Plummer-like Softening#

To avoid numerical instability at short range, we use a softened force law: \( \mathbf{F}_{ij} = k\, q_i q_j\, \frac{\mathbf{r}_{ij}}{\left(\|\mathbf{r}_{ij}\|^2 + \epsilon^2\right)^{3/2}}. \)

This corresponds to a softened potential \( \Phi(r) = -\,\frac{k\,q}{\sqrt{r^2 + \epsilon^2}},\quad \mathbf{F} = -\nabla\Phi. \)

ElectroSim uses two softening choices:

  • Pairwise forces: \(\epsilon = f_\mathrm{soft}(r_i + r_j)\) (contact-based).

  • Field-at-point: \(\epsilon_j = f_\mathrm{soft} r_j\) (source-based).

Rationale:

  • Contact-based softening scales with geometric particle size to mitigate infinite forces during near-contact.

  • Field visualization uses per-source radius to keep the mapping simple and stable.

Trade-offs:

  • Larger \(\epsilon\) increases stability but biases the force at small separations.

  • Smaller \(\epsilon\) increases fidelity but can require smaller DT_S and can cause extreme accelerations.

References: :cite:plummer1911.

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