VOID in Multidisciplinary Research

• VOID is a concept defined by its boundary conditions rather than absolute emptiness, spanning materials science, cosmology, and machine learning.
• Studies reveal that void dynamics control key processes such as fracture in metals, hotspot formation in explosives, and hierarchical cosmic structure evolution.
• In machine learning, the VOID framework disrupts latent diffusion by amplifying encoding uncertainty and counteracting guidance signals to secure model outputs.

“Void” denotes several technically distinct objects across contemporary research. In materials science it refers to cavities or pore-like defects whose growth, interaction, and coalescence control fracture, rupture, or hotspot formation; in cosmology it denotes large underdense regions of the cosmic web, including nested subvoids, walls, and filaments; and in machine learning it also appears as the acronym VOID, a defense framework against unauthorized mimicry in latent diffusion models (Holte et al., 2020, Rieder et al., 2014, Jain et al., 2018, Roy et al., 2019, Qiu et al., 10 Jun 2026). Taken together, these usages suggest that a void is rarely treated as absolute emptiness. Rather, it is usually defined by its boundary conditions, its embedding field, and the dynamics induced by stress, flow, geometry, or inference.

1. Technical meanings and recurring structure

In porous metals, a void is a micron-scale cavity embedded in a plastic matrix. The decisive region is often not the cavity interior itself but the inter-void ligament, the last load-bearing bridge between neighboring voids. Localized plastic flow in that ligament governs void coalescence and thus ductile fracture (Holte et al., 2020). In amorphous graphene, the term denotes a mesoscopic hole-like defect with a perimeter of distorted rings and localized elastic energy; it is explicitly distinguished from a single missing atom or point defect (Jain et al., 2018). In shocked HMX, voids are pores whose collapse produces hotspots that regulate ignition and growth (Roy et al., 2019).

In cosmology, voids are large underdense regions traced by galaxies, halos, or the matter field. They are not empty in any absolute sense. High-resolution simulations and local-universe reconstructions show that voids can contain tenuous walls and filaments, and that the void network is interconnected through passages rather than partitioned into isolated bubbles (Rieder et al., 2014, Tully et al., 2019). Void-finding methodology therefore depends strongly on whether the target object is a spherical underdensity, a watershed basin, a tessellation-defined region, or a probabilistic catalog sample.

A recurring feature across these literatures is that void behavior is boundary-dominated. In solids, line tension, strain gradients, and ligament hardening determine whether a cavity remains stable or coalesces. In cosmology, density compensation, tidal environment, and tracer definition determine whether a region is an expanding void-in-void system or a partially collapsing void-in-cloud system. In recent probabilistic treatments of cosmic voids, the relevant object is not a single catalog but the conditional distribution $P(\{v_\alpha\}|\{g_\beta\})$, reflecting that sparse galaxy data do not uniquely determine “the” voids (Thiele, 16 Feb 2026).

2. Inter-void localization in ductile solids

The ductile fracture of porous metals is strongly affected by both inter-void spacing and material size effects. Three-dimensional finite-element unit-cell calculations with a discrete spherical void in a strain-gradient-enhanced matrix show that imposed loading can be parameterized by fixed stress triaxiality and Lode parameter, with

$$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$

and macroscopic stress ratios

$$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$

The gradient-enhanced constitutive law introduces plastic strain gradients through

$$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$

so that small ligaments can harden through a length-scale-dependent mechanism (Holte et al., 2020).

Without strain-gradient effects, decreasing the inter-void ligament size increases the propensity for void coalescence. Smaller ligaments generate higher local stress concentration, earlier plastic localization, easier coalescence, and lower critical load for failure. The critical equivalent stress generally increases with ligament size, and this size effect becomes stronger as stress triaxiality increases. Lode-parameter effects are modest overall, but for small ligaments $L=-1$ gives the lowest coalescence stress, whereas for larger ligaments $L=0$ can become the most critical. For large or cubic cells, localization can transition away from the smallest ligament and instead develop along approximately $45^\circ$ to the loading axis, especially for $L=0$ (Holte et al., 2020).

With a strain-gradient-enhanced matrix, the interaction changes qualitatively. Increasing $L_D/r_0$ raises the critical coalescence stress because the strain gradients harden the inter-void ligament and resist localization. The effect is strongest at high triaxiality and for small ligaments. As gradient effects increase, dependence on ligament size weakens, the influence of the Lode parameter decreases, and deformation becomes more spatially homogenized. In the large-$L_D/r_0$ limit, the matrix hardens sufficiently that the whole cell yields more uniformly, and the effects of ligament size, triaxiality, and Lode parameter fade toward a common upper bound (Holte et al., 2020).

A common oversimplification is therefore that “smaller void spacing always means easier failure.” The results show that this statement is accurate for conventional plasticity but softened once plastic strain gradients are admitted, because the same geometric confinement that promotes localization also increases gradient-induced hardening (Holte et al., 2020).

3. Nucleation and rupture in amorphous graphene

In amorphous graphene, void energetics are described by a competition between perimeter cost and pressure-driven area gain. For a void of radius $$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$0, the free energy is

$$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$1

where $$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$2 is the line tension and $$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$3 is the applied stretching pressure (Jain et al., 2018). This is a classical nucleation-theory structure: small voids are penalized by line tension, but beyond a critical size the area term dominates and growth becomes irreversible.

When line tension is approximated by its thermodynamic-limit value $$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$4, the critical radius is

$$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$5

At $$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$6, the reported thermodynamic-limit line tensions are $$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$7 for flat graphene and $$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$8 for buckled graphene, yielding critical radii of $$T=\frac{\Sigma_h}{\Sigma_e}, \qquad \Sigma_h=\frac{1}{3}(\Sigma_{11}+\Sigma_{22}+\Sigma_{33}),$$9 and $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$0, respectively. The slightly lower buckled value is attributed to additional relaxation in the third dimension (Jain et al., 2018).

The line tension is not constant at finite size. The dominant correction is inversely proportional to void radius,

$$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$1

with fitted values $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$2 for flat graphene and $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$3 for buckled graphene. The simulations and a polygon model of the void perimeter both support this $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$4 scaling. The paper also argues against the conventional intuition that bulk compressibility is the decisive lower bound: instead, the shear modulus sets the lower limit of the line tension, because the dominant energetic contribution at the boundary comes from the angular three-body term associated with shear rigidity (Jain et al., 2018).

This framework is presented as relevant not only for graphene but also for other two-dimensional amorphous materials such as h-BN, phosphorene, borophene, and transition metal dichalcogenides. The broader implication is that rupture in these systems can be organized around the same boundary-versus-area competition, with finite-size corrections and shear elasticity controlling the onset of runaway void growth (Jain et al., 2018).

4. Hotspot formation from collapsing voids in HMX

In shocked HMX, voids are meso-scale heterogeneities whose collapse localizes energy into hotspots. The ignition-and-growth response is decomposed into a single-cylindrical-void baseline multiplied by correction factors for void shape and void-void interactions. These modifiers are learned from high-fidelity reactive simulations using Modified Bayesian Kriging, with shape parameterized by aspect ratio $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$5 and orientation $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$6, and interactions by void fraction $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$7 (Roy et al., 2019).

Void shape is a first-order control on ignition. For elongated voids with $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$8 and $$\Sigma_{22}=\rho_2\Sigma_{11},\qquad \Sigma_{33}=\rho_3\Sigma_{11}.$$9, ignition is amplified relative to the cylindrical baseline. For $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$0, the modifier decreases sharply, and for $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$1 and $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$2, the ignition modifier is approximately zero, meaning that the elongated void may fail to ignite even when the area-matched cylindrical void would be critical. The largest reported growth modifier is $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$3, indicating that growth is shape-sensitive but less dramatically so than ignition (Roy et al., 2019).

Void-void interactions are governed by blast-wave reinforcement and chemical feedback. At $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$4, $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$5 and $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$6: ignition resembles isolated-void behavior, while growth is reduced because the shock attenuates as it traverses the void bed. At $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$7, the modifiers rise to $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$8 for ignition and $$(\dot E^p)^2=\frac{2}{3}\dot{\varepsilon}^p_{ij}\dot{\varepsilon}^p_{ij}+L_D^2\dot{\varepsilon}^p_{ij,k}\dot{\varepsilon}^p_{ij,k},$$9 for growth. In this regime, inter-void spacing is about $L=-1$0, local pressures reach about $L=-1$1–$L=-1$2, hotspot temperatures reach about $L=-1$3–$L=-1$4, and the interacting collapse sequence can produce a self-sustaining meso-scale detonation front (Roy et al., 2019).

The reactive–inert comparison is central. Shock focusing alone does not reproduce the strong intensification observed in the reactive case; chemical heat release is identified as the predominant mechanism driving the strengthening of void-void interactions, with shock focusing acting as an important precursor but not a sufficient cause (Roy et al., 2019).

5. Cosmic voids as hierarchical underdensities

Cosmic voids are the large underdense regions that occupy most of the volume of the late-time cosmic web. Their internal structure is hierarchical rather than featureless. In the CosmoGrid $L=-1$5CDM simulation, void regions embed tenuous walls, which in turn embed tenuous filaments; walls have a typical thickness of about $L=-1$6, and the short filaments identified in void systems are typically no longer than $L=-1$7 with a diameter of about $L=-1$8 (Rieder et al., 2014). The Local Void, reconstructed from Cosmicflows-3 peculiar velocities, begins very near the Local Group, subtends about $L=-1$9 of the sky, is connected to the void in front of Perseus-Pisces, and links through passages to the Hercules and Sculptor voids. Its dynamical influence on the Local Group is estimated at about $L=0$0–$L=0$1 (Tully et al., 2019).

A longstanding theoretical dichotomy distinguishes void-in-void from void-in-cloud evolution. In shell-based analyses of simulated voids, R-type voids occupy underdense surroundings and expand more coherently, while S-type voids are embedded in overdense environments and show broader, more structured shell velocity fields with infall patches. The linear prediction

$L=0$2

tracks the qualitative outflow–infall trend, but departures depend on both environment and tracer density. Velocity smoothness increases with void radius, and voids around $L=0$3 lie in a transition regime between overprediction and underprediction by linear theory (Ruiz et al., 2015).

Excursion-set modeling of the void-in-cloud process remains contested. The two-barrier Sheth–van de Weygaert model uses $L=0$4 and $L=0$5 to suppress small voids, but high-resolution $L=0$6CDM simulations show that spherical-void abundances agree with the model only down to roughly $L=0$7. Below that scale, the measured abundance exceeds the prediction by up to about two orders of magnitude. Density and velocity profiles indicate that many small voids are not crushed out of existence but persist as partially collapsing underdensities. This supports the view that standard void-in-cloud suppression is too strong and too simplified (Chan et al., 2019).

Galaxy populations inside voids are likewise structured rather than anomalous in a categorical sense. The Void Galaxy Survey sample of 59 galaxies at $L=0$8 finds that void galaxies are generally disk dominated and star forming, with Sérsic indices $L=0$9 in both $45^\circ$0-band and $45^\circ$1, and star-formation rates often below $45^\circ$2. In $45^\circ$3–$45^\circ$4 space they overlap dwarf irregulars and spirals, and their specific star formation rate and star formation efficiency follow trends similar to galaxies in average-density environments (Beygu et al., 2015). At finer environmental resolution, the observational “bridge effect” reports that void galaxies in straighter filaments are on average more luminous, with $45^\circ$5 for 148 low-redshift void filaments and stronger correlations for richer systems (Shim et al., 2015). A separate SDSS study of AGN in 323 voids finds larger fractions of AGN and star-forming galaxies in void interiors than in the field, stronger nuclear activity in voids, and a decrease of retired galaxies by about $45^\circ$6 in voids relative to denser environments (Ceccarelli et al., 2021).

Two misconceptions are therefore directly contradicted by the literature. First, a cosmic void is not simply an empty cavity: it can host walls, filaments, aligned galaxy systems, and substantial galaxies (Rieder et al., 2014, Tully et al., 2019). Second, void residency does not by itself imply a special galaxy class. The strongest result from the VGS is demographic rather than ontological: voids favor low-mass, late-type, gas-rich systems, but do not force a fundamentally different star-formation mode (Beygu et al., 2015).

6. Detection, catalog construction, and precision statistics

Because voids are tracer-defined and boundary-sensitive, their identification is algorithmically nontrivial. AVISM combines geometry and dynamics by selecting candidate cells with $45^\circ$7 and $45^\circ$8, using the default threshold $45^\circ$9, then growing cube-based regions until a density-gradient, divergence, or overdensity condition is met. The default stopping parameters are $L=0$0, $L=0$1, and $L=0$2. A hierarchical refinement over nested grids yields explicit voids-in-voids and substructure. In comparison with DIVE and ZOBOV on mini-UCHUU halos, all three methods recover similar total underdense volume, with AVISM giving a volume filling fraction of about $L=0$3 (Monllor-Berbegal et al., 29 Sep 2025).

Tessellation-based methods introduce an additional source of uncertainty: the tessellation itself is unstable under small perturbations in point positions or number density. An optimization scheme based on repeated subsampling and averaging reduces this scatter for the void size function, void two-point correlation function, and void power spectrum. For ZOBOV/VIDE voids, the reported improvements in $L=0$4 constraints are about $L=0$5–$L=0$6 from the optimized void size function and about $L=0$7–$L=0$8 from the optimized void power spectrum. For large Delaunay-tessellation voids, the optimized correlation function can yield BAO signal-to-noise ratios that outperform halos (Liu et al., 20 Jun 2026).

Recent work also reframes void finding as a probabilistic inverse problem. A graph-neural-network flow-matching model evolves test particles $L=0$9 to sample from $L_D/r_0$0, rather than outputting a single deterministic catalog. The model is trained on simulated data, can emulate deterministic teachers such as VIDE, and is explicitly designed to represent stochasticity, void exclusion, and catalog-level correlations (Thiele, 16 Feb 2026). A related methodological warning comes from neutrino-mass inference: once one spurious highly non-spherical VIDE void is removed, there is no evidence that the void-halo mass function contains information beyond the ordinary halo mass function, whereas the proposed VorHMF statistic, which splits halos by Voronoi cell volume, may retain environmental information that the global HMF averages away (Bayer et al., 2024).

Void observables are also becoming cross-probes. Roman-like mock catalogs combined with CMB lensing forecasts indicate that the Void $L_D/r_0$1 CMB lensing signal is less sensitive to mock-construction choices than void counts or void-galaxy clustering. The highest forecast signal-to-noise is obtained for 2D voids with rescaled profiles: about $L_D/r_0$2 with Planck, $L_D/r_0$3 with Simons Observatory, and $L_D/r_0$4 with CMB-S4-like experiments. This suggests that, although the cosmological dependence still requires fuller quantification, Void $L_D/r_0$5 CMB lensing is emerging as a precision observable rather than merely a detection target (Sar et al., 21 May 2026).

7. VOID in latent diffusion security

In machine-learning security, VOID is the name of a defense framework for defeating unauthorized mimicry in latent diffusion models. The threat model includes both training-based personalization attacks—full fine-tuning, DreamBooth, LoRA, SVDiff, and Textual Inversion—and inference-based editing attacks such as SDEdit, inpainting, ControlNet, DDIM inversion, and DiffEdit (Qiu et al., 10 Jun 2026). The paper’s central critique of prior defenses is that they rely on a semantic-steering premise: small perturbations are expected to keep redirecting generation despite VAE compression, diffusion noise injection, and the model’s restoration prior. The reported conclusion is that this premise fails because latent diffusion models tend to restore the protected identity.

VOID therefore shifts from semantic steering to semantic corruption. Its first component, Encoding Uncertainty Amplification, exploits the asymmetry between the VAE mean and variance branches. The reported average Lipschitz constants are $L_D/r_0$6 and $L_D/r_0$7, so the variance branch is about eight times more sensitive. The defense amplifies latent uncertainty in the reparameterized representation

$L_D/r_0$8

to destabilize semantic content at the bottleneck. Its second component, Guidance Signal Counteraction, suppresses the classifier-free-guidance signal used during denoising, thereby weakening the model’s ability to restore identity (Qiu et al., 10 Jun 2026).

Visual utility is preserved by HVS-Guided Perturbation Masking and Timestep-Partitioned Optimal Selection. The perturbation is kept within an $L_D/r_0$9 budget and projected into regions estimated to be visually tolerant under a Top-Down Just Noticeable Difference model. On the reported benchmark of 24 defenses against 10 mimicry attacks over 5 datasets, VOID increases average FID from $L_D/r_0$0 for the strongest prior defense baseline to $L_D/r_0$1, described as a $L_D/r_0$2 improvement. At the same time, the protected images retain PSNR $L_D/r_0$3, SSIM $L_D/r_0$4, and LPIPS $L_D/r_0$5 (Qiu et al., 10 Jun 2026).

The acronym therefore represents a specialized contemporary usage of “void”: not a physical cavity or underdensity, but a deliberate collapse of semantic recoverability inside a stochastic generative pipeline. This suggests a broader lexical pattern already visible in the physical sciences: a void is often defined less by literal absence than by the failure of a system to maintain or restore its usual structure under the constraints imposed on it.