $$
 
 
 Note that a **RestShape**PolynomialSpringsForceField does exist. It will compute the same non-linear force with regards to the rest shape of one single object. To avoid Nan problems when a spring has a zero length, an exponential addition to the denominator has been added. As a result, the stress simulation is shifted compared with polynomial values, but it keeps its nonlinearity:
 
 $$
-J_F(u)=\left(S\frac{\partial&space;\sigma}{\partial&space;L}\cdot\frac{(1-sc\cdot&space;e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)})}{l_0}-S\sigma\cdot\frac{(1-sc\cdot&space;e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)})}{\sqrt{\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2+e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)}}}&space;\right)\cdot\frac{1}{\sqrt{\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2+e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)}}}\cdot\begin{bmatrix}\Delta&space;x^2&\Delta&space;x\Delta&space;y&\Delta&space;x\Delta&space;z\\\Delta&space;y\Delta&space;x&\Delta&space;y^2&\Delta&space;y\Delta&space;z\\\Delta&space;z\Delta&space;x&\Delta&space;z\Delta&space;y&\Delta&space;z^2\end{bmatrix}+S\sigma\cdot\frac{1}{\sqrt{\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2+e^{sh-sc(\Delta&space;x^2+\Delta&space;y^2+\Delta&space;z^2)}}}\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}
+J_F(u)=\left(S\frac{\partial \sigma}{\partial L}\cdot\frac{(1-sc\cdot e^{sh-sc(\Delta x^2 + \Delta y^2 + \Delta z^2)})}{l_0}-S\sigma\cdot\frac{(1-sc\cdot e^{sh-sc(\Delta x^2 + \Delta y^2 + \Delta z^2)})}{\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 + e^{sh-sc(\Delta x^2 + \Delta y^2 + \Delta z^2)}}} \right)\cdot\frac{1}{\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 + e^{sh-sc(\Delta x^2 + \Delta y^2 + \Delta z^2)}}}\cdot\begin{bmatrix}\Delta x^2&\Delta x\Delta y&\Delta x\Delta z\\\Delta y\Delta x&\Delta y^2&\Delta y\Delta z\\\Delta z\Delta x&\Delta z\Delta y&\Delta z^2\end{bmatrix} + S\sigma\cdot\frac{1}{\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 + e^{sh-sc(\Delta x^2 + \Delta y^2 + \Delta z^2)}}}\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}
 $$
 
 More details were given in the pull-request [#1342](https://github.com/sofa-framework/sofa/pull/1342).