periodic2 electrostatics

For an x/y-periodic, z-nonperiodic slab, field_boundary.mode="periodic2" computes finite images, infinite-periodic k\ne0, and plane-average k=0 separately and adds each component exactly once.

The zero in zero mode refers to the x/y wave numbers $k_x=k_y=0$. This component is the height-dependent plane-average field, determined by the lower boundary condition and Gauss’s law. It is one of the physical components handled by the field solver.

Specify domain topology and field closure

[domain]
box_min = [0.0, 0.0, 0.0]
box_max = [1.0, 1.0, 1.0]
periodic_axes = ["x", "y"]

[field_boundary]
mode = "periodic2"

Periodicity belongs to [domain] topology. [field_boundary] selects the free or periodic2 field closure applied to that cell. The current periodic2 implementation accepts only x/y periodic and z nonperiodic; a species or [particle_boundary] cannot override periodic axes.

Decompose the field into three components

Component	Physical meaning	Production path
Primary and near images	Strong local field near the primary cell	Finite-image Direct/FMM sum
Far k\ne0	Infinite-periodic far field varying in x/y	cached_kneq0 operator
Surface k=0	Plane-average field from total charge below each height	Triangle-height cumulative polynomial

cached_kneq0 returns only the nonzero modes; field composition adds the physical boundary-conditioned surface k=0 exactly once. See Finite-image periodic2 configuration for a complete run configuration.

Select the computation path for nonzero modes

Nonzero-mode construction	Use	Constraint
Finite images	Small comparison and finite-image model	Contains nothing outside the image range
panel_spectral_reference	Small triangle-P0 reference	Direct and mode/quadrature convergence
cached_kneq0	Infinite-periodic nonzero modes for FMM production	x/y periodic, z nonperiodic, and exclude_k0

field_periodic_far_correction="auto" has compatibility behavior equivalent to none. Infinite-periodic production must select cached_kneq0 explicitly. Startup validation checks high-level settings against typed [periodic2] configuration and rejects contradictory zero-mode ownership. Removed [outer_plasma] and [coupling] tables are unknown input. m2l_root_oracle has been removed and is rejected at startup.

The Ewald2P teacher, root-multipole-to-local operator, cache, and FMM-state connection are documented separately in periodic2 Far Correction. This page treats that implementation as the nonzero component of the complete field.

Define the range included by a finite-image sum

Sources are added explicitly for primary cell and configured image layer $N$,

$$(i,j)\in[-N,N]^2.$$

This evaluates near interactions with the original kernel but omits the smooth field of infinitely many images outside the range. Therefore field_periodic_far_correction="none" is a finite-image model and must not be interpreted as an infinite-periodic solution without convergence as $N$ increases.

The near-image layer in FMM must match the shell subtracted when fitting the far operator. The cache fingerprint includes image layer for this reason.

Separate the infinite-periodic far field with Ewald2P

cached_kneq0 applies the difference between an Ewald2P teacher and finite-image shell as an operator and removes the teacher’s symmetric k=0. Field composition adds the selected physical k=0 exactly once:

$$K_\mathrm{surface} =\left(K_\mathrm{shell}+R_\mathrm{Ewald}^{\mathrm{full}}-K_0^\mathrm{sym}\right) +K_0^\mathrm{physical}.$$

The expression in parentheses belongs to the nonzero backend. zero_mode_policy="exclude_k0" prevents double counting; it does not discard the mean field. See periodic2 Far Correction for Ewald splitting, operator fitting, the FMM insertion point, and cache lifecycle.

Add the physical k=0 component exactly once

For triangle total charge $q_i$, let $F_i(z)$ be the fraction of its area at or below height $z$. Plane-average cumulative charge is

$$C(z)=\sum_iq_iF_i(z).$$

$F_i$ is piecewise quadratic between the three vertex heights. A geometry plan sorts all vertex heights into breakpoints and stores quadratic coefficients contributed by each triangle in each interval. Horizontal triangles are separate sheet charges; evaluation on a sheet distinguishes minus trace, plus trace, and principal value.

When $q_i$ changes, stored geometry coefficients are multiplied by charge into interval differences, then a prefix sum forms

$$C(z)=a_0+a_1z+a_2z^2$$

and its primitive. Geometry plan is not rebuilt.

Derive zero-mode field and potential from Gauss’s law

For cell area $A=L_xL_y$ and lower far field $E_\mathrm{bottom}$, Gauss’s law gives

$$E_0(z)=E_\mathrm{bottom}+\frac{C(z)}{\epsilon_0A}.$$

Potential from gauge point $(z_g,\phi_g)$ is

$$\phi_0(z)=\phi_g-E_\mathrm{bottom}(z-z_g) -\frac1{\epsilon_0A}\int_{z_g}^zC(\zeta)\,d\zeta.$$

A binary search locates the breakpoint interval and evaluates its quadratic and cubic primitive, giving $O(\log N_z)$ per point.

A nonneutral cell can retain constant far field and linearly growing potential in z. Zero mode is a physical Gauss-law component, not a numerical term that may be deleted.

Close the mean field with a z-boundary condition

For total surface charge $Q=\sum_iq_i$, current choices are:

lower_boundary_model	$E_\mathrm{bottom}$	$E_\mathrm{top}$	Meaning
symmetric_vacuum	$-Q/(2\epsilon_0A)$	$+Q/(2\epsilon_0A)$	Equal vacuum half-spaces above and below with no external field
e_bottom_zero	$0$	$Q/(\epsilon_0A)$	Legacy closure fixing lower flux to zero

Neither solves dielectric screening or polarization. symmetric_vacuum is the minimal symmetric closure without an additional interface or permittivity; e_bottom_zero exists for legacy reproduction and is not a universal physical default.

With matching_plane_quasistatic, BEACH passes $D_H=\epsilon_0E_\mathrm{bottom}+Q/A$, derived from the same lower closure and complete surface charge, to the outer response. The returned $(H,\Phi_H)$ becomes the zero-mode gauge point. This preserves the field and Gauss’s law while making the inner potential continuous with the outer sheath; it does not add the outer field profile inside BEACH. See Quasistatic Matching-Plane Coupling for configuration and the height-dependence test.

Search periodic images reachable by particle trajectories

Field targets are wrapped into the primary periodic cell, while physical trajectory-event positions are retained. Mesh collision searches geometric periodic images required by a segment. Field near-image layer and collision image bound are distinct numerical concepts and do not share one fixed count. See Particle collision and boundary events.

Converge each component separately

• Increase image layers and converge the observable for a finite-image model.
• For cached models, compare cache miss/hit and thread/MPI configurations.
• Vary Ewald $\alpha$, real/reciprocal layers, and proxy/check settings to bound teacher/operator error.
• Compare against an oracle to exclude duplicate primary, near, far, symmetric k=0 subtraction, or physical k=0 addition.
• Inspect Gauss residual and lower/upper closure.
• Do not interpret a finite-height potential difference in a nonneutral cell directly as escape energy to infinity.

See FMM internals for internal Ewald formulas and operator APIs.

Code reference

• Periodic FMM plan, state, and evaluation: bem_coulomb_fmm_core.f90
• Ewald teacher and cached root operator: bem_coulomb_fmm_periodic_root_ops.f90
• Cached symmetric k=0 subtraction: bem_coulomb_fmm_eval_ops.f90
• Surface zero-mode plan/state: bem_periodic_zero_mode_plan.f90
• Zero-mode evaluation: bem_periodic_zero_mode_eval.f90
• Nonzero Fourier reference evaluation: bem_coulomb_fmm_periodic_nonzero_reference.f90
• Upper-vacuum Fourier evaluation: bem_coulomb_fmm_periodic_nonzero_upper_vacuum.f90
• Component ownership and snapshot composition: bem_electrostatic_snapshot.f90