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C and Gp — the Laplace solver

lineforge's bitmap C/Gp pipeline solves the 2D electrostatic Laplace equation on a Cartesian pixel grid:

$$ \nabla \cdot (\varepsilon_r \nabla V) = 0 $$

with Dirichlet boundary conditions at conductor pixels (V = +1, −1, or 0) and homogeneous Dirichlet at the outer boundary (V = 0 at infinity, approximated via the open-boundary grid extension).

5-point FD stencil with εr coupling

For a uniform Cartesian grid of pixel side $h$, the standard 5-point FD scheme with εr varying per cell is:

$$ V_{i,j},(\alpha_E + \alpha_W + \alpha_N + \alpha_S) = \alpha_E V_{i,j+1} + \alpha_W V_{i,j-1} + \alpha_N V_{i-1,j} + \alpha_S V_{i+1,j} $$

where each $\alpha_X$ is the εr at the cell-edge between (i,j) and the named neighbor — lineforge uses the arithmetic mean of the two adjacent pixels (matching atlc v1):

$$ \alpha_E = \tfrac{1}{2}(\varepsilon_{r,(i,j)} + \varepsilon_{r,(i,j+1)}) $$

(and similarly for W, N, S).

For piecewise-constant material maps, the arithmetic-mean stencil produces a solution accurate to second-order in $h$ inside each material region, and first-order at material interfaces.

Successive over-relaxation (SOR)

The iterative scheme:

$$ V_{i,j}^{(k+1)} = V_{i,j}^{(k)} + \omega \left( \frac{\alpha_E V_{i,j+1}^{(k)} + \alpha_W V_{i,j-1}^{(k)} + \alpha_N V_{i-1,j}^{(k)} + \alpha_S V_{i+1,j}^{(k)}} {\alpha_E + \alpha_W + \alpha_N + \alpha_S}

  • V_{i,j}^{(k)} \right) $$

Optimal ω for an $N \times N$ Dirichlet problem is

$$ \omega_{\text{opt}} = \frac{2}{1 + \sin(\pi/N)} \approx 2 - \frac{2\pi}{N} $$

For typical PCB usermaps ($N \sim 100$–$1000$), lineforge's default $\omega = 1.9$ is close to optimal.

Algebraic multigrid

For grids ≳ $1000 \times 1000$, SOR's iteration count grows as $\mathcal{O}(N^2)$, which becomes prohibitive. lineforge falls back to PyAMG's smoothed-aggregation multigrid, which converges in $\mathcal{O}(N)$ work — typically a 5–10× speedup on big grids.

Capacitance from the field

After convergence, the per-unit-length capacitance is extracted via the energy integral:

$$ C = \frac{\varepsilon_0}{V^2} \iint \varepsilon_r |\mathbf{E}|^2 , dA \quad [\mathrm{F/m}] $$

where $V$ is the applied voltage difference (2.0 for a ±1 driven line). The gradient $\mathbf{E} = -\nabla V$ is computed via central differences on the solved $V$ field.

Effective εr, Z₀, and L

A second Laplace solve is run with $\varepsilon_r \equiv 1$ everywhere (vacuum). Call the resulting capacitance $C_\text{vacuum}$. Then:

$$ L = \frac{\mu_0 \varepsilon_0}{C_\text{vacuum}}, \qquad \varepsilon_\text{eff} = \frac{C}{C_\text{vacuum}} $$

$$ v_p = \frac{c}{\sqrt{\varepsilon_\text{eff}}}, \qquad Z_0 = \frac{1}{v_p \cdot C} = \sqrt{\frac{L}{C}} $$

This is the same trick atlc v1 uses, and matches Wadell's textbook approach.

Dielectric loss (Gp)

For a lossy dielectric with loss tangent $\tan\delta$, the per-unit-length shunt conductance at angular frequency $\omega$ is:

$$ G_p = \omega \cdot \frac{\varepsilon_0}{V^2} \iint \varepsilon_r \tan\delta , |\mathbf{E}|^2 , dA \quad [\mathrm{S/m}] $$

Frequency-dependent. For a uniform dielectric this simplifies to $G_p = \omega C \tan\delta$.

Open boundary

Phase 2 extends the usermap by edge replication out to a 3200×3200 effective area, then uses zero-Dirichlet at the outer edge. Phase 4's polish work implements atlc2's progressive 8×-coarse-pixel extension, which reduces the equation count for very large unshielded simulations without sacrificing accuracy in the field-energy regions that matter for C.

References