Friction¶
Smooth Mollifier¶
- ipctk.smooth_friction_f0(y: float, eps_v: float) float¶
Smooth friction mollifier function.
\[f_0(y)= \begin{cases} -\frac{y^3}{3\epsilon_v^2} + \frac{y^2}{\epsilon_v} + \frac{\epsilon_v}{3}, & |y| < \epsilon_v \newline y, & |y| \geq \epsilon_v \end{cases}\]
- ipctk.smooth_friction_f1(y: float, eps_v: float) float¶
The first derivative of the smooth friction mollifier.
\[f_1(y) = f_0'(y) = \begin{cases} -\frac{y^2}{\epsilon_v^2}+\frac{2 y}{\epsilon_v}, & |y| < \epsilon_v \newline 1, & |y| \geq \epsilon_v \end{cases}\]
- ipctk.smooth_friction_f2(y: float, eps_v: float) float¶
The second derivative of the smooth friction mollifier.
\[f_2(y) = f_0''(y) = \begin{cases} -\frac{2 y}{\epsilon_v^2}+\frac{2}{\epsilon_v}, & |y| < \epsilon_v \newline 0, & |y| \geq \epsilon_v \end{cases}\]
- ipctk.smooth_friction_f1_over_x(y: float, eps_v: float) float¶
Compute the derivative of the smooth friction mollifier divided by y (\(\frac{f_0'(y)}{y}\)).
\[\frac{f_1(y)}{y} = \begin{cases} -\frac{y}{\epsilon_v^2}+\frac{2}{\epsilon_v}, & |y| < \epsilon_v \newline \frac{1}{y}, & |y| \geq \epsilon_v \end{cases}\]