PARCEL: Unit-Based Abstraction in Science & Industry

• PARCEL is a polysemous term denoting unit-based abstractions, such as Lagrangian aggregates, shipment units, land parcels, and model architectures.
• In CFD, parcels represent multiple physical particles to reduce computational load while preserving essential kinematic and dynamical states.
• In logistics and remote sensing, parcels are optimized as analytical units that drive routing, sortation, and spatial classification in multimodal systems.

to=arxiv_search.search 乐亚json {"query":"PARCEL arXiv PARCEL Pool-Anchored Resampling conditioned elastic queries (Kuzucu et al., 28 May 2026)", "max_results": 5} to=arxiv_search.search 荣富json {"query":"(Marzouk et al., 2024) Effects of Turbulence Modeling and Parcel Approach on Dispersed Two-Phase Swirling Flow", "max_results": 3} to=arxiv_search.search 开号地址json {"query":"terraced parcel dataset GTPBD (Zhang et al., 19 Jul 2025) GTPBD-MM (Zhang et al., 14 Apr 2026) online parcel assignment PPO-OPA (Zeng et al., 2021)", "max_results": 10} Parcel is a polysemous technical term whose meaning depends strongly on disciplinary context. In computational fluid dynamics, a parcel is a Lagrangian aggregate representing many identical physical particles; in logistics, it is the shipment unit that drives assignment, sortation, routing, storage, forecasting, and loss-analysis models; in geospatial analysis, it is a bounded land unit such as a road-enclosed urban polygon or a terraced farmland instance; and in machine learning and medical imaging, PARCEL is also used as an acronym for distinct architectures and training paradigms (Marzouk et al., 2024, Long et al., 2014, Kuzucu et al., 28 May 2026, Wang et al., 2022). The term therefore does not denote a single ontology but a family of unit-based abstractions that support decomposition, aggregation, and decision-making at different scales.

1. Definitions and semantic range

Across the cited literature, “parcel” is defined operationally rather than lexically. In Euler–Lagrange CFD, a computational parcel $p$ represents $W_p$ identical physical particles, carries combined mass and momentum, and exists to reduce the number of Lagrangian trajectories that must be integrated (Marzouk et al., 2024). In urban GIS, a parcel is “a continuous built-up area enclosed by roads,” generated as a polygonal gap in a processed road network (Long et al., 2014). In terraced remote sensing, a parcel is a connected region of terraced farmland enclosed by field ridges, with boundary, mask, and parcel-instance labels defined separately (Zhang et al., 19 Jul 2025). In logistics, a parcel is the physical shipment unit associated with product, customer, order, routing, or destination attributes; several formulations treat parcels as agents, commodities, or assignment requests rather than merely as objects (Zeng et al., 2021, Leeuw et al., 2023, Kato et al., 13 May 2026).

This semantic range is not accidental. In every case, the parcel is the smallest unit at which a system preserves meaningful state. In CFD that state is kinematic and dynamical; in logistics it is transactional and spatiotemporal; in remote sensing it is topological and geometric. A plausible implication is that the term persists because it balances fidelity and tractability: finer than a coarse continuum, but structured enough to be aggregated.

2. Computational parcels in dispersed two-phase flow

The most explicit mathematical treatment of a parcel appears in the Euler–Lagrange simulation of a co-axial particle-laden swirling air flow in a vertical circular pipe (Marzouk et al., 2024). There, a computational parcel $p$ represents $W_p$ identical particles, with

$W_p = N_{\text{real}}/N_{\text{parcel}},$

and parcel mass

$$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$

Its trajectory satisfies $dX_p/dt=u_p$, while parcel momentum is driven by drag and buoyancy-corrected gravity. The drag force is scaled directly by $W_p$, and two-way momentum coupling enters the gas-phase momentum equation through a cell source term

$$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$

The implementation is fully specified. Parcels are injected at the primary nozzle with a prescribed mass flow rate and a log-normal size distribution of mean $d=45\,\mu\text{m}$. Each parcel is assigned inlet position, axial speed $W_p$0, mass $W_p$1, and turbulent time scales $W_p$2 and $W_p$3. Position is updated explicitly, momentum is integrated implicitly, turbulent dispersion is modeled by a Discrete Random Walk using $W_p$4, and wall interaction is treated as specular elastic collision with tangential friction neglected (Marzouk et al., 2024).

The surrounding turbulence-model comparison clarifies what the parcel abstraction does and does not control. Among three $W_p$5-$W_p$6 variants, the standard model achieved the best overall performance against experimental mean velocity profiles; the realizable model was unable to satisfactorily predict the radial velocity and was the most computationally expensive; the RNG model predicted additional recirculation zones (Marzouk et al., 2024). This is significant because it separates turbulence-closure error from parcel-grouping error.

The performance results are especially instructive. Solving $W_p$7 of physical time on identical dual-core AMD Opteron 265, $W_p$8 machines, the particle approach with $W_p$9 injected $p$0 parcels and required approximately $p$1, whereas the parcel approach with $p$2 injected $p$3 parcels and required approximately $p$4 (Marzouk et al., 2024). Reducing parcel count by two orders of magnitude therefore produced only a $p$5 CPU-time reduction, because the dominant cost remained the gas-phase RANS/turbulence solver, although Lagrangian-array memory dropped by approximately $p$6. Mean gas-phase axial, radial, and tangential velocity profiles differed negligibly, by less than $p$7, across $p$8, but at $p$9 particle-phase detail was smoothed, with slightly reduced heavy-parcel penetration and weakened small-particle entrainment (Marzouk et al., 2024). This directly counters a common misconception that parceling automatically yields major wall-clock speedups: at low loading, memory savings can be dramatic while CPU savings remain modest.

3. Parcels as optimization objects in logistics networks

In logistics research, the parcel is often the atomic unit of stochastic decision-making. In online parcel assignment, each arriving parcel $W_p$0 has candidate routes $W_p$1, route costs $W_p$2, and interactions with capacity and proportion constraints; the resulting Online Assignment MDP uses parcel observations $W_p$3, constraint states $W_p$4, route-selection actions $W_p$5, and a reward that trades off cost against state-dependent violation risk (Zeng et al., 2021). The PPO-OPA algorithm uses attention networks for policy evaluation and produced results comparable to the Primal-Dual algorithm without assuming total parcel volume is known in advance. On one Cainiao dataset with capacity constraints, PPO-OPA achieved average cost $W_p$6, IP gap $W_p$7, and violation $W_p$8, versus $W_p$9, $W_p = N_{\text{real}}/N_{\text{parcel}},$0, and $W_p = N_{\text{real}}/N_{\text{parcel}},$1 for PDO and $W_p = N_{\text{real}}/N_{\text{parcel}},$2, $W_p = N_{\text{real}}/N_{\text{parcel}},$3, and $W_p = N_{\text{real}}/N_{\text{parcel}},$4 for the proportional baseline; on a proportion-constraint dataset, PPO-OPA obtained $W_p = N_{\text{real}}/N_{\text{parcel}},$5, $W_p = N_{\text{real}}/N_{\text{parcel}},$6, and $W_p = N_{\text{real}}/N_{\text{parcel}},$7 (Zeng et al., 2021).

A more structural formulation appears in parcel sortation. The model in “Fast Combinatorial Algorithms for Efficient Sortation” treats parcel flows as commodities on a directed graph $W_p = N_{\text{real}}/N_{\text{parcel}},$8, with feasibility defined in the transitive closure and objective

$W_p = N_{\text{real}}/N_{\text{parcel}},$9

the busiest node’s sort-point load (Dyk et al., 2023). Determining whether $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$0 is NP-hard even when $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$1 is a star, via reduction from Hitting Set. The paper also develops witness-set lower bounds, gives an exact $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$2 algorithm for single-source out-trees, a polynomial-time additive-$$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$3 approximation for multi-source out-trees, and a $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$4-approximation on stars (Dyk et al., 2023). Here the parcel matters not as an item description but as a contributor to combinatorial congestion and sorter capacity.

Parcel routing and consolidation are also coupled at network scale. In hyperconnected megacity parcel logistics, commodities traverse hub networks where individual parcels may either be sorted or bypass intermediate sorting in sealed containers. A path-based integer program jointly chooses routing and containerized consolidation so as to minimize expected total pickup-to-delivery time under sorting, cross-docking, and vehicle-capacity constraints (Kaboudvand et al., 2021). On a synthetic $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$5 urban grid, the study reports approximately $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$6 average transit-time savings and approximately $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$7 average handling-time savings, with up to approximately $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$8 transit-time savings and up to approximately $$m_p = W_p \, m_{\text{particle}} = W_p \, (\pi \rho_p d_p^3/6).$$9 handling-time savings depending on configuration and demand pattern (Kaboudvand et al., 2021). This suggests that, in logistics network models, a parcel is simultaneously a demand unit and a handleable flow element whose treatment determines both travel time and hub workload.

4. Transport systems, handling platforms, and parcel analytics

Several works treat parcels as physical units that constrain platform morphology, routing feasibility, or operational risk. In a reconfigurable multicopter, the parcel itself becomes the drone’s body, while modular propulsion units attach in counter-rotating symmetric pairs on opposite parcel faces (Schiano et al., 2022). The configuration generator takes parcel dimensions $dX_p/dt=u_p$0 and mass $dX_p/dt=u_p$1, enumerates feasible module counts, scores them by

$dX_p/dt=u_p$2

then produces layout files, geometry files, a PX4 mixer, and PID parameters derived from simulation-in-the-loop. Indoor experiments used parcels A–D with masses $dX_p/dt=u_p$3–$dX_p/dt=u_p$4, sizes $dX_p/dt=u_p$5 and $dX_p/dt=u_p$6, and $dX_p/dt=u_p$7–$dX_p/dt=u_p$8 modules; overall position RMSE was $dX_p/dt=u_p$9 and yaw RMSE was approximately $W_p$0, while outdoor missions were completed with RTK-based waypoint accuracy within a few tens of centimeters (Schiano et al., 2022). The main trade-offs are explicit: manual reconfiguration time, control complexity for asymmetric or low-module-count layouts, and box-shaped aerodynamic inefficiency.

Warehouse-control problems impose additional structure. In conveyor parcel routing with order-contiguous arrivals, parcels are agents in an online MAPF-OC problem on a directed graph $W_p$1, with the extra requirement that arrivals at each destination form contiguous blocks by order label (Kato et al., 13 May 2026). The proposed Dual-Ordering Prioritized Planning algorithm maintains order-level and agent-level partial orders and uses prioritized planning with destination blocking; under a well-formed setting it is complete and polynomial-time. Experiments on layouts derived from actual warehouses show that Vanilla DOPP remains under $W_p$2 even for $W_p$3 on Medium and Large maps, and that Level3-NS can yield up to $W_p$4 further makespan reduction (Kato et al., 13 May 2026). In a related but distinct warehouse setting, multi-robot parcel sorting decomposes into bin assignment and decentralized path planning over a strongly connected directed road network; GA and MIP reduce expected travel cost by about $W_p$5 over random assignment, while PRYP and EPRY achieve near-centralized throughput with per-step runtimes in the millisecond range (Guo et al., 2023).

Parcel handling is also studied at the perception and manipulation layer. The Parcel-Suction-Dataset contains $W_p$6 synthetic cluttered scenes, $W_p$7 parcel asset models, and $W_p$8 million precision-annotated suction grasp poses, with candidate labels defined as

$W_p$9

Diffusion-Suction reformulates suction grasp prediction as conditional denoising diffusion over point-cloud-conditioned score maps and reports Top-50 AP $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$0 and Top-1 AP $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$1 on Parcel-Suction-Dataset, outperforming Normal STD, DexNet 3.0, and Cao et al. baselines (Huang et al., 11 Feb 2025).

Operational analytics extends the concept further. For parcel pick-up points, a Markov jump process models parcel life cycles from order confirmation through in-transit states to “delivered at PUP and awaiting customer pick-up,” producing one- to four-day-ahead load forecasts; on a real B2C dataset from Roussillon, the approach achieved MAE $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$2 parcels and MAPE approximately $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$3 at $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$4, and MAE $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$5 and MAPE approximately $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$6 at $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$7 (Nguyen et al., 2024). For last-mile loss prediction, Belgian shipment data with $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$8 parcel records and $$S_i^p = -\frac{1}{V_{\text{cell}}}\sum_{p\in\text{cell}}F_{\text{drag},p,i}.$$9 losses supported both balanced supervised learning and a deep hybrid ensemble in which an autoencoder-derived reconstruction-error vector is classified by a random forest; the AE-RF model achieved precision approximately $d=45\,\mu\text{m}$0, recall approximately $d=45\,\mu\text{m}$1, balanced accuracy approximately $d=45\,\mu\text{m}$2, and ROC-AUC approximately $d=45\,\mu\text{m}$3, while SHAP analysis identified quantity, stock value, size length, depots, regions, and delivery mode as prominent risk factors (Leeuw et al., 2023).

5. Parcel as a land unit in urban and terraced remote sensing

In spatial analysis, parcel-level representation is used to bridge geometric detail and administrative relevance. For urban mapping, parcels are generated from a nationwide road network by connecting segments within $d=45\,\mu\text{m}$4, trimming cul-de-sacs shorter than $d=45\,\mu\text{m}$5, assigning road widths by hierarchy, buffering processed roads, and erasing them from the study area; this yielded $d=45\,\mu\text{m}$6 parcels across $d=45\,\mu\text{m}$7 Chinese cities (Long et al., 2014). A vector cellular automata model then classifies each parcel as urban or non-urban using parcel size, compactness, distance to CBD, normalized POI density, neighborhood state within $d=45\,\mu\text{m}$8, spatial constraints such as slope $d=45\,\mu\text{m}$9 and water bodies, and a stochastic term. Calibrated on $W_p$00 Beijing parcels, the model explained $W_p$01 of parcel-level variance and achieved $W_p$02 overall classification accuracy against a planner-prepared reference; overlap with external datasets was $W_p$03 for DMSP/OLS, $W_p$04 for GLOBCOVER, $W_p$05 for sub-district population density, and $W_p$06 for road-intersection kernel density (Long et al., 2014).

Terraced agriculture requires a different parcel formalization. GTPBD defines a parcel as a connected terraced-farmland region whose topological boundary coincides with annotated ridge lines, and provides $W_p$07 non-overlapping $W_p$08 high-resolution tiles with three-level labels: $W_p$09-pixel-wide boundary labels, mask labels, and parcel-instance labels (Zhang et al., 19 Jul 2025). The dataset covers seven major geographic zones in China and terraced regions across $W_p$10 countries, includes more than $W_p$11 manually annotated complex terraced parcels, and supports semantic segmentation, edge detection, terraced parcel extraction, and unsupervised domain adaptation. Benchmark results show, for example, Mask2Former with best semantic-segmentation IoU $W_p$12 and F1 $W_p$13; REAUNet-Sober with best edge-detection ODS $W_p$14, OIS $W_p$15, and AP $W_p$16; HBGNet with lowest object-level GTC $W_p$17; and SegFormer with best pixel-level parcel-extraction IoU $W_p$18 and F1 $W_p$19 (Zhang et al., 19 Jul 2025).

GTPBD-MM extends this framework by aligning optical imagery, structured text, and DEM data. Built on GTPBD, it covers more than $W_p$20 of terraced farmland from $W_p$21 countries, with $W_p$22–$W_p$23 optical imagery and co-registered DEM, and evaluates image-only, image+text, and image+text+DEM settings (Zhang et al., 14 Apr 2026). Its baseline model, ETTerra, uses a text-guided semantic branch with cross-attention between SAM image features and CLIP text features, plus a DEM-guided geometric branch that modulates image features by elevation-derived scale and shift. On the test set, ETTerra achieved Recall $W_p$24, F1 $W_p$25, OA $W_p$26, mIoU $W_p$27, OIS $W_p$28, ODS $W_p$29, and GTC $W_p$30, outperforming HBGNet, PixelLM, and FSVLM baselines (Zhang et al., 14 Apr 2026). The paper also states the key limitation clearly: coverage remains limited to well-known terraced areas, and only optical, text, and DEM modalities are included.

6. PARCEL as an acronym in machine learning and medical imaging

PARCEL also appears as a method name in at least two unrelated domains. In large vision–LLMs, PARCEL stands for Pool-Anchored Resampling with Conditioned Elastic Queries, a visual tokenization architecture for elastic compression under multiple visual-token budgets (Kuzucu et al., 28 May 2026). The method partitions representation into grid-aligned pool tokens $W_p$31, which act as low-frequency spatial anchors, and elastic query tokens conditioned on those anchors through Pool-Conditioned Query Resampling. It uses two routing regimes: for $W_p$32, a $W_p$33 pool with $W_p$34; for $W_p$35, an $W_p$36 pool with $W_p$37. Trained from the PaliGemma-2 Stage-1 recipe for $W_p$38 samples and evaluated on $W_p$39 benchmarks, PARCEL retained $W_p$40 of uncompressed PG2 accuracy on images and $W_p$41 on videos at $W_p$42, versus approximately $W_p$43/$W_p$44 for MQT and approximately $W_p$45/$W_p$46 for M$W_p$47; at $W_p$48, retention remained $W_p$49 for images and $W_p$50 for videos (Kuzucu et al., 28 May 2026). The ablations attribute gains to dynamic budget routing and sequential pool-conditioned query resampling rather than extra capacity alone.

In accelerated multi-coil MRI, PARCEL stands for Physics-bAsed unsupeRvised Contrastive rEpresentation Learning (Wang et al., 2022). The method trains two MoDL-style unrolled networks on differently re-undersampled views of the same multi-coil k-space data, with a co-training loss comprising undersampled calibration, reconstructed calibration, and contrastive representation terms. Each network uses $W_p$51 stages, a $W_p$52-layer CNN denoiser with $W_p$53 convolutions and $W_p$54 channels, and $W_p$55 conjugate-gradient iterations per data-consistency block. The system was evaluated on fastMRI knee data and in-house brain data under $W_p$56, $W_p$57, and $W_p$58 sampling regimes, outperforming SENSE, Variational-Net, U-Net-256, and SSDU, and approaching fully supervised MoDL within $W_p$59–$W_p$60 PSNR and $W_p$61–$W_p$62 SSIM (Wang et al., 2022). An ablation showed that adding the contrastive loss raised PSNR by approximately $W_p$63 on knee $W_p$64 and by approximately $W_p$65 on brain $W_p$66.

The coexistence of these two expansions underscores a final point. “PARCEL” is not a stable acronym attached to one canonical method family; it is reused for technically disjoint contributions in vision–language compression and MR reconstruction (Kuzucu et al., 28 May 2026, Wang et al., 2022). This suggests that, for researchers, disambiguation by full title and domain is essential whenever the term appears in citation, retrieval, or implementation contexts.