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Uniform-gradient Hex8 development boundary

The experimental elements.uniform_strain_hex8(...) policy lowers ordinary model.step() to small-strain elasticity or a bounded finite-strain history provider, both with owned-cell MPI assembly and same-partition restart. Nonmatching interface composition remains serial. The private elements._uniform_hex.UniformHex8 remains its local verification oracle. This is not advertised as an Abaqus C3D8R reproduction.

Ordinary workflow

policy = elements.uniform_strain_hex8(
    hourglass_modulus=material_shear_scale,
    hourglass_scale=0.1,
)
step = model.step(target=u, element_policy=policy, dt="auto", steps=100)
result = step.solve_result(
    output="fields.xdmf", field_variables=("S", "E", "MISES", "SENER")
)

Optional cell fields come directly from the uniform-gradient constitutive response: S is physical stress, E is small strain (both 3-by-3 tensors), MISES is von Mises equivalent stress and SENER is physical strain-energy density. They use DG0 without smoothing across cells or material interfaces. Artificial hourglass energy remains separate; no artificial stress is added to S. Von Mises output is not an anisotropic failure criterion. Units follow the model's consistent unit system and are not inferred. Processing provenance and cell location remain in SimulationResult with or without file output. Omitting field_variables preserves the existing primary-field default; an empty tuple requests no derived fields. Unsupported fields are rejected before stepping.

Bounded finite-strain workflow

Set kinematics="finite_strain" on the same element policy. Register one full-domain material with positive density, the existing dP/dF convention, stored energy/dissipation and an explicit initial_array_response(count). The native logarithmic J2 material accepts optional density=; density affects inertia, not the constitutive equations. For example:

material = constitutive.finite_strain_j2_logarithmic(
    young=100.0, poisson=0.3, yield_stress=1.0,
    hardening_modulus=5.0, density=2.0,
)
model.material(material)
policy = elements.uniform_strain_hex8(
    hourglass_modulus=40.0, hourglass_scale=0.1,
    kinematics="finite_strain",
)
step = model.step(
    target=u, element_policy=policy, dt=1e-4, steps=100,
    omega_squared_bound=1e8,
    maximum_negative_growth_per_increment=0.1,
)
result = step.solve_result(field_variables=("S", "P", "F", "SENER", "PEEQ"))

The numerical values above are demonstration choices, not universal stable defaults. omega_squared_bound is a caller-declared complete-path bulk and hourglass ceiling. The current signed tangent is screened at each endpoint; dt="auto" only selects from the declared ceiling, not from an automatic nonlinear wave-speed estimate. Negative curvature is rejected unless its growth-resolution policy is explicitly supplied, as described below.

S is accepted Cauchy stress, P first Piola stress, F the accepted mean deformation gradient, and SENER material stored energy per reference volume. Declared material state output names are also available (FP, PEEQ, PDENER for native J2). All derived fields are unsmoothed DG0 views of accepted state; output never advances the material. Artificial energy stays in its own ledger. As in the small-strain route, total_mechanical_energy excludes artificial hourglass energy; total_discrete_energy includes it. The work-balance quantity accounted_internal_kinetic_energy additionally includes accumulated material dissipation. These three quantities must not be used interchangeably. Strong prescribed motion and displacement-independent reference body/traction loads are supported. Starts are undeformed with virgin history; restarts reuse the accepted state. The admitted interface special case remains isotropic, undamaged and serial. A separate bounded serial route admits one registered frictionless rigid plane, fixed or in prescribed translation (see below). Regional history materials, follower loads, general contact, damage/deletion, mass scaling and cross-partition restart are not admitted.

examples/finite_hex_extension.py demonstrates the installed-use workflow and its same-partition checkpoint. An independent test material (compressible Neo-Hookean with a named peak-energy history) exercises the same entry point, array protocol, fields and restart without changing the core or invoking a per-point Python update. This is protocol integration evidence, not validation of arbitrary external materials. A separate ordinary-Step regression combines finite bulk and unequal-mesh Q1 elastic bonding and verifies interrupted recovery against continuous execution.

The October 10 clean-source wheel (7abe50be) passed 233 selected serial tests and 50 selected tests per rank on two MPI ranks, with pytest source-path injection disabled. The installed example completed 1,000 increments on two ranks and its published result passed artifact-integrity verification. The same MPI selection also passed with an explicitly selected TCP provider; this local runtime comparison does not imply multi-node or cross-partition support. The archive evidence/hex8/2026-10-10-installed-acceptance.json identifies the wheel and test boundaries. These are acceptance checks, not a published package release or a complete industrial validation.

After the input-admission and initialization safeguards, clean numerical source 7ad7a067 was rebuilt and independently installed again. Its 29 focused public-Step, external-material, energy and cyclic/restart tests passed, with source-path injection disabled; strict documentation construction also passed. evidence/hex8/2026-10-10-installed-admission-serial.json records that exact wheel. The subsequent material-description guard passes 15 related serial tests; new rank-local description/input-preparation fault tests are retained in bounded remote MPI release CI. Earlier wheel results are not substituted for that final-revision check.

Small-strain elastic workflow

The model uses a 3D studies.dynamic_solid() Study, continuous Q1 hexahedra, registered constant isotropic or anisotropic elasticity, positive density and ordinary strong constraints. Fixed material orientation and complete disjoint material regions are supported. The engineering-shear order is shared with the existing elasticity module. A prescribed time increment above the conservative bulk-plus-hourglass bound is rejected. dt="auto" uses that bound with safety 0.8; it does not tune an adaptive nonlinear increment.

The bound uses small Gram spectra rather than a coarse trace estimate: with C=L L^T, physical eigenvalues are those of V L^T B M^-1 B^T L (6-by-6); hourglass eigenvalues are obtained from coefficient * gamma^T M_node^-1 gamma (4-by-4). Adding the two largest eigenvalues bounds their combined cell stiffness. Positive assembled mass then makes the maximum cell bound conservative for the body. Optional interface stiffness is added separately. All temporary arrays are chunked.

In regular and perturbed single-cell checks this permits about 2.97x and 2.95x larger stable steps than the former trace bound, with the same safety factor. Twenty independently distorted, positive-definite anisotropic cell checks compare the bound against full 24-by-24 spectra. Stability is not temporal accuracy: wave/load time resolution can still require a smaller user-selected dt; no physical time-step convergence claim follows from increasing the bound.

The result distinguishes strain_energy, kinetic_energy, hourglass_energy, physical total_mechanical_energy and total_discrete_energy. These components feed the existing accepted-path DynamicEnergyLedger: natural-load and strong prescribed-motion work are integrated every accepted increment, independent of history output cadence. The discrete balance explicitly includes artificial hourglass energy, not just material storage. Rigid acceleration and a deforming single-cell oscillator supply independent analytic/discrete references; this is not a blanket accuracy guarantee for arbitrary loads. Serial and distributed same-partition interrupted/continuous runs agree; changed operator identity rejects a checkpoint atomically. Public portable restart, finite deformation, evolving material orientation, damage/deletion, eigenstrain, additional contact operators and operator-changing time inputs are not admitted by this provider.

An optional cohesive_force now composes the fixed-reference, undamaged elastic Q1 interface. Its independent nonmatching traces retain original bilinear interpolation. The conservative stable step includes both interface and bulk stiffness; interface stored energy and failed-increment/restart state share the existing lifecycle. See nonmatching interfaces.

See examples/uniform_hex_wave.py for a runnable ordinary-workflow example. On 2026-10-09 a candidate wheel was built without isolation downloads, installed into a separate environment and used outside the source directory to execute this 100-step example. Field output and the result manifest were written; the original installed package was not replaced. This was a local candidate acceptance, not a public version release.

The subsequent owned-cell MPI candidate (commit f4f6f79f) was also installed outside the source tree. Its two-rank 100-step wave publishes one collective manifest, which passes integrity verification; 13 distributed Hex8 tests pass against the installed wheel with source-path injection disabled. Collective publication uncovered and fixed a duplicate rank-local timing record and an unstable Python-object representation in Step metadata. Timing now uses the existing min/mean/max performance evidence, while residual identity is explicit.

Candidate 8ab39cd3 additionally passes installed ordinary-Step/interface tests (14 selected tests), plus a two-rank 100-step wave with constitutive DG0 fields and a collectively published, integrity-verified manifest. This checks packaging and publication, not independent physical validation. Serial field regressions separately verify tensor shear, material stress jumps and the integral of physical energy density against the accepted energy history.

For imported C3D8R, the migration report continues to say topology_only. It points to this explicitly selected alternative but never silently substitutes it for the source formulation, material updates or section controls.

Formulation

Finite-deformation contribution and staged evidence

elements._finite_uniform_hex.FiniteUniformHexBatch separates total-Lagrangian kinematics from material updates. It computes F = I + sum(u_a outer grad_X N_a) using reference-volume-average gradients, consumes first Piola stress and reference-volume stored-energy density, and maps dP/dF to a matrix-free nodal tangent action. It does not treat Cauchy stress as Piola stress or embed a new material law. This mean-gradient approximation is not a selective volumetric F-bar correction.

The fixed-reference artificial potential uses the existing affine-annihilating mode vectors. Since their contraction with reference coordinates is zero, their displacement contraction rotates with the current geometry; its squared norm is objective. The coefficient remains a declared fixed reference value, not a damage-updated modulus. Current-cell Bernstein Jacobian admission and positive mean deformation determinant are checked separately.

Local tests cover finite affine deformation, large rigid rotations, superposed rotation of already deformed/hourglassed cells, resultant force and current moment, energy derivatives and tangent finite differences. An independent compressible Neo-Hookean oracle checks the conservative case. The existing finite-strain J2 batch protocol separately supplies Cauchy stress converted as P = J sigma F^-T and its declared dP/dF; the resulting element tangent agrees with fixed-old-state force differences. Plastic stored energy is not treated as an incremental potential or a complete dissipation balance.

The private DOLFINx bridge now reuses MaterialQuadratureResponse at one material point per cell, with fixed committed history during trial evaluations. Downstream element/scatter failures discard trial state and restore scratch stress/tangent fields. The caller, not the element, commits accepted material state. A 12-cell global J2 patch checks interior equilibrium, boundary Piola traction, current-configuration moment, global tangent differences, and four loading/unloading increments. Its tiny dense Newton loop is a test oracle, not a second production solver. Shared Q1 layout preparation is also consumed by the existing small-strain operator.

Private serial material checkpoints use an optional typed numeric-array payload instead of expanding integration-point arrays into JSON lists. The v6 manifest binds its payload by size and SHA-256; shape, dtype and finite values are checked before nodal assignment, and material identity is checked within joint rollback. Reading uses allow_pickle=False. The previous v5 JSON auxiliary representation remains readable. Failed payload/manifest publication preserves the previous checkpoint and removes the new generation's unpublished files when possible. The composed Procedure capability explicitly refuses rank-count portability: portable nodal fields alone do not make material or interface history portable. The subsequent v7 encoding adds rank-local arrays for same-partition MPI restart; it does not make these arrays portable across partitions.

The private assembly exposes one open trial scope through material evaluation, force scatter and the Procedure's final checks, avoiding three nested copies of the same scratch fields. Downstream exceptions (including interruption) restore those fields and discard material trial state. Fixed-reference spectral geometry is cached compactly per cell; material-dependent spectra are still recomputed. No stale tangent or unchecked current-cell geometry is reused.

Stored-energy availability is explicit in the shared material batch result: missing optional energy is not a physically defined zero. This route rejects providers without stored energy, and rejects a material-requested increment reduction instead of ignoring it. Empty MPI material partitions preserve their component schema and neutral time-scale summary; owned-cell MPI assembly and empty-rank restoration are checked separately.

An instantaneous spectral screen bounds the declared nodal tangent through small Gram matrices, checked against an explicit 24-by-24 matrix. It requires a symmetric material tangent and rejects unsupported asymmetry. The default rejects negative curvature; an explicit signed growth-resolution option is described below. This is not a general elastoplastic wave-speed policy or a guarantee over a future increment.

The signed report separately encloses positive and negative eigenvalues using the positive/negative parts of the symmetric material tangent. It does not replace that tangent by its absolute value. A regression demonstrates that an isochoric J2 patch can have negative material curvature while its assembled, constrained free-DOF matrix is positive definite. Thus a local curvature finding is not reported as a structural instability verdict.

The private residual lifecycle now runs under the existing central- difference Procedure with joint nodal/material rejection, accepted-time guards, and same-partition checkpoint recovery. Tests inject failure even after material commit and reproduce the uninterrupted trajectory after retry. A serial user interruption likewise restores the accepted nodal, material and interface station before propagating cancellation; this is not coordinated recovery from an arbitrary MPI process failure. Cancellation after auxiliary checkpoint restoration also restores the complete pre-load station before propagating, and a subsequent continuation reproduces an uninterrupted run. A constrained one-cell elastic Hencky bar is checked against an independently integrated ODE, m_eff u'' + C log(1+u)/(1+u) = 0, with extension exceeding 10% and second-order time refinement over three increment sizes. This checks temporal integration of the one-cell model, not spatial continuum convergence or plastic dynamics.

The finite route requires a caller-declared complete-path spectral ceiling and checks each endpoint against the signed symmetric-tangent screen. Ordinary isochoric J2 extension can have negative material curvature; this is not by itself a constitutive bug or proof of global instability. General wave-speed/curvature treatment remains a gate, not an absolute-value workaround. A private composition now admits the already existing elastic nonmatching interface only when all three separation stiffnesses are equal. Its reference-area potential is K * |jump|^2 / 2; traction is parallel to the jump. Checked coplanar common refinement and Q1 trace interpolation are retained. Tests rotate an already opened interface, check current force/moment balance, and exercise joint explicit commit, injected failure and interrupted recovery. The combined stable bound adds the interface contribution; it is not the minimum of the independent bulk/interface time limits. This is a bounded isotropic elastic special case, not a convected anisotropic/damaging interface. Unequal normal/tangential stiffness is explicitly rejected. The restriction is consistent with the frame-indifference/angular-momentum analysis of Ottosen, Ristinmaa and Mosler (2016); their general surface-deformation-gradient extension is not implemented here. Finite-strain public Step lowering and same-partition distributed bulk restart are implemented within the bounded workflow above. Cross-partition restart, general finite-deformation interface kinematics and distributed contact composition remain unimplemented. The bounded serial translating-plane composition is described below. The default small-strain policy is unchanged. The batch stores compact geometry and evaluates forces/tangent actions in bounded chunks without retaining dense 24-by-24 element matrices. The measured columnar-history speedup below applies to its stated workload; newer spectral and kinematics optimizations require their own controlled measurements.

Reference: Sierra/SM Theory Manual, ยง15.1.4 motivates an objective reference-configuration hourglass potential; our existing normalization and coefficient remain explicitly declared, not Sierra defaults.

Public small-strain formulation

Basix supplies Hex8 node order, trilinear interpolation and geometric quadrature. Reference preparation integrates volume and shape gradients once. The material subsequently sees one volume-average engineering strain vector in the existing elasticity order (xx, yy, zz, 2yz, 2xz, 2xy). A symmetric positive-definite six-by-six elasticity tensor maps that vector to physical stress components. The preparation quadrature does not imply multiple constitutive updates.

Four reference hourglass vectors are normalized by sqrt(8). The corrected vectors are gamma = Gamma - grad_mean @ (X_centered.T @ Gamma), annihilating affine displacement. The candidate's explicitly chosen stiffness energy is scale * modulus * V^(1/3) * sum((gamma.T @ u)^2) / 2. This normalization and coefficient are declared choices, not commercial code defaults; bending and wave evidence must constrain their appropriate use. Physical and artificial energies remain distinct. Consistent row-sum masses are integrated rather than assuming distorted cells have equal nodal mass.

Geometry admission expands the triquadratic Jacobian determinant in a tensor Bernstein basis, using 27 Basix evaluations. Positive control coefficients bound the determinant throughout the reference cell. Ambiguous cells undergo adaptive half-subdivision, limited to depth 5 and 4,096 visited boxes per cell. Unresolved cells are rejected explicitly, not mislabeled as inverted. Preparation remains chunked; ordinary affine cells take the vectorized fast path.

The check includes a scale-relative floating-point margin; it is not an interval-arithmetic certificate, a guarantee of good approximation quality, or a test for overlaps between separate cells. A regression fixture passes the old corner/Gauss samples but has a negative determinant elsewhere and is now refused. Distortion-dependent accuracy still requires mesh convergence evidence.

Local evidence

Targeted tests verify regular/distorted affine response, infinitesimal rigid translation/rotation, six physical stiffness modes, twelve controlled hourglass modes, six remaining rigid null modes, energy-gradient/tangent consistency, positive mass, geometric scaling, and independent higher-order integration of mean gradients and mass. General positive-definite orthotropic elasticity is exercised. The local oracle retains a mass-scaled stiffness row-sum bound; compact execution uses the sharper Gram bound described above and composes it into the whole-model time-step policy.

No finite-rotation objectivity, material damage, deletion, viscous/relaxation control or MPI capability follows from these local checks. Distributed assembly and same-partition restart are tested separately through the ordinary Step.

Compact execution and serial global evidence

UniformHexBatch retains mean gradients, four corrected hourglass vectors, positive nodal masses and coefficients. Shared material stiffness is stored once; heterogeneous tensors are supported. Geometry preparation and response evaluation are chunked. No per-cell 24-by-24 matrices are retained. Response iteration can consume chunks without storing all output fields. Global residual evaluation also gathers displacement through the node map one chunk at a time; it no longer creates a full (cells, 8, 3) displacement copy each step.

A 2,048-cell, single-thread local microbenchmark on 2026-10-09 compared five response repetitions against the scalar reference operator. Median response times were 0.00192 s (batch) and 0.01785 s (scalar), approximately 9.3x. Compact stored arrays occupied 1,081,632 bytes versus 18,874,368 bytes for two dense cell matrices alone. This is not a complete solver speedup or peak-memory measurement; global DOFs, connectivity and temporary gathers add storage.

A separate 20,000-cell gather comparison (three repetitions, same thread limit) measured traced transient allocations of 5,155,736 bytes for the full displacement gather versus 1,512,088 bytes for chunk gathering. Median response time changed from 0.02053 s to 0.02119 s; energies were identical. This optimization reduces temporary allocation, not measured CPU time, and does not represent process RSS.

Energy monitoring now uses an energy-only evaluation of the same cell kinematics; it does not recompute unused nodal forces or skip accepted increments. Residual assembly reuses bounded, chunk-local reduction maps, with PETSc still responsible for ghost accumulation. An alternating five-repeat comparison on 8,192 cells and 50 steps measured median run time 0.85270 s before these two changes and 0.62880 s after (1.36x throughput, about 26% less run time). Physical energy and kinetic energy agree to roundoff. This is one local no-field-I/O elastic workload, not a claim about nonlinear material, contact or industrial models.

tools/benchmark_uniform_hex.py measures ordinary-Step preparation, run time, peak process RSS, and the existing collective performance evidence. A bounded 262,144-cell/20-step single-thread run (energy-only path before scatter reduction) took 7.04 s preparation and 8.75 s advancement, with 1,258,487,808 bytes peak RSS. RSS includes mesh/runtime/preparation storage and is not the compact kernel size. The automatically selected step is stable for this declared linear operator, not an accuracy criterion. No field I/O, contact or material history is included.

The private UniformHexResidual uses DOLFINx blocked Q1 DOF maps and the existing central-difference integrator. Serial affine internal forces agree with independent fully integrated UFL assembly. Separate resultant and moment checks pass. An analytic longitudinal standing wave converges under mesh refinement, with a bounded energy error. The conservative time-step bound uses positive cell mass and stiffness Rayleigh bounds; additional interface/contact stiffness is not included and must be composed separately.

The operator has two-/four-rank owned-cell assembly evidence: reverse ghost accumulation for mass and internal force, global physical and artificial energy, a global maximum stability bound, empty local partitions, and collective rejection of rank-local invalid material/non-finite fields. Distributed nodal wave response agrees with the serial formulation. Ordinary model.step also passes distributed natural-load and prescribed-motion work, field output, and same-partition interrupted/continuous runs. A two-rank test includes an empty owned-cell partition. Rank-local time-input/kinematic failures are delivered before field communication and accepted state is restored. These tests do not enable nonmatching interface MPI or cross-partition restart.

A slender cantilever (length 10, unit square section, E=100, nu=0, uniformly distributed end shear) was checked against Euler-Bernoulli deflection with the rectangular-section Timoshenko shear correction. This is a beam-theory reference, not an exact three-dimensional solution or a commercial-code run.

Hourglass scale 20x2x4 relative tip error 40x4x8 relative tip error Fine-mesh artificial energy / total
0.05 4.883% 1.177% 0.375%
0.10 3.275% 0.799% 0.747%
0.20 0.205% 0.051% 1.484%

All three choices improve under refinement. This does not select a universal default coefficient or establish accuracy for a single element through the thickness. Tests are in test_uniform_hex.py and test_uniform_hex_global.py.

With deterministic interior-node perturbations up to 20% of each grid spacing (seed 1729; the box and boundary planes retained), corresponding fine-mesh tip errors are 0.995%, 0.532%, and 0.341%; artificial-energy fractions are 0.475%, 0.897%, and 1.707%. Every error decreases from the coarse mesh. This is bounded distortion evidence, not admission of arbitrary poor-quality meshes.

Nonuniform finite-deformation spatial check

tools/verify_finite_hex_manufactured.py uses an independent compressible Neo-Hookean test provider (mu=30, lambda=40) and a manufactured displacement 0.08*sin(pi*x)*sin(pi*y)*sin(pi*z)*(1, 0.4, -0.2) on the unit cube. UFL differentiates the exact first-Piola field to generate the body force; degree-ten integration assembles that load and the displacement error. All boundary displacements are zero. A test-only Newton/CG oracle solves the private contribution; it is not a new public nonlinear Procedure.

Cells per edge Relative displacement L2 error Hourglass / physical energy
2 34.46% 14.92%
4 5.868% 2.019%
8 1.340% 0.447%
12 0.586% 0.194%
16 0.327% 0.109%

The refined displacement error approaches second-order convergence, while artificial energy decreases. Equilibrium residuals are below 1e-9 relative. The coarse mesh is visibly inadequate: patch-test success must not be confused with accuracy for nonuniform deformation. This is regular-mesh hyperelastic verification, not plastic localization, industrial forming, or Abaqus agreement.

With deterministic interior-node perturbations of up to 20% of grid spacing (seed 1729; boundary retained), errors on 4/8/12/16 cells per edge are 6.483% / 1.596% / 0.732% / 0.417%; corresponding artificial/physical energy ratios are 2.052% / 0.488% / 0.221% / 0.126%. Full-cell geometric admission is still required. This bounded perturbation test does not admit arbitrary distortion. Raw regular/distorted records, including clean source revision 341060e3, are retained under evidence/hex8/2026-10-09-finite-spatial-*.json. Their wall times overlap a separate endurance run and are not used as performance evidence.

Remaining implementation gates

Bounded serial capacity evidence

The private affine-dilation oracle completed 262,144 Hex8 cells and 500 accepted increments at clean source aca01be0. Peak process RSS was 3,074,506,752 bytes (about 3.07 GB); final absolute displacement/stress/stored-energy errors were 6.38e-16, 1.50e-12 and 1.13e-12 against the independent homogeneous solution. The native finite-strain J2 provider remained elastic. The record is evidence/hex8/2026-10-09-finite-affine-capacity.json.

This establishes bounded serial capacity and repeated state advancement, not plastic dynamics, spatial convergence, a complete energy balance or industrial forming. The run predates columnar transport; its elapsed time overlapped other checks and is not used to quantify that optimization. Its explicit 8 GiB budget was an observed-RSS stop at reporting stations, not an operating-system limit.

The subsequent clean 5bb568a3 plastic-path capacity run completed 262,144 cells and 1,000 increments through 20% axial extension. Its independent proportional logarithmic-J2 oracle checks stress, stored energy, PEEQ and irreversible dissipation, not only displacement. The final absolute stress and PEEQ errors are 1.03e-11 and 1.01e-13; the maximum sampled relative work/energy residual is 1.92e-7. Peak process RSS is 3,383,967,744 bytes (3.15 GiB). The 200 reporting stations are preserved in evidence/hex8/2026-10-10-finite-plastic-capacity.json.

This run used the private verification entry point and predates the subsequent geometry/tangent optimizations. Its approximately 114-minute run overlapped short acceptance checks and is not a controlled performance baseline. The ordinary Step is tested separately. A homogeneous capacity test still does not establish nonuniform accuracy, contact forming, or a general stable-step estimator; it retains a caller-declared path ceiling and explicit signed-curvature growth-resolution policy.

The optimized ordinary model.step() entry point then completed the same 262,144-cell, 1,000-increment oracle at clean source 7ad7a067. Preparation took 8.66 s; the monitored solve/oracle loop took 3,864.74 s (64.4 minutes), with peak RSS 3,435,347,968 bytes (3.20 GiB). Final stress/PEEQ errors and maximum sampled relative energy residual retain the values above. Raw evidence, including all 200 stations and entrypoint="ordinary_model_step", is in evidence/hex8/2026-10-10-finite-public-plastic-capacity.json. These capacity timings overlap brief acceptance checks and must not be used to compute an old/new speedup. The separate paired 4,096-cell plastic measurement in performance evidence provides that comparison. The large run does not measure field export, durable checkpoint throughput, distributed scaling or the tester's industrial geometry.

Public and private promotion boundaries

Compact preparation and serial affine/bending/wave checks are implemented. DOLFINx retains mesh/DOF ownership; Procedure retains time integration and stability composition. The experimental policy, ordinary Step, same-partition restart and separate artificial energy are implemented. Bounded distortion is tested; remaining work includes more severe and application-specific mesh-quality accuracy limits, richer loading evidence and distributed nonmatching interface composition. No implicit numerical equivalence to imported commercial reduced-integration elements is assumed.

Accepted finite-explicit energy and signed-curvature policy

The finite route now reuses DynamicEnergyLedger with cached accepted force and material energy. Initial energy is declared explicitly by the material (the native logarithmic J2 law supplies a virgin response); no artificial time increment is used to obtain an initial stress. The same declaration populates the initial quadrature response fields. Sampling and restart do not reintegrate the material. Bulk stored energy, artificial hourglass energy, isotropic reference-interface energy, cumulative material dissipation and kinetic energy remain separate. Prescribed-motion work uses the existing reaction-path ledger. Accepted values participate in the same atomic rollback/checkpoint transaction.

Independent proportional finite-stretch checks cover elastic and plastically yielding logarithmic J2 response, stored energy and yield-stress-times-PEEQ dissipation. Time refinement reduces the work/energy residual. The interface composition has interrupted/continuous restart and cached-energy checks. These tests do not establish arbitrary plastic loading, localization or forming.

The ordinary Step also has two complete tension/compression cycles with an independent scalar coaxial logarithmic-plasticity recurrence. It keeps signed plastic strain distinct from accumulated PEEQ, checks stress/storage/dissipation, refines the accepted-path work residual, and restores exactly after a reversal. This is rate-independent proportional-direction cyclic evidence, not a nonproportional or fatigue constitutive validation.

Independent nonuniform plastic manufactured runs are archived in evidence/hex8/2026-10-10-finite-plastic-manufactured.json and its -fine-time.json companion. Five axial meshes (8 to 128 cells, two cells in each transverse direction) use analytically derived forcing with a lumped nodal load rule. At 128 axial cells and 8,000 increments, relative displacement, cell stress and PEEQ errors are approximately 0.00459%, 0.119% and 0.131%. Reducing the increment fourfold barely changes the spatial errors. The moving yield front prevents claiming a uniform second-order stress convergence rate. These clean-revision records retain their original field names: in those older records total_mechanical_energy included the separately reported artificial energy, which is below 1e-27 here. Current output separates mechanical and discrete totals explicitly; the raw historical evidence is not rewritten.

The default still rejects negative material curvature. An explicit option may instead require dt * sqrt(negative_bound) <= eta, with 0 < eta <= 0.25, while retaining the positive-frequency ceiling separately. For a frozen scalar negative mode, central difference has growth rate 2*asinh(omega*dt/2)/dt, rather than the exact omega; this motivates a growth resolution limit, not a physical stability certificate. Negative material curvature is not sufficient to diagnose a constrained global instability. No negative eigenvalues are silently replaced by their absolute values in the physical residual or tangent. This opt-in and the signed spectrum are included in restart identity/evidence. Nonsymmetric tangents remain rejected.

Remaining general finite-explicit gate

The finite bulk assembly now retains owned-cell force/energy contributions and uses DOLFINx quadrature maps to exchange deformation gradients for ghost points. Nonuniform finite plastic trajectories agree with serial execution on the tested partitions; an empty owned-cell partition is valid. Rank-local geometry, force, spectrum and accepted-energy failures reject collectively. Explicit Procedure now synchronizes a completed material commit before any rank enters subsequent monitoring, including an injected failure after one rank committed. The shared transient schema v7 stores numeric auxiliary state per rank, while v6 remains readable for serial arrays and v5 for JSON state. Finite same-partition MPI restart now has interrupted/continuous, empty-rank, corrupt-payload, swapped-partition and failed-publication checks. A rank's material/energy payload is bound to its local identity in addition to the collective identity. Cross-partition recovery and distributed nonmatching interfaces remain separate gates. The material driver may update visible ghost points, but these never contribute a second time to force, energy or negative-curvature cell counts.

The signed screen distinguishes the highest oscillatory frequency, negative curvature and the declared complete-path ceiling. It never takes absolute eigenvalues as a replacement constitutive tangent. Automatic material-owned wave-speed/effective-modulus admission remains a separate gate.

The practical design sequence is: material-owned effective-modulus evidence; Operator-owned geometric/mass conversion and additive interface contribution; Procedure-owned increment selection and rejection; State-owned rollback. Initial/accepted response sampling must not manufacture a positive time increment to query a rate-dependent law. Energy output must consume an explicit accepted response, including its initial energy, before claiming a global balance.

Abaqus documents separate user-supplied effective bulk/shear moduli when its automatic estimate is not conservative for a highly nonlinear user material. This supports a distinct stability contract rather than assuming that every algorithmic tangent is a wave-speed model. Its explicit analysis guide also distinguishes element-wise and global frequency estimates. These are design references, not claims that AgentFEM implements those algorithms: VUMAT effective moduli, explicit stability estimation.

Material-owned stability envelopes (experimental)

The fixed-reference Hex8 operator can now convert declared symmetric material bounds -c_minus I <= dP/dF <= c_plus I into separate positive-frequency and negative-curvature bounds. It uses the reference-volume-scaled 3x3 mass-gradient Gram, adds positive hourglass stiffness only to the upper bound, and retains the maximum element bound for assembled positive lumped mass. Dense single-element and shared-node assembly oracles check this conversion, including anisotropic signed tangents and a bounded Neo-Hookean deformation domain. It does not infer material bounds or certify their validity domain.

An external material may implement explicit_stability_envelope() returning constitutive.FirstPiolaTangentEnvelope(positive_modulus=..., negative_modulus=..., minimum_stretch=..., maximum_stretch=..., source=...). Both moduli are in the model's consistent stress units and bound symmetric dP_iJ/dF_kL in reference configuration. The provider owns the derivation over all admissible internal states and inputs, throughout the principal-stretch domain. State-restricted or nonsymmetric tangents cannot be silently represented by this contract.

The ordinary finite Hex8 model.step() may then omit omega_squared_bound; dt="auto" selects a fixed initial increment from the domain bound plus hourglass/interface contributions. A numerical omega_squared_bound explicitly selects the existing caller-ceiling route instead. Providers without a reviewed envelope, including the present native finite J2 provider, still need that ceiling.

Initial and trial endpoint responses are checked for domain and signed-tangent violations; failure uses existing full rollback. Envelope identity enters restart compatibility, including its domain, moduli and source. Results identify provider-declared evidence, not automatic scientific verification. An independent Neo-Hookean test provider exercises equivalence with the explicit-ceiling route, restart, changed-identity rejection and domain-failure rollback. New distributed envelope tests await their separate MPI acceptance; existing caller-ceiling MPI evidence does not establish the new route.

Candidate eb75f6c5 passes 37 isolated installed-wheel checks with source-path injection disabled; all 299 packaged Python modules match that clean source. The 108 source-level focused checks overlap that selection and are not additive. The exact wheel hash and scope are in 2026-10-10-material-envelope-installed.json.

No variable time integration, skipped tangent, nonlinear stability guarantee between endpoints or general finite-material wave-speed claim follows from this route. A sampled initial modulus is not a whole-domain envelope.

Contact composition groundwork

The existing contact residual now delegates numeric checkpoint and rollback snapshots to its nested material provider, retaining rank-local arrays instead of expanding/gathering them as JSON. Two nested contact wrappers around a real finite Hex8 material pass a shared binary-archive round trip; malformed outer identity is rejected before material mutation. The contact layer inherits the material's same-partition restriction rather than claiming portable restart. An additional private serial composition now advances a moving plane against a bounded Neo-Hookean body with the summed material/contact spectral ceiling. It uses the existing Procedure and energy ledger, not a special forming solver. At time increments 2e-4, 1e-4, 5e-5, final relative balance errors are 5.005e-6, 1.251e-6, 3.128e-7; displacement differences also decrease by approximately four on halving the increment. The test restores mid-path state exactly and rolls back an injected failure after both material/contact commits. Accepted reaction sampling reuses the material's cached force rather than performing a zero-increment constitutive update.

Ordinary translating-plane composition (candidate)

Register one RigidContactPair with model.add_boundary_model(pair) and use the same finite-Hex model.step(...). The lowering keeps the constitutive residual as the owner of fields and material energy while the composed residual owns total force and contact State. It adds the contact spectral bound once to the bulk/hourglass/interface ceiling before selecting a fixed increment; a caller ceiling or a reviewed material envelope is still required for the bulk.

This admission supports one frictionless plane, fixed or in prescribed translation, in serial or distributed execution, with an optional elastic bond. Rotation, curved tools, multiple pairs and friction reject before stepping. Bonding on a separate trace is bounded as described below; no bond-failure/contact switching is admitted. The penalty potential uses the reference-surface quadrature weights; it must not be interpreted as an independently validated current-area pressure formulation.

An independent one-element uniaxial-strain oracle solves (rho*V/2)*q'' = -A*P11(1+q) - k*A*positive(q-s) with a refined scalar RK4 reference. For compressible Neo-Hookean material, P11(a) = mu*(a-1/a) + lambda*log(a)/a. Displacement, velocity and integrated tool work converge at approximately second order under time refinement. The oracle never reads the FEM tangent or residual. This checks the lumped one-element model, not spatial accuracy or industrial forming.

examples/finite_hex_moving_plane.py uses ordinary registered assets and native finite J2. Its 1,000-increment source run reports positive material dissipation and tool work, with final relative work/energy residual about 2.03e-7. Interrupted/continuous material State and nodal displacement match exactly. These are small serial acceptance gates; general tool geometry and the tester's actual finite orthotropic material remain separate requirements.

The focused two-rank gate tests/test_parallel_finite_hex_contact.py compares the energy/work history with serial execution and checks exact same-partition restart, an intentionally empty rank, collective rejection of inconsistent contact descriptions or corrupt outer records, and rollback after a rank-local post-commit failure. Nested material State remains rank-local; contact work records remain canonical. This does not establish repartitioned finite-material restart. Distributed nonmatching bonding has its separate acceptance below.

Clean candidate afef243a passes 29 isolated installed-wheel checks with source injection disabled; all 299 packaged Python modules match source. The installed example also completes and its output passes artifact-integrity verification. The wider 142-test source selection overlaps these checks, not an additional independent sample. Exact wheel identity and limits are recorded in 2026-10-10-finite-contact-installed.json.

Bond plus external tool contact

The same ordinary finite-Hex Step admits the reviewed equal-stiffness elastic reference bond together with one external translating frictionless plane. The contact boundary and either bond trace must have disjoint nodal support; shared edges are conservatively refused as well. This intentionally excludes contact on the bonded interface and any automatic transition after failure.

The finite-Hex composition also passes two-rank elastic/J2 bulk comparison, ordinary result output, stable-step equality, exact same-partition restart, corrupt interface-state rejection and post-commit rollback. Reference geometry is replicated; each integration point is owned once and force exchange is sparse. See tests/test_parallel_nonmatching.py. Cross-partition recovery, distributed implicit interfaces and damage remain outside this admission.

Bulk/hourglass, interface and contact contributions enter the common spectral sum once. The existing material State owns the bond transaction; the contact wrapper adds tool state without replacing that owner. The energy ledger reports bulk storage, bond storage, artificial energy, contact potential, material dissipation and tool path work separately. No additional integrator is introduced.

tests/test_finite_hex_bond_contact.py checks a coarse/fine two-block specimen against an independently integrated four-mass axial Neo-Hookean system, with three time increments and second-order convergence. It also checks plastic loading, complete post-commit rollback, exact interrupted restart and changed bond-identity rejection. The lumped reference is not a continuum mesh-convergence or Abaqus comparison. examples/finite_hex_bond_contact.py supplies a Gmsh/native J2 demonstration alongside its geometry helper nonmatching_hex_bond.py. The 35-cell, 1,000-increment Gmsh demonstration reports tool work 0.00678778226, bond energy 4.14790767e-8, material dissipation 6.26068213e-5 and final relative energy residual 1.85668401e-7 in its declared consistent model units.

An independent external test provider additionally uses a restricted orthotropic Green-strain energy, with fixed reference material axes. Analytic dP/dF and energy derivatives are checked by finite differences, then by superposed rigid rotation. The same provider runs and restarts the combined Step without any material-name branch in core. See tests/test_finite_hex_orthotropic_provider.py. This test law is not shipped as the tester's material or claimed reliable in arbitrary large compression. FEBio's orthotropic hyperelastic discussion motivates the energy-based reference, not a claim of a FEBio cross-validation.

The design follows explicit penalty-contact practice in keeping additional contact stiffness in the time-step restriction, and fixed initial bonding distinct from evolving contact; see Abaqus contact enforcement and Sierra interface interactions. These are algorithm-design references, not equivalence claims.

Accepted contact-history performance

The accepted tool-work ledger caches its incremental sum and rolling history fingerprint. In-memory rollback shares immutable accepted stations rather than re-encoding the whole work history; durable checkpoints still retain the complete history. Restores recompute and validate the ledger before committing it.

On the measured 12-cell, 1,000-increment serial workload, three alternating runs reduced median elapsed time from 4.342 s to 1.282 s (3.39x, 70.5% less time). Displacements, velocities, material state and complete recorded histories are identical. This is a history-overhead-dominated workload, not a general solver or industrial-model speedup. The benchmark reinstates the previous full-history integration/serialization path without changing physics or tolerances. See tools/benchmark_finite_contact_history.py and the versioned evidence/hex8/2026-10-10-contact-history-performance.json record.

Sources

  • Johnen, Weill and Remacle, 2017, Robust and efficient validation of the linear hexahedral element, https://arxiv.org/abs/1706.01613. Admission uses the Bernstein convex-hull and subdivision construction, not the paper's optimized 20-tetrahedron implementation or its performance claim.
  • Flanagan and Belytschko, 1981, A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, https://doi.org/10.1002/nme.1620170504.
  • Sandia, Sierra/SM Theory Manual, section 15.1, uniform-gradient Hex8 and orthogonal hourglass control: https://www.sandia.gov/files/sierra/SM_Theory_5_20/main/element_formulations.html. The candidate uses the affine-removal construction but explicitly documents its own stiffness normalization rather than claiming full Sierra equivalence.
  • Sandia, Sierra/SM Theory Manual, Dynamics, element eigenvalue bounds: https://www.sandia.gov/files/sierra/SM_Theory_5_30/main/dynamics.html. The small-Gram calculation above applies this bound to the declared fixed linear operator; it does not implement a global Lanczos estimator.
  • PETSc, VecGhostUpdateBegin, reverse-add accumulation of ghost contributions: https://petsc.org/main/manualpages/Vec/VecGhostUpdateBegin/.
  • Abaqus theory, Energy balance, separating work, kinetic energy and physical internal-energy channels: https://docs.software.vt.edu/abaqusv2025/English/SIMACAETHERefMap/simathe-c-energybalance.htm. AgentFEM's present discrete ledger and its tested material decomposition are bounded independently; this reference does not imply Abaqus equivalence.
  • Abaqus verification documentation, Performance of continuum and shell elements for linear analysis of bending problems: https://docs.software.vt.edu/abaqusv2025/English/SIMACAEBMKRefMap/simabmk-c-linbending.htm. Motivation for thickness refinement, hourglass-energy inspection and not equating patch-test success with reliable bending performance.