3D Periodic Space Sampling Methods

Updated 27 September 2025

Three-dimensional periodic space sampling is a technique for discretizing functions, fields, or geometries within periodic domains using methods like windowed lattice sums and adaptive algorithms.
It enhances computational accuracy and efficiency in applications ranging from wave scattering and particle simulations to crystallography and adaptive mesh generation.
The approach leverages specialized strategies—such as particle-mesh methods, manifold tracing, and Poisson–disk sampling—to optimize performance under periodic constraints.

A three-dimensional periodic space sampling method refers to any mathematically rigorous, numerically efficient, and practically implementable procedure for sampling, discretizing, or representing functions, fields, or geometric objects that possess, are embedded within, or interact with a three-dimensional periodic structure. Applications span wave scattering, particle and field simulations, mesh generation, crystallography, and constrained random geometry. The following sections organize, systematize, and classify the main principles and representative methods found within recent research for this domain.

1. Periodization of Green Functions and Lattice Sum Convergence

Central to numerous scattering and field computation problems is the evaluation of Green functions adapted to periodic domains. For the Helmholtz equation in three dimensions, the quasi-periodic Green function

$G^{\mathrm{qper}}(\mathbf{x}) = \frac{1}{4\pi} \sum_{m, n \in \mathbb{Z}} \frac{e^{ik r_{m,n}}}{r_{m,n}}\, e^{-i\mathbf{k}\cdot(m\mathbf{v}_1 + n\mathbf{v}_2)}$

with $r_{m,n} = \sqrt{ |(x, y) + m\mathbf{v}_1 + n\mathbf{v}_2|^2 + z^2 }$ exhibits extremely slow, conditional convergence due to the algebraic decay of the free-space Green function (Bruno et al., 2013).

To overcome this, the smoothly-windowed lattice sum technique multiplies each term by a $C^\infty$ cutoff function $\chi_a(|m\mathbf{v}_1 + n\mathbf{v}_2|)$ that transitions from $1$ to $0$ at a prescribed truncation parameter $a$ : $G^a(\mathbf{x}) = \frac{1}{4\pi} \sum_{m,n} \frac{e^{ik r_{m,n}}}{r_{m,n}}\, e^{-i\mathbf{k}\cdot(m\mathbf{v}_1 + n\mathbf{v}_2)}\, \chi_a(|m\mathbf{v}_1 + n\mathbf{v}_2|).$ A rigorous proof establishes "superalgebraic" convergence: for any integer $n>0$ ,

$|G^a(\mathbf{x}) - G^{\mathrm{qper}}(\mathbf{x})| < C_n/a^n,$

in sharp contrast with classical windowing or Ewald methods, and enabling highly accurate kernel evaluations for boundary-integral methods.

2. Discretization Strategies and Particle–Mesh Methods in Periodic Domains

Many space sampling problems seek efficient discretizations (on grids, meshes, or basis expansions) for sources, potentials, or densities that respect periodicity. The particle-mesh method achieves this by:

Binning discrete particles onto an equidistant, three-dimensional mesh,
Computing the field via discrete convolution (FFT-accelerated) with either the free-space or periodically-extended Green function,
Employing "switch" functions to blend near- and far-field kernel evaluations for improved accuracy and computational efficiency (Dohlus et al., 2015).

For singly-periodic source distributions, the periodic Green function is

$G_p(\mathbf{r}) = \sum_{n=-\infty}^\infty \left[ G(\mathbf{r} + n \mathbf{r}_p) - G(\mathbf{r}_0 + n \mathbf{r}_p) \right],$

where normalization at a reference point is necessary for convergence. Only one period of the physical particle distribution is simulated, with the periodic replication—in Fourier space or via direct sums—incorporating the contributions from the infinite lattice while restricting the computational domain, yielding orders-of-magnitude performance boosts for microstructured or modulated beam simulations.

3. Manifold Tracing and Adaptive Sampling in Dynamical Systems

Sampling of low-dimensional, but geometrically intricate, invariant manifolds within three-dimensional periodic flows requires methods that adjust sampling density dynamically. The adaptive manifold tracing algorithm operates by launching orbits from initial conditions near an unstable periodic orbit (UPO) and inserting new orbits wherever the inter-orbit separation exceeds a set threshold (Taborda et al., 2016).

A pseudocode summary of the refinement step:

For all pairs of neighboring sampled orbits:
    If the distance between orbits at a given "time" > d_max:
        Insert new orbit at midpoint initial condition
        Update ordering tags

To achieve computational efficiency, a further approximation "jumps" over long stretches of the manifold using cubic interpolation between previously sampled points, reducing the typically O(N²⁾ cost to O(N log N), without sacrificing resolution or ordering—essential for the numerical paper of high-resolution phase space structures, Lagrangian coherent structures, or chaotic mixing boundaries.

4. Sampling-Based and Volume Computation Methods for Constrained Spaces

When sampling in three-dimensional configuration spaces defined by distance constraints (e.g., for molecular free energy landscapes), direct uniform sampling is computationally prohibitive due to effective dimensionality and topological complexity. The Cayley coordinate method addresses this by constructing a branched covering over a convex, lower-dimensional "base space" parameterized by selected distances (Zhang et al., 29 Aug 2024).

The workflow:

Uniformly sample in the Cayley base space,
Use the efficient inverse Cayley mapping to reconstruct preimages in Cartesian coordinates,
Traverse epsilon-sized hypercubes in the Cartesian domain using a "frontier hypercube" data structure, visiting only those that intersect the feasible region, achieving linear runtime in the number of relevant grid cells.

This is highly advantageous in configurational entropy computations, ligand binding assessments, or soft matter assembly modeling—domains where volume of feasible space is of central interest.

5. Mesh Generation and Variable Resolution Sampling

For geometric applications, such as modeling discrete fracture networks or complex microstructures, generating an adaptive mesh that resolves periodic (or locally periodized) features demands a robust space sampling method. The maximal Poisson–disk sampling approach generates points in the domain satisfying non-overlap (inhibition radius) conditions, allowing variable local density via a spatially varying radius $\rho(\mathbf{x})$ (Krotz et al., 2021): $\|\mathbf{p} - \mathbf{q}\| \geq \min( \rho(\mathbf{p}),\ \rho(\mathbf{q}) ).$ A two-stage algorithm—(1) Poisson–disk sampling, (2) conforming Delaunay tetrahedralization—ensures that mesh resolution is high near critical features (e.g., fracture intersections) and coarse elsewhere. Postprocessing steps remove "sliver" tetrahedra (low-quality elements), ensuring geometrically acceptable three-dimensional meshes.

6. Lattice Invariants, Metrics, and Space Parameterization

In crystallography and material science, the classification and systematic sampling of three-dimensional periodic lattices up to isometry (LISP: Lattice Isometry SPace) is captured by the obtuse superbase formalism (Bright et al., 2021). Each lattice (up to rigid motion) is uniquely specified (within a finite ambiguity) by a six-tuple of "root products" $r_{ij} = \sqrt{ -\mathbf{v}_i \cdot \mathbf{v}_j }$ from the superbase: $\mathrm{RF}(\Lambda) = (r_{23}, r_{13}, r_{12}, r_{01}, r_{02}, r_{03}).$ Defining distances between lattices with

$\mathrm{RM}_d(\Lambda, \Lambda') = \min_{\sigma \in S_4} d(\mathrm{RF}(\Lambda),\ \sigma(\mathrm{RF}(\Lambda'))),$

where $d$ is any metric on $\mathbb{R}^6$ , yields a complete, continuous, and computationally tractable framework for exploring and sampling this moduli space—a prerequisite for data-driven crystallography and the systematic classification of real structures from large databases.

7. Basis Expansions and Sampling of Bandlimited Volumetric Data

For volumetric data sampled on periodic grids, especially when representing functions concentrated in both space and frequency, expansions in three-dimensional generalized prolate spheroidal wavefunctions (GPSWFs) provide near-optimal basis representations (Katz et al., 2018). The function $f$ supported (or concentrated) in the ball $R$ , and bandlimited to a ball of radius $c$ , admits expansion: $f(\mathbf{x}) = \sum_{n,m,k} a_{n,m,k} \psi_{n,m,k}(\mathbf{x}),$ with the GPSWFs $\psi_{n,m,k}$ forming an orthogonal (and space/frequency-concentrated) basis and the coefficients computable as weighted sums of grid samples. A truncation rule based on the eigenvalues of the associated integral operator ensures computational tractability and a provable approximation error, facilitating robust reconstructions in tomographic imaging, microscopy, and computational physics.

These advanced periodic space sampling methods share several unifying themes: encoding periodicity at the kernel, grid, or coordinate level; using structure-adapted discretizations (windowed lattice sums, adaptive or variable-resolution grids, basis functions maximally localized under constraints); and leveraging mathematical reductions (symplectic and combinatorial geometry, moduli parameterizations, convexification of configuration spaces) for either computational efficiency or analytical control. The development of such methods has enabled quantitatively precise, scalable, and physically meaningful simulations and analyses across physical, engineering, and mathematical disciplines.