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This paper presents a systematic classification for detachable dry joints suitable for the structural reuse of reinforced concrete elements, together with an analytical evaluation of representative configurations. The classification, named JOINTS and applicable to concrete connections beyond the reuse context, organizes the design space along five categories whose initials spell JOINT: the Juxtaposition (spatial arrangement of the paired elements), the Origin (their prior structural function), the Interface geometry, the Newly introduced materials at the joint, and the force-Transfer mechanism. A sixth category, the Score, completes the acronym and evaluates each configuration for sustainability, cost-efficiency, and mechanical performance. From this design space, three representative configurations are evaluated analytically under active post-tensioning, differing in both their joint geometry (planar, stepped, and angled) and their tendon inclination. The evaluation covers failure of the anchorage of the post-tensioning member, the contact limit of the interface, and the material limit of the concrete adjacent to the interface. Although complex joint geometries theoretically enhance interlocking, their benefit is limited by the compressive-shear strength of the concrete. The reinforcement of the reused elements may be damaged near the cut, and the splitting reinforcement that a conventional design would require is not guaranteed to exist. The present analysis therefore conservatively assumes that the concrete at the joint face alone resists the combined compression and shear, neglecting any residual contribution from the existing reinforcement. This analytical evaluation represents a first step toward quantifying the score over a parametric range of connections, here focusing on the mechanical criterion.

Graph-based collaborative filtering learns user and item representations by propagating embeddings over a user--item interaction graph. We investigate a modular, sheaf-inspired scoring extension placed on top of a three-layer LightGCN backbone. The motivation comes from cellular sheaves, where local vector spaces are related by linear restriction maps; here, this idea motivates structured, pair-dependent transformations of user and item embeddings rather than a change to graph propagation. We compare the identity inner-product scorer, diagonal, rank-one low-rank, and full matrix residual scorers, and a parameter-matched residual MLP baseline (\texttt{mlp\_full}) on Amazon-Book, Gowalla, MovieLens-1M, and Yelp2018. We report ranking quality and catalog coverage at cutoffs 10 and 20. The selected structured variants show modest, dataset-dependent changes in ranking quality and sometimes substantial changes in coverage. Our method is inspired by sheaf geometry but does not construct a cellular sheaf or implement sheaf-Laplacian diffusion; the experiments therefore do not establish a causal benefit from sheaf geometry itself.

Graph neural surrogates for granular flow can resolve the motion of individual grains, but propagating information across a large amount of grains requires repeated, costly updates on the full particle graph. This study introduce a Hierarchical Coarse-Grained Graph Neural Network (HCG-GNN) that couples direct interactions between neighboring grains with information exchange on two coarser graphs of occupied spatial cells. In discrete element method (DEM) benchmarks of granular flows down incline planes, HCG-GNN captured key flow characteristics, including front propagation, lateral spreading, deposition morphology, free surface evolution, internal shear behavior, and energy conversion. Results also show that preserving grain-specific features as coarse information returned to the particle graph and integrating predicted accelerations to advance particle motion both improved multi-step accuracy. With 5 of its 7 message-passing steps performed on coarse graphs, HCG-GNN achieved accuracy comparable to a conventional particle-level GNN with a 6.47-fold training speedup and 46.2% lower GPU memory use. A model trained on up to 5000 grains predicted a 2 s release of 10000 grains in 31 s, compared with 2000 s for DEM under the benchmark conditions. This combination of speed and grain-scale resolution supports particle-resolved inverse design, with prescribed runout or discharge targets guiding the search for suitable geometries and grain dynamics revealing the basis for differences in performance.

Neural surrogates for computational fluid dynamics (CFD) are trained and ranked on datasets that each vary one design family in one flow domain. Such datasets cannot test the least that an aerodynamic foundation model must do: predict CFD-quality incompressible flow in seconds for new shapes in new enclosures. On DrivAerML, larger models and longer training barely lower Transolver's error; doubling the training cases cuts it by about 12% up to 400 cases, exhausting the dataset. We introduce AeroZoo, a benchmark and training corpus of 3,815 time-mean solutions to the unsteady Reynolds-averaged Navier--Stokes (RANS) equations with a transition model, across seven geometry families, eight enclosure configurations and Reynolds numbers from $10^3$ to $10^8$. Each sample poses a well-posed PDE problem in the form a solver receives it: the boundary meshes grouped by condition type, the prescribed values and the fluid properties. The 2,816 samples with a stable time mean form the canonical benchmark: two tasks, engineering metrics and 22 frozen splits, disjoint in geometry, that hold out geometry families, enclosures, boundary-condition layouts and Reynolds ranges. On the random split, boundary-condition tokens cut GeoTransolver's median velocity error from 76.9% to 8.4% of the inlet speed in single-seed runs (uniform inlet flow: 11.0%). ISLA, a transformer for boundary-driven PDEs, reaches 5.3% over four seeds, about 4 times the change from a finer mesh. On the same test samples, a held-out geometry family barely changes ISLA's error relative to uniform flow, and a held-out enclosure or boundary-condition layout raises it 2.4 to 9.2 times. The data and the scoring code will be released.

A foundational principle of physics is that all coordinate systems are equally valid: the behavior of a PDE-governed system does not depend on the origin or orientation of the frame, nor on the units or mesh used to write it down, and neither should a surrogate model. Yet the neural surrogates with the lowest errors on today's aerodynamics benchmarks lack these invariances and fail under simple transformations of the same problem: for models trained on PDEs of cars in one pose (i.e., translation and rotation) and from one mesher, rotating the full PDE raises the surface-pressure error from 4% to more than 100%, and re-sampling its boundary raises it to about 30%. We introduce ISLA (Invariant Slice Attention), a slice-attention transformer for steady, boundary-driven PDEs in which no learned operation receives a coordinate and every sum over the boundary is a quadrature. Positions enter only as distances and relative angles to learned slice anchors, lengths are measured in a unit computed from the boundary, and interior points query the boundary through a tree-accelerated learned kernel without influencing one another. With every boundary cell a token and an exact boundary read, ISLA is equivariant to rigid motions and reflections, invariant to the units of its declared data, and consistent under mesh refinement, while at every setting each query is independent of all other queries. On AeroZoo, 2,816 flows that vary in shape, size, speed, and boundary conditions, its mean reference-scaled velocity and pressure errors are at least 3.1 and 3.3 times lower than any baseline's. On AirfRANS its errors are second only to those of a boundary-integral model (GLOBE) and at least 1.6 times lower than those of Transolver and GeoTransolver.