MicroPlate: Benchmark for Solid Mechanics Models

• MicroPlate is a benchmark of architected plate problems that evaluates dynamic stress and deformation via explicit and implicit simulation regimes.
• It employs large 3D unstructured meshes, detailed lattice resolutions, and user-controlled autoregressive loading to capture nonlinear constitutive behavior.
• The benchmark guides surrogate-based design by comparing metrics like displacement error and von Mises stress correlations to rank stress-accurate simulators.

MicroPlate is a benchmark of architected plate problems introduced to evaluate learned world models for nonlinear solid mechanics, particularly under conditions that combine large three-dimensional unstructured meshes, explicit lattice microstructure, path-dependent constitutive response, inertia, and user-controlled autoregressive loading. It was created to assess whether a model such as LEIA can predict both deformation and stress fields in real time across two complementary regimes: architected lattices whose microstructure is resolved directly in geometry, and a homogeneous visco-hyperelastic plate whose microstructural change is represented implicitly through internal degrees of freedom (Yang et al., 27 May 2026).

1. Definition and motivation

MicroPlate was designed around two engineering difficulties that conventional surrogates and existing datasets seldom address simultaneously. The first is accurate stress prediction on very large three-dimensional unstructured meshes that explicitly resolve lattice microstructure. The second is path-dependent stress behavior in a homogenized continuum setting where displacement snapshots alone are insufficient because stress depends on internal state and loading history (Yang et al., 27 May 2026).

The benchmark therefore emphasizes the variables that matter operationally in architected-material design loops: mesh scale, geometric detail, nonlinear constitutive response, and temporal rollout under stepwise boundary-condition control. In this setting, stress accuracy is not a secondary diagnostic. It is a primary target, because ranking candidate architected materials by mechanical performance requires reliable stress fields rather than displacement fidelity alone.

A central feature of MicroPlate is that it pairs two regimes whose difficulties are complementary rather than redundant. The explicit regime stresses geometric resolution and scale; the implicit regime stresses constitutive memory, inertia, and latent internal state. This pairing makes MicroPlate less a single dataset than a structured testbed for probing failure modes of learned simulators under distinct but practically relevant mechanical conditions (Yang et al., 27 May 2026).

2. Two benchmark regimes

MicroPlate contains two classes of problems.

Regime	Physical character	Contents
Explicit microstructure lattices	Quasi-static, compressible Neo-Hookean, lattice geometry resolved in 3D	63 lattices; meshes from 71,000 to 442,000 nodes and about 250,000 to 2,020,000 tetrahedra; 8,836 trajectories of 30 steps
Implicit visco-hyperelastic plate	Dynamic finite viscoelasticity with inertia and internal variables	363-node mesh; 1,000 trajectories of 100 steps each

The explicit regime models architected lattices as a $5 \times 5 \times 1$ tiling of a unit-cell lattice whose beams are expanded under the full cubic symmetry group. Stress depends only on the current deformation because the material law is quasi-static compressible Neo-Hookean. This regime targets the challenge of resolving struts, junctions, and stress concentrations on meshes that can exceed two million tetrahedra (Yang et al., 27 May 2026).

The implicit regime uses a homogeneous plate in which microstructural change is encoded by symmetric positive-definite tensorial internal variables $A^{(i)}$ associated with visco-hyperelastic branches. Here, stress depends on both the current deformation gradient and the internal variables, so displacement fields do not uniquely determine stress. The trajectories are deliberately history-sensitive: each consists of 50 random loading steps followed by their exact reverse for unloading (Yang et al., 27 May 2026).

The dataset splits also reflect these objectives. In the explicit regime, 55 lattices are used for training and 8 lattices are held out for generalization. For the 55 training lattices, the first 100 trajectories, totaling 5,500, are used for training and trajectory 101 is reserved for evaluation. In the implicit regime, the 1,000 trajectories are split 800/100/100 for train/validation/test with seed 42 (Yang et al., 27 May 2026).

3. Geometry, meshing, and loading protocols

In the explicit regime, each lattice begins from a compact graph of nodes and beams with radii. Every beam is replicated under all 48 operations of the octahedral group $O_h$, that is, six coordinate permutations combined with eight sign combinations. The symmetric unit cell is then tiled $5 \times 5 \times 1$. The plate geometry is obtained by wrapping coordinates into the unit cell through a signed distance field, evaluated at 30 voxels per unit cell and assembled into a $[10 \times 10 \times 2]$ plate geometry via modular coordinate mapping. Smooth-min blending joins beam capsules at junctions, marching cubes extracts the surface, and TetGen generates tetrahedral meshes. A representative MicroPlate lattice contains 301,565 nodes and 1,382,904 tetrahedra (Yang et al., 27 May 2026).

Boundary conditions are standardized across the benchmark. The left face, $x=x_{\min}$, is clamped with zero displacement, while the right face, $x=x_{\max}$, serves as the controlled grip. The remaining faces are traction-free. This induces a clear separation between plate-scale actuation and microstructure-scale response.

The action space comprises four boundary-condition actions: stretch, twist, and shear in two orthogonal directions. Actions are discrete at each step with values in $\{-1,0,+1\}$, and cumulative boundary-condition values condition the dynamics. In the lattice regime, loading is quasi-static and incremental over 30 steps, with per-step magnitudes of 0.15 for stretch, shear$_y$, and shear$_z$, and 0.08 for twist. In the visco-hyperelastic regime, loading is dynamic with inertia, using Newmark-beta integration with $A^{(i)}$0, $A^{(i)}$1, $A^{(i)}$2, and 5 FEM substeps per action step (Yang et al., 27 May 2026).

All simulations start from rest. Displacement is initialized as $A^{(i)}$3. For the visco-hyperelastic plate, velocity and acceleration are zero, internal variables satisfy $A^{(i)}$4, and the reference pressure is set to enforce near-incompressibility. The governing balance in the reference configuration is

$A^{(i)}$5

with the quasi-static lattice case obtained by taking $A^{(i)}$6 (Yang et al., 27 May 2026).

4. State variables, constitutive laws, and discretization

MicroPlate encodes both kinematic and stress information. The benchmark uses

$A^{(i)}$7

with distortional tensor $A^{(i)}$8. Stress is represented through the first Piola–Kirchhoff tensor

$A^{(i)}$9

the Cauchy stress

$O_h$0

and the von Mises stress

$O_h$1

where $O_h$2 (Yang et al., 27 May 2026).

For the explicit lattices, the free energy per unit reference volume is compressible Neo-Hookean,

$O_h$3

with $O_h$4 and $O_h$5. No internal variables appear, so the constitutive response is memoryless apart from the geometric nonlinearity.

For the implicit regime, the benchmark uses finite viscoelasticity with inertia. The free energy is decomposed as

$O_h$6

The equilibrium part is Arruda–Boyce with chain stretch $O_h$7 and locking stretch $O_h$8, while each non-equilibrium branch is neo-Hookean relative to $O_h$9:

$5 \times 5 \times 1$0

The internal variables evolve according to

$5 \times 5 \times 1$1

The parameters are $5 \times 5 \times 1$2, $5 \times 5 \times 1$3, $5 \times 5 \times 1$4, $5 \times 5 \times 1$5, and $5 \times 5 \times 1$6 non-equilibrium branches with $5 \times 5 \times 1$7, $5 \times 5 \times 1$8, and $5 \times 5 \times 1$9 (Yang et al., 27 May 2026).

The discretizations are regime-specific. The lattice regime uses legacy FEniCS (dolfin 2019.1.0) with CG1 linear Lagrange tetrahedral elements and quasi-static Newton iteration solved by MUMPS with $[10 \times 10 \times 2]$0, $[10 \times 10 \times 2]$1, and maximum 15 iterations. The visco-hyperelastic regime uses FEniCSx (DOLFINx 0.8) with a mixed Taylor–Hood formulation: CG2 for displacement, CG1 for pressure, internal variables at quadrature points, and Newton iteration with MUMPS LU using $[10 \times 10 \times 2]$2 and maximum 50 iterations (Yang et al., 27 May 2026).

5. Benchmark protocol, baselines, and metrics

MicroPlate is not only a collection of trajectories; it is also an evaluation protocol for stress-accurate learned simulators. Output fields are displacement and stress at mesh nodes. During tokenizer training, displacement and stress are normalized per component to zero mean and unit variance. For large meshes, training uses furthest-point sampling for uniform spatial coverage, while decoding reconstructs fields at all nodes (Yang et al., 27 May 2026).

The benchmark defines two principal metrics. The first is per-frame displacement relative $[10 \times 10 \times 2]$3 error,

$[10 \times 10 \times 2]$4

reported as the median per trajectory. The second is the Pearson correlation between predicted and FEM-derived von Mises stress fields, again summarized by the median per trajectory. For models without a stress head, stress is computed post hoc from predicted displacement using tetrahedral shape-function differentiation and the constitutive law, followed by volume-weighted averaging from elements to nodes (Yang et al., 27 May 2026).

The baseline set comprises MeshGraphNets, UPT, Transolver, and a PointNet-style per-frame set encoder. In the lattice regime, models may predict displacement only, with post-hoc stress, or predict stress directly using a stress head. In the visco-hyperelastic regime, the comparison is more stringent: baselines are evaluated both with indirect stress prediction, in which internal variables and pressure are inferred and mapped to stress, and with direct prediction of displacement plus the six independent stress components. Without a stress head, outputs scale to 22 dimensions in the visco case; with a stress head, the output dimension is 9 (Yang et al., 27 May 2026).

MicroPlate also standardizes the autoregressive setting. The central question is not whether a model can fit one-step teacher-forced targets, but whether it can maintain deformation and stress fidelity under rollout. This suggests that the benchmark is optimized for interactive or control-oriented deployment rather than static surrogate regression.

6. Quantitative findings and surrogate-guided design

The benchmark results show that stress prediction strategy strongly affects performance. In the large explicit-lattice regime, post-hoc stress from displacement alone yields weak von Mises correlation: without gradient supervision, median displacement errors are 10.83% in-distribution and 14.60% on held-out lattices, while VM correlation is only 0.244 and 0.210. Tet Sobolev gradient supervision improves these values to 7.08% and 7.89% displacement error, with VM correlation 0.410 and 0.388, but adds 53% training overhead. A direct stress head achieves much higher stress fidelity, with VM correlation 0.870 in-distribution and 0.654 held-out at only 6% overhead. The best reported configuration combines a stress head, pushforward dynamics, and velocity input, reaching 4.68% displacement error in-distribution, 5.39% held-out, and VM correlation 0.942 in-distribution and 0.662 held-out (Yang et al., 27 May 2026).

In the implicit visco-hyperelastic regime, the gap between indirect and direct stress prediction is sharper. Under teacher forcing, all baselines without a stress head remain at VM correlations of at most 0.26, whereas with a stress head all baselines exceed 0.94. Under autoregressive rollout over 100 steps, LEIA achieves displacement error 5.99% and VM correlation 0.986. UPT yields 31.8% and 0.882; PointNet 2.87% and 0.938; Transolver 2.42% and 0.947. MeshGraphNets is identified as a failure case under long rollout, collapsing to 241% displacement error and 0.012 VM correlation without a head, and recovering only to 9.94% and 0.225 with a head (Yang et al., 27 May 2026).

MicroPlate is also used for surrogate-guided search. Starting from a seed lattice topology represented as a graph of nodes and beams with radii, candidates are mutated by six operators, meshed through the MicroPlate pipeline, and evaluated under stretch and shear actions. Candidates are ranked by the stress-based design metric

$[10 \times 10 \times 2]$5

where $[10 \times 10 \times 2]$6 is work computed by trapezoidal integration of $[10 \times 10 \times 2]$7force$[10 \times 10 \times 2]$8 over $[10 \times 10 \times 2]$9displacement$x=x_{\min}$0 and $x=x_{\min}$1 is normalized volume fraction. In the reported search, 553 candidates were evaluated in 30 minutes of wall-clock time, with per-candidate evaluation approximately 100–300 times faster than FEM depending on mesh resolution. Starting from seed mg_046 with volume fraction 6.6%, the search found a 10-strut design, reduced from 15 struts, with volume fraction 2.1% and a 3.6-fold improvement in the design metric, validated by finite element ground truth (Yang et al., 27 May 2026).

These results make MicroPlate notable for linking benchmark performance directly to downstream design quality. A plausible implication is that the benchmark was constructed not merely to compare simulators, but to test whether stress-accurate rollouts are sufficient for design ranking under realistic mesh and constitutive complexity.

7. Scope, limitations, and future directions

MicroPlate is intentionally narrow in some respects. The architected lattices are restricted to cubically symmetric unit cells expanded under $x=x_{\min}$2 and tiled $x=x_{\min}$3. The action space is limited to four loading modes. Broader geometries, arbitrary three-dimensional shapes, and additional microstructure families are not included. Nor does the benchmark cover multiphysics couplings such as fluid–structure interaction, thermal transport, or electrochemistry (Yang et al., 27 May 2026).

The benchmark also exposes model-specific and regime-wide failure modes. In the visco-hyperelastic setting, indirect stress recovery through internal variables is broadly inadequate. In long autoregressive rollouts, message-passing temporal models may collapse without sufficiently strong conditioning. Even LEIA, which performs best overall, degrades on out-of-distribution mutated lattices. The paper notes that simple tokenizer round-trip checks or latent cycle-consistency checks miss some inaccurate but in-distribution dynamics, motivating a learned confidence head (Yang et al., 27 May 2026).

From a reproducibility perspective, code is stated to be available at the LEIA repository together with MicroPlate and baseline implementations, but the text does not specify dataset file formats, licensing terms, or direct archive layout. Training hyperparameters, however, are documented in detail, including tokenizer dimensions, optimizer settings, pushforward rollout length, and hardware based on $x=x_{\min}$4 H100 80GB GPUs. This suggests that MicroPlate is positioned as a benchmark for technically intensive, large-scale model development rather than a lightweight entry-level dataset.

In the literature on learned mechanics, MicroPlate occupies a specific niche: it couples explicit microstructure resolution with implicit internal-state dynamics, and it evaluates models by stress-aware autoregressive performance rather than by displacement snapshots alone. That combination makes it a demanding reference problem for action-conditioned simulators intended for interactive exploration and architected-material design (Yang et al., 27 May 2026).