Friction¶
Smooth Mollifier¶
- double ipc::smooth_friction_f0(const double y, const double eps_v);¶
Smooth friction mollifier function.
\[ f_0(y)= \begin{cases} -\frac{y^3}{3\epsilon_v^2} + \frac{y^2}{\epsilon_v} + \frac{\epsilon_v}{3}, & |y| < \epsilon_v \newline y, & |y| \geq \epsilon_v \end{cases} \]
- double ipc::smooth_friction_f1(const double y, const double eps_v);¶
The first derivative of the smooth friction mollifier.
\[ f_1(y) = f_0'(y) = \begin{cases} -\frac{y^2}{\epsilon_v^2}+\frac{2 y}{\epsilon_v}, & |y| < \epsilon_v \newline 1, & |y| \geq \epsilon_v \end{cases} \]
- double ipc::smooth_friction_f2(const double y, const double eps_v);¶
The second derivative of the smooth friction mollifier.
\[ f_2(y) = f_0''(y) = \begin{cases} -\frac{2 y}{\epsilon_v^2}+\frac{2}{\epsilon_v}, & |y| < \epsilon_v \newline 0, & |y| \geq \epsilon_v \end{cases} \]
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double ipc::smooth_friction_f1_over_x(
const double y, const double eps_v);¶ Compute the derivative of the smooth friction mollifier divided by y ( \(\frac{f_0'(y)}{y}\)).
\[ \frac{f_1(y)}{y} = \begin{cases} -\frac{y}{\epsilon_v^2}+\frac{2}{\epsilon_v}, & |y| < \epsilon_v \newline \frac{1}{y}, & |y| \geq \epsilon_v \end{cases} \]
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double ipc::smooth_friction_f2_x_minus_f1_over_x3(
const double y, const double eps_v);¶ The derivative of f1 times y minus f1 all divided by y cubed.
\[ \frac{f_1'(y) y - f_1(y)}{y^3} = \begin{cases} -\frac{1}{y \epsilon_v^2}, & |y| < \epsilon_v \newline -\frac{1}{y^3}, & |y| \geq \epsilon_v \end{cases} \]