Field evaluation

At the start of each batch, BEACH constructs a field from the current element charges, q_elem, and uses that same field for all particles tracked in the batch. Charge carried to a surface is committed at the end of the batch, so it changes the field starting with the next batch.

Element charge is always represented as a constant density over its triangle. Users select how interactions are evaluated and which field boundary is applied.

Choice	Role	Values
element kernel	Charge distribution on one element	triangle_p0 (fixed)
solver	How interactions from many sources are accumulated	direct / treecode / fmm / auto
field boundary	How periodic images and the far field are included	free / periodic2

Select a solver from problem size

Solver	Main use	Field boundary	Approximation
Direct	Small problems, reference results, and split references	free, constrained periodic2	Evaluates the analytic panel kernel for every element
Treecode	Medium free-space problems	free	Uses monopoles for far nodes and the analytic panel kernel in near leaves
FMM	Large problems and many targets	free, periodic2	Approximates far interactions with multipole and local expansions
auto	Select by element count for a free boundary	free	Direct or FMM

With auto, Direct is used when nelem < tree_min_nelem and FMM is used at and above the threshold. The default threshold is 256. Runtime depends not only on the element count, but also on the number of particles, number of steps, and target distribution, so measure a reduced case representative of the production run.

Solver and Field-Boundary Compatibility

This table is the canonical compatibility reference for solvers and field boundaries.

Solver	free	periodic2
direct	Supported	Split reference only. Requires periodic2.nonzero_mode_backend="panel_spectral_reference", zero_mode_policy="exclude_k0", and a supported lower-boundary model
treecode	Supported	Unsupported
fmm	Supported	Supported. Infinite-periodic production runs use cached_kneq0
auto	Supported	Unsupported

periodic2 requires a [domain] box, periodic_axes=["x", "y"], and a nonperiodic z axis. The Direct split reference is for reduced validation cases; the normal periodic2 production path uses FMM.

Separate domain topology from the field boundary

[domain]
box_origin = [0.0, 0.0, 0.0]
box_size = [1.0, 1.0, 1.0]
periodic_axes = ["x", "y"]

[field_boundary]
mode = "periodic2"

[domain] owns the simulation cell and periodic topology; [field_boundary] owns field closure. mode="free" selects a free-space field, while mode="periodic2" uses the domain’s x/y periodicity. The current periodic2 implementation accepts only x/y periodic and z nonperiodic. Species boundary settings may select open, reflect, or redistributed_reflect on nonperiodic faces, but cannot add, remove, or override periodic axes.

Discretize element charge with triangle P0

BEACH distributes each element’s total charge with constant density over its triangle. triangle_p0 is the only element kernel and is implicit rather than selected in a [field] table. It requires finite non-degenerate triangles and a resolved vacuum side for every element. Treecode evaluates near leaves with the analytic panel kernel and distant nodes as monopoles for both field and potential. See Direct, Treecode, and FMM for details.

The former [field] table and sim.softening key have been removed. Leaving either in an input fails as an unknown table or key; BEACH does not reinterpret it as another element model.

Normalize length to control numerical scale

sim.field_normalization selects the representative internal length $L_0$. It does not change input or output units.

Value	$L_0$	Origin $\mathbf{x}_0$
si	1 m	0
length	field_length_scale	0
box	Largest extent in [domain]	domain.box_min
mesh	Largest mesh-bounding-box extent	Mesh-bounding-box minimum

The internal calculation uses

$$\mathbf{x}'=\frac{\mathbf{x}-\mathbf{x}_0}{L_0},$$

then multiplies electric field by $k_c/L_0^2$ and potential by $k_c/L_0$ to restore SI units. box requires a [domain] with positive extents. mesh falls back to field_length_scale only when the mesh is empty.

Combine the solver with boundary components in periodic2

periodic2 combines a finite- or infinite-image nonzero mode with a physical zero mode. Production uses FMM; reduced split references use a Direct panel spectral backend. See periodic2 electrostatics for component ownership.

Measure solver and mesh-discretization errors separately

For a new mesh, first compare a reduced identical case against Direct. Then vary the source-mesh resolution to measure discretization error and vary solver controls to measure approximation error. Compare not only fields at a few points, but also the observables used in the study, such as impact locations, accumulated charge, and conserved quantities. See Validating results for the complete workflow.

Code reference

• Initialization and automatic selection: bem_field_solver_config.f90
• Field and potential evaluation: bem_field_solver_eval.f90
• Treecode/FMM charge refresh: bem_field_solver_tree.f90
• Configuration values: Configuration parameters