Ramp model

The Ramp model is described by:

u(r)=\left\{{\begin{array}{ll}\infty &{\rm {if}}\;r<\sigma \\W_{r}-(W_{r}-W_{a}){\frac {r-\sigma }{d_{a}-\sigma }}&{\rm {if}}\;\sigma \leq r\leq d_{a}\\W_{a}-W_{a}{\frac {r-d_{a}}{d_{c}-d_{a}}}&{\rm {if}}\;d_{a}<r\leq d_{c}\\0&{\rm {if}}\;r>d_{c}\end{array}}\right.

where $W_{r}>0$ and $W_{a}<0$ .

Graphically, one has:

where the red line represents an attractive implementation of the model, and the green line a repulsive implementation.

References