forked from 1b-agent/fieldcraft-upstream
verification: hyperelastic and thermal oracle cases and documentation
Five cases against closed forms: a Mooney-Rivlin block in Treloar's uniaxial, equibiaxial and pure-shear modes and a Yeoh block in uniaxial tension (the compressible energy's stresses, exact); a stretched Mooney-Rivlin tube inflated by a follower pressure (Rivlin's incompressible solution, 0.6 % at nu = 0.4995); free and fully held thermal expansion at a thermal stretch of 1.2 (exact); a bar held at its ends heated past yield (exact). docs/fem.md, the literature, ROADMAP (M15, M17), parity and the oracle page follow. Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
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@@ -41,9 +41,9 @@ pressure) is a degree of freedom of the existing DOF framework, not a separate s
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| M12 CFD: lattice Boltzmann | D3Q19/D3Q27 on grids voxelised from the face-tagged surface; TRT, regularised and cumulant collision; interpolated walls, inlets and outlets; LES (Smagorinsky, WALE) with wall models; thermal LBM and conjugate heat transfer with the structure's thermal solver; non-Newtonian viscosity (power-law, Carreau–Yasuda, Cross, Herschel–Bulkley, Bingham) and temperature-dependent viscosity; porous media (Brinkman, Darcy–Forchheimer); multiphase and free surface (M13); validation by the lid-driven cavity, backward-facing step, cylinder wake (Strouhal) and turbulent channel; octree refinement shared with the hex mesher; wgpu backend (deterministic per device, checked against the CPU) (CPU core: done — `fieldcraft-lbm` with D3Q19 TRT and Guo forcing, halfway bounce-back and moving walls, slip and periodic faces, Zou–He velocity and pressure faces, exact voxelisation of obstacle bodies, momentum-exchange forces per face; fluid materials, fluid domains and fluid steps in the model; `model.fluid_domain.*`, `solve.fluid`; verified by Poiseuille channel and pipe, Couette, Taylor–Green decay, sphere-array Stokes drag and mass conservation; Schäfer–Turek 2D-1 as a benchmark) | started |
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| M13 Fluid–structure interaction | Immersed-boundary coupling of the lattice Boltzmann fluid with the explicit or implicit structure; interface force transfer; sub-cycling; Aitken then interface quasi-Newton coupling; free-surface flow | planned |
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| M14 Blast and impact fluids | EOS-driven fluids in the explicit solver: ALE multi-material or SPH, with fluid–structure contact | planned |
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| M15 Nonlinear implicit core | Newton–Raphson with line search, automatic increments and cutbacks, convergence on force, displacement and energy; geometric nonlinearity (total and updated Lagrangian, large rotations for shells and beams, follower pressure); every `fem` material behind one interface with its consistent tangent (plasticity, damage and thermal strain from the explicit solver reused implicitly); arc-length (Riks) for snap-through; linear buckling (eigen) with preload stress stiffening, and imperfections from buckling modes; nonlinear stabilisation; restart. Verified by the elastica, Lee's frame, the snap-through arch, the hinged cylindrical shell and plastic collapse loads (solids core: done — `nonlinear_static` steps with automatic increments and cutbacks, Newton–Raphson with the consistent tangent, force, displacement and energy convergence and an optional line search; a material point interface (Kirchhoff stress and spatial tangent from F, with history) with St Venant–Kirchhoff and compressible neo-Hookean; total Lagrangian tet4, tet10, wedge6, hex8, hex20 (full and reduced), hybrid hex8/hex20/tet10 by the three-field mixed form, plane strain and axisymmetric elements; follower pressure with its symmetrised load stiffness; Cauchy stress and logarithmic strain, increment history and frames, `solve.nonlinear`; Abaqus and LS-DYNA decks; verified by the patch and objectivity tests, uniaxial stretch, the elastica and tube inflation by a follower pressure (finite-strain Cook's membrane pending its transcribed reference; J2 plasticity on the interface: Simo's exponential return map at finite strain and the radial return at small strain, both with consistent tangents and the return shared with explicit dynamics, bilinear isotropic and kinematic (small strain), tabulated and Johnson–Cook (rate-independent) hardening, hybrid elements included, load paths for cyclic loading, a tangent predictor for path-dependent materials, verified by uniaxial tension to large strain, Hill's thick cylinder (plastic zone and collapse pressure), a cantilever's collapse and the Bauschinger effect, with quadratic Newton convergence. Next: Mooney–Rivlin and Yeoh, thermal strain, incompatible modes (EAS), shells and beams, arc-length) | started |
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| M15 Nonlinear implicit core | Newton–Raphson with line search, automatic increments and cutbacks, convergence on force, displacement and energy; geometric nonlinearity (total and updated Lagrangian, large rotations for shells and beams, follower pressure); every `fem` material behind one interface with its consistent tangent (plasticity, damage and thermal strain from the explicit solver reused implicitly); arc-length (Riks) for snap-through; linear buckling (eigen) with preload stress stiffening, and imperfections from buckling modes; nonlinear stabilisation; restart. Verified by the elastica, Lee's frame, the snap-through arch, the hinged cylindrical shell and plastic collapse loads (solids core: done — `nonlinear_static` steps with automatic increments and cutbacks, Newton–Raphson with the consistent tangent, force, displacement and energy convergence and an optional line search; a material point interface (Kirchhoff stress and spatial tangent from F, with history) with St Venant–Kirchhoff and compressible neo-Hookean; total Lagrangian tet4, tet10, wedge6, hex8, hex20 (full and reduced), hybrid hex8/hex20/tet10 by the three-field mixed form, plane strain and axisymmetric elements; follower pressure with its symmetrised load stiffness; Cauchy stress and logarithmic strain, increment history and frames, `solve.nonlinear`; Abaqus and LS-DYNA decks; verified by the patch and objectivity tests, uniaxial stretch, the elastica and tube inflation by a follower pressure (finite-strain Cook's membrane pending its transcribed reference; J2 plasticity on the interface: Simo's exponential return map at finite strain and the radial return at small strain, both with consistent tangents and the return shared with explicit dynamics, bilinear isotropic and kinematic (small strain), tabulated and Johnson–Cook (rate-independent) hardening, hybrid elements included, load paths for cyclic loading, a tangent predictor for path-dependent materials, verified by uniaxial tension to large strain, Hill's thick cylinder (plastic zone and collapse pressure), a cantilever's collapse and the Bauschinger effect, with quadratic Newton convergence; Mooney–Rivlin and Yeoh options (decoupled, hybrid elements), verified in Treloar's three modes and by a stretched tube's inflation; thermal strain by the multiplicative split at finite strain and the additive one at small strain, with temperatures from temperature loads or the step's own conduction, verified by free and constrained expansion and a heated bar held past yield. Next: incompatible modes (EAS), kinematic hardening at finite strain, shells and beams, linear buckling, arc-length with an indefinite factorization) | started |
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| M16 Structural dynamics | Implicit transient (Newmark, HHT-α, generalised-α); modal superposition transient; steady-state harmonic (direct and modal), with Rayleigh, modal, structural and material damping; random vibration from PSDs and response spectrum (SRSS, CQC); complex eigenvalues with damping; rotating machinery (Campbell diagrams, gyroscopic terms); substructures (Craig–Bampton) | planned |
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| M17 Constitutive library and material library | Elastic: transversely isotropic, fully anisotropic (21 constants), temperature-dependent. Elastomers: hyperelastic neo-Hookean, Mooney–Rivlin, Yeoh, Ogden, Arruda–Boyce, Gent, Marlow, with Mullins softening and finite-strain viscoelasticity (Prony series; parallel networks after Bergström–Boyce), foams. Viscoelasticity: small- and finite-strain Prony series, time–temperature superposition (WLF, Arrhenius), storage and loss moduli for harmonic steps. Viscoplasticity and creep: Norton power law, time and strain hardening, Perzyna, Anand, Chaboche combined hardening. Plasticity: Hill anisotropic, Drucker–Prager, Mohr–Coulomb, Cam-clay, crushable foam, concrete damaged plasticity, Gurson. Fluids: Newtonian and non-Newtonian viscosity shared with M12. Material library: curated, cited data sets (steels, aluminium and titanium alloys, polymers, elastomers, composites, tissues) with units and temperature dependence; calibration from test data (uniaxial, biaxial, shear, relaxation, DMA) by least squares with Drucker stability checks; user materials as a safe Rust trait | planned |
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| M17 Constitutive library and material library | Elastic: transversely isotropic, fully anisotropic (21 constants), temperature-dependent. Elastomers: hyperelastic neo-Hookean, Mooney–Rivlin, Yeoh (these three done, in nonlinear steps), Ogden, Arruda–Boyce, Gent, Marlow, with Mullins softening and finite-strain viscoelasticity (Prony series; parallel networks after Bergström–Boyce), foams. Viscoelasticity: small- and finite-strain Prony series, time–temperature superposition (WLF, Arrhenius), storage and loss moduli for harmonic steps. Viscoplasticity and creep: Norton power law, time and strain hardening, Perzyna, Anand, Chaboche combined hardening. Plasticity: Hill anisotropic, Drucker–Prager, Mohr–Coulomb, Cam-clay, crushable foam, concrete damaged plasticity, Gurson. Fluids: Newtonian and non-Newtonian viscosity shared with M12. Material library: curated, cited data sets (steels, aluminium and titanium alloys, polymers, elastomers, composites, tissues) with units and temperature dependence; calibration from test data (uniaxial, biaxial, shear, relaxation, DMA) by least squares with Drucker stability checks; user materials as a safe Rust trait | planned |
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| M18 Composites and anisotropy | Material orientations (rectangular, cylindrical, spherical, from geometry edges and faces, per-element fields, draping of fibres over curved parts); layups on shells and continuum shells, ply stacks on swept hex; classical lamination theory checks; ply stresses and strains through the thickness; failure indices (Tsai–Wu, Tsai–Hill, Hashin, Puck, LaRC) and progressive damage; homogenisation (rule of mixtures, Mori–Tanaka, RVEs with periodic boundary conditions); delamination through M20's cohesive zones. Verified by laminate closed forms and the NAFEMS composite benchmarks | planned |
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| M19 Coupled fields | On the DOF framework, each a degree of freedom beside the displacements: piezoelectricity (electric potential, d/e coupling and permittivity matrices, voltage and charge conditions, resonance and impedance through M16 harmonic steps); poroelasticity (Biot, u–p, Darcy flow, consolidation; Terzaghi and Mandel verification; soils and biphasic tissue); electro-thermal (Joule heating) and thermo-electro-mechanical coupling; acoustics (Helmholtz elements, structure–acoustic coupling, absorbing boundaries); low-frequency electromagnetics later | planned |
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| M20 Damage, fracture, fatigue and adhesion | Continuum damage beyond Johnson–Cook (Lemaitre, ductile and shear criteria) for implicit and explicit; fracture mechanics (J-integral and stress intensity factors, crack-front meshes from the hex mesher; XFEM later); fatigue (S–N and ε–N with mean-stress corrections, rainflow counting, Paris crack growth); adhesion and debonding by cohesive zones (bilinear and exponential traction–separation, mixed mode with Benzeggagh–Kenane) as cohesive elements and cohesive contact; adhesive joints. Verified by DCB, ENF and MMB delamination tests and published fatigue data | planned |
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@@ -78,7 +78,7 @@ What exists and what is missing, by area; each gap is in a milestone above.
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| Solution procedures | Linear static, nonlinear static (solids with geometric and material nonlinearity: hyperelasticity, J2 plasticity), frequency (Lanczos), steady and transient heat, coupled temperature–displacement, explicit dynamics, LBM flow | Nonlinear shells and beams, arc-length, buckling (M15); implicit, harmonic, random and spectrum dynamics (M16) |
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| Contact, friction, adhesion | Explicit planar rigid walls (frictionless); couplings, springs, dashpots | Implicit and explicit deformable contact, tie, friction laws, thermal contact, wear (M7); cohesive adhesion and debonding (M20) |
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| Materials | Isotropic and orthotropic elasticity, density, thermal, J2 plasticity (bilinear, table, Johnson–Cook), Johnson–Cook damage and erosion, compressible neo-Hookean hyperelasticity, Newtonian fluids | Elastomers beyond neo-Hookean (Mooney–Rivlin and the rest), viscoelasticity, viscoplasticity and creep, anisotropic and pressure-dependent plasticity, non-Newtonian viscosity, a cited material library and calibration (M17) |
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| Materials | Isotropic and orthotropic elasticity, density, thermal, J2 plasticity (bilinear, table, Johnson–Cook), Johnson–Cook damage and erosion, compressible neo-Hookean, Mooney–Rivlin and Yeoh hyperelasticity, Newtonian fluids | Elastomers beyond Yeoh (Ogden, Arruda–Boyce and the rest), viscoelasticity, viscoplasticity and creep, anisotropic and pressure-dependent plasticity, non-Newtonian viscosity, a cited material library and calibration (M17) |
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| Composites and anisotropy | Orthotropic constants | Orientations, layups, ply results, failure criteria, homogenisation (M18) |
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| Coupled fields | Thermal–structural (one way and coupled) | Piezoelectric, poroelastic, electro-thermal, acoustics (M19) |
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| Failure | Johnson–Cook damage and element erosion (explicit) | Fracture mechanics, fatigue, cohesive zones, composite progressive damage (M18, M20) |
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@@ -171,6 +171,7 @@ converges, so a cut-back increment leaves no trace. The materials:
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| Elastic (isotropic or orthotropic), no option | St Venant–Kirchhoff: S = D E, E = ½(FᵀF − I). Right for large rotations with small strains; it softens without bound in large compression. |
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| Isotropic elastic with `"hyperelastic": {"type": "neo_hookean"}` | Compressible neo-Hookean with the isochoric–volumetric split, W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)², μ and κ from E and ν (C₁₀ = μ/2, D₁ = 2/κ). Its small-strain limit is the linear material, which linear steps use. |
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| Isotropic elastic with `{"type": "mooney_rivlin", "c01": …}` or `{"type": "yeoh", "c20": …, "c30": …}` | Decoupled polynomials W̄ = C₁₀ (Ī₁ − 3) + C₀₁ (Ī₂ − 3) + C₂₀ (Ī₁ − 3)² + C₃₀ (Ī₁ − 3)³ with the same volume κ/2 (J − 1)²: Mooney–Rivlin with C₁₀ = μ/2 − C₀₁ (0 ≤ C₀₁ ≤ μ/2), Yeoh with C₁₀ = μ/2, so 2(C₁₀ + C₀₁) is the elastic μ and the small-strain material stays the elastic one. Stress and tangent from the fictitious τ̄ and c̄ (Holzapfel 2000, eqs. 6.168, 6.193). Abaqus `*HYPERELASTIC, MOONEY-RIVLIN` / `YEOH`; LS-DYNA refuses them by name. |
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| Isotropic elastic with `"plastic"` (bilinear isotropic or kinematic, table, Johnson–Cook) | J2 plasticity (`fieldcraft_fem::elastoplastic`): at finite strain Simo's multiplicative model (CMAME 99, 1992) with a Hencky elastic energy, τ = K ln J I + 2G dev ½ ln bₑ, the history Cₚ⁻¹ and the equivalent plastic strain, integrated by the exponential return map on the principal logarithmic strains; at small strain the additive radial return with the back stress. Both call `J2::radial_return`, the update of explicit dynamics, and its consistent tangent `J2::tangent`. |
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@@ -261,8 +262,20 @@ summary), every increment's load factor, iterations and residuals, and a displac
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converged increment (`results.frame`, the slider in the results panel).
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**Couplings** stay the linear constraints of the linear solver (small rotations of the
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reference points), with a warning; springs act along their fixed directions. Temperature and
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centrifugal loads are refused by name in nonlinear steps.
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reference points), with a warning; springs act along their fixed directions. Centrifugal loads
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are refused by name in nonlinear steps.
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**Thermal strain.** A nonlinear step's temperatures are its temperature loads, or the steady
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conduction of its temperature boundary conditions and heat loads solved first (one-way, as the
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linear coupled temperature–displacement step); they ramp with the load factor from each
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material's reference temperature. With nonlinear geometry F = F_m F_θ, F_θ = diag(1 + αᵢ ΔT) at
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each integration point (V. A. Lubarda, Appl. Mech. Rev. 57:95–108, 2004): the material sees F_m,
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the loops run on the expanded volume J_θ dV₀ (mixed elements project det F_m), and the reported
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strain is the total logarithmic one; the linear stretch 1 + α ΔT keeps α the engineering
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coefficient, so a free body grows by exactly α ΔT. Without nonlinear geometry the split is
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additive (ε_m = ε − α ΔT), and elastic elements on their linear stiffness take the equivalent
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nodal forces as a dead load. Johnson–Cook plasticity softens at the point's temperature. The
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expansion is constant in temperature (the model has no temperature-dependent coefficient yet).
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**Decks**: Abaqus `*STEP, NLGEOM=YES|NO, INC=` with `*STATIC` (initial, period 1, minimum,
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maximum) and `*CONTROLS, PARAMETERS=LINE SEARCH`; the neo-Hookean option as `*HYPERELASTIC, NEO
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@@ -293,4 +306,12 @@ cylinder, the bore displacement at a plastic radius of 1.5a (0.009 %) and the co
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(2·10⁻⁸); a cantilever's collapse 3.1 % above beam theory's hinge, converged under refinement;
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Newton's order with the consistent tangent 1.9 (small strain) and 2.3 (finite strain) in a
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plastic cantilever; and the refusals and the softening material that stops at the minimum
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increment.
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increment. Mooney–Rivlin and Yeoh (`nl-mooney-rivlin-modes`, `nl-yeoh-uniaxial`,
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`nl-mooney-rivlin-tube`): tangents by central differences (material point and hybrid
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elements), Treloar's uniaxial, equibiaxial and pure-shear modes against the compressible closed
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form (exact), and a tube inflated by a follower pressure while stretched axially to λ_z = 1.5
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(Rivlin's incompressible solution, 0.6 % at ν = 0.4995). Thermal strain (`nl-thermal-expansion`,
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`j2-heated-bar`, `crates/solver/tests/nonlinear_thermal.rs`): free expansion and a fully held
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block at a thermal stretch of 1.2 (exact, full and hybrid), a bar held at its ends heated past
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yield (exact), the linear limit against the linear and coupled solvers, and tangents with a
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temperature field.
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+27
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Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the policy is [verification.md](verification.md). Every case is our own model, built with FieldCraft commands from the problem's published description, and compared with the reference value its `expected.json` cites, within the stated tolerance and for the stated reason. Accepted results are frozen in each case's `fixture.json`: a later change that moves one beyond its band (10⁻⁹ of its size) fails CI even while the reference still passes. No reference value was produced with a commercial product.
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**149 of 155 quantities pass** (6 known failure(s)) in 44 active cases: 35 theory, 9 published benchmark, 0 experiment; 9 pending cases wait for capabilities still being built.
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**163 of 169 quantities pass** (6 known failure(s)) in 49 active cases: 40 theory, 9 published benchmark, 0 experiment; 9 pending cases wait for capabilities still being built.
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## Theory: closed-form solutions, patch tests and mesh invariants
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@@ -31,8 +31,10 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
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| [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | sixth frequency (the last rigid-body mode) | 0 | ≈ 0 | ≈ 0 | ± 8.1264e-5 | pass | matches |
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| [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | first free–free bending frequency (mode 7) | 0.081264 | 0.080618 | -0.795% | 1.5% | pass | matches |
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| [free-beam-frequencies](../tests/oracle/free-beam-frequencies/) | total mass | 20 | 20 | < 1e-7% | 1e-12 rel. | pass | matches |
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| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | axial stress back at zero strain after a pull to ε_a = 0.01 | -247.52 | -247.52 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | accumulated equivalent plastic strain | 0.016089 | 0.016089 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | axial stress back at zero strain after a pull to ε_a = 0.01 | -247.52 | -247.52 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [j2-bauschinger](../tests/oracle/j2-bauschinger/) | accumulated equivalent plastic strain | 0.016089 | 0.016089 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [j2-heated-bar](../tests/oracle/j2-heated-bar/) | axial stress, ΔT = 300 (α ΔT = 3.6·10⁻³, 2.9 times the yield strain) | -254.65 | -254.65 | < 1e-7% | 1e-8 rel. | pass | blessed |
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| [j2-heated-bar](../tests/oracle/j2-heated-bar/) | equivalent plastic strain | 0.0023267 | 0.0023267 | < 1e-7% | 1e-7 rel. | pass | blessed |
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| [j2-uniaxial-explicit](../tests/oracle/j2-uniaxial-explicit/) | axial Cauchy stress at a logarithmic strain of 0.2 | 643.56 | 643.57 | 0.000174% | 1% | pass | matches |
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| [j2-uniaxial-explicit](../tests/oracle/j2-uniaxial-explicit/) | equivalent plastic strain | 0.19678 | 0.19678 | -0.00000113% | 1% | pass | matches |
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| [kirsch-plate](../tests/oracle/kirsch-plate/) | σ_xx at the top of the hole (0, 1, 0) over the remote stress | 3 | 3.0121 | 0.402% | 2% | pass | matches |
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@@ -59,8 +61,15 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
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| [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 5 | 0.38763 | 0.38955 | 0.495% | 2% | pass | matches |
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| [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip deflection δ_v/L at P L²/(E I) = 10 | 0.81061 | 0.81421 | 0.444% | 1% | pass | matches |
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| [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip shortening δ_h/L at P L²/(E I) = 10 | 0.555 | 0.558 | 0.541% | 2% | pass | matches |
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| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | axial Cauchy stress at a logarithmic strain of 0.2 | 642.74 | 642.74 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | equivalent plastic strain | 0.19678 | 0.19678 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | axial Cauchy stress at a logarithmic strain of 0.2 | 642.74 | 642.74 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/) | equivalent plastic strain | 0.19678 | 0.19678 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | axial Cauchy stress, uniaxial stretch λ = 2 | 2.7366 | 2.7366 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | lateral displacement at (1, 1, 1), uniaxial: l − 1 | -0.28647 | -0.28647 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | in-plane Cauchy stress, equibiaxial stretch λ = 1.5 | 2.8427 | 2.8427 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | Cauchy stress along the stretch, pure shear λ = 2 | 3.5261 | 3.5261 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/) | Cauchy stress across, in the held direction, pure shear λ = 2 | 1.5067 | 1.5067 | < 1e-7% | 1e-6 rel. | pass | blessed |
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| [nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/) | radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 at λ_z = 1.5 | 5 | 5.0311 | 0.622% | 1% | pass | blessed |
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| [nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/) | radial displacement of the outer wall (20, 0, 0): b − B, a small difference of radii | 0.61553 | 0.64339 | 2.8e-2 | ± 0.04 | pass | blessed |
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| [nl-neo-hookean-uniaxial](../tests/oracle/nl-neo-hookean-uniaxial/) | axial Cauchy stress at the corner (1, 1, 1), stretch λ = 2 | 8.0601 | 8.0601 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-neo-hookean-uniaxial](../tests/oracle/nl-neo-hookean-uniaxial/) | lateral displacement u_y at the corner (1, 1, 1), l − 1 | -0.18686 | -0.18686 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-patch-hex8](../tests/oracle/nl-patch-hex8/) | u_x at inner node 22 | 0.13583 | 0.13583 | < 1e-7% | 1e-8 rel. | pass | matches |
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@@ -76,6 +85,11 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
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| [nl-rigid-rotation](../tests/oracle/nl-rigid-rotation/) | u_y at inner node 22: the rotation's | 0.078591 | 0.078591 | -5.6e-17 | ± 1.0000e-9 | pass | matches |
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| [nl-rigid-rotation](../tests/oracle/nl-rigid-rotation/) | u_z at inner node 22: the rotation's | -0.067441 | -0.067441 | -1.2e-16 | ± 1.0000e-9 | pass | matches |
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| [nl-rigid-rotation](../tests/oracle/nl-rigid-rotation/) | von Mises stress at inner node 22 | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-8 | pass | matches |
|
||||
| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | displacement of the corner (1, 1, 1) along x, free, α ΔT = 0.2 | 0.2 | 0.2 | < 1e-7% | 1e-8 rel. | pass | blessed |
|
||||
| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | stress of the free body (σ_xx at the corner) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-8 | pass | blessed |
|
||||
| [nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/) | stress of the held block, σ_xx (hydrostatic) | -3.5108 | -3.5108 | < 1e-7% | 1e-8 rel. | pass | blessed |
|
||||
| [nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/) | axial Cauchy stress, stretch λ = 2 | 3.1743 | 3.1743 | < 1e-7% | 1e-6 rel. | pass | blessed |
|
||||
| [nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/) | lateral displacement at (1, 1, 1): l − 1 | -0.28545 | -0.28545 | < 1e-7% | 1e-6 rel. | pass | blessed |
|
||||
| [patch-hex20](../tests/oracle/patch-hex20/) | u_x at the corner (1, 1, 1) | -2.5000e-4 | -2.5000e-4 | < 1e-7% | 1e-9 rel. | pass | matches |
|
||||
| [patch-hex20](../tests/oracle/patch-hex20/) | u_y at the corner (1, 1, 1) | 0.001 | 0.001 | < 1e-7% | 1e-9 rel. | pass | matches |
|
||||
| [patch-hex20](../tests/oracle/patch-hex20/) | u_z at the corner (1, 1, 1) | 0.00225 | 0.00225 | < 1e-7% | 1e-9 rel. | pass | matches |
|
||||
@@ -125,9 +139,9 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
|
||||
| [patch-wedge6](../tests/oracle/patch-wedge6/) | σ_yy at the inner node (0.46, 0.55, 0.53) | 2 | 2 | < 1e-7% | 1e-9 rel. | pass | matches |
|
||||
| [patch-wedge6](../tests/oracle/patch-wedge6/) | σ_zz at the inner node (0.46, 0.55, 0.53) | 3 | 3 | < 1e-7% | 1e-9 rel. | pass | matches |
|
||||
| [patch-wedge6](../tests/oracle/patch-wedge6/) | σ_xy at the inner node | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-9 | pass | matches |
|
||||
| [plastic-hinge-cantilever](../tests/oracle/plastic-hinge-cantilever/) | tip force at a tip deflection of 15 (4.5 times the first yield's) | -312.5 | -322.61 | -3.23% | 5% | pass | blessed |
|
||||
| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | bore displacement at the pressure that takes the plastic zone to c = 15 (a = 10, b = 20, σ_y = 250) | 0.024355 | 0.024358 | 0.0089% | 0.2% | pass | blessed |
|
||||
| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | radial force on the bore (per radian times 2π, height 1) at a bore displacement of 0.3, 24 times the first yield's: the collapse pressure 2k ln 2 = 200.094 times 2πa | 12572 | 12572 | 0.00000178% | 0.1% | pass | blessed |
|
||||
| [plastic-hinge-cantilever](../tests/oracle/plastic-hinge-cantilever/) | tip force at a tip deflection of 15 (4.5 times the first yield's) | -312.5 | -322.61 | -3.23% | 5% | pass | matches |
|
||||
| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | bore displacement at the pressure that takes the plastic zone to c = 15 (a = 10, b = 20, σ_y = 250) | 0.024355 | 0.024358 | 0.0089% | 0.2% | pass | matches |
|
||||
| [plastic-thick-cylinder](../tests/oracle/plastic-thick-cylinder/) | radial force on the bore (per radian times 2π, height 1) at a bore displacement of 0.3, 24 times the first yield's: the collapse pressure 2k ln 2 = 200.094 times 2πa | 12572 | 12572 | 0.00000178% | 0.1% | pass | matches |
|
||||
| [pure-bending-coupling](../tests/oracle/pure-bending-coupling/) | bending stress σ_xx at mid-span, top fibre (10, 0.5, 1) | 15 | 15 | -0.0000103% | 0.5% | pass | matches |
|
||||
| [pure-bending-coupling](../tests/oracle/pure-bending-coupling/) | rotation of the reference point about y | 0.3 | 0.30355 | 1.18% | 1% | known failure: 1.2% high: the distributing coupling weights every node of the end face equally, corner and mid-side nodes of hex20 alike, which is not the consistent linear traction; the end face deforms locally and the fitted rotation includes it. Area-consistent weights would remove it | — |
|
||||
| [rotating-cylinder-axisymmetric](../tests/oracle/rotating-cylinder-axisymmetric/) | radial stress σ_r on the axis (r = 0) | 0.38571 | 0.38608 | 0.0958% | 1% | pass | matches |
|
||||
@@ -181,6 +195,7 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
|
||||
- **[cdt-stepped-block](../tests/oracle/cdt-stepped-block/)** (theory): Constrained Delaunay mesh of a stepped block: exact volume, valid elements. Source: mesh invariants: a tetrahedralization of a polyhedron fills its volume exactly (Si, TetGen paper ACM TOMS 41(2) 2015 for the definition of a CDT; Diazzi et al., ACM TOG 42(6) 2023); the volume is 60·40·10 + 40·40·10 + 20·40·10 − 10·10·10. Tolerances — volume: a planar boundary is tessellated exactly and the constrained Delaunay mesh fills it exactly: only rounding; invalid: every element of a valid mesh has positive volume; undetermined: a linear tetrahedron's sign is exact
|
||||
- **[free-beam-frequencies](../tests/oracle/free-beam-frequencies/)** (theory): Natural frequencies of a free beam: rigid-body modes and the first bending mode. Source: Euler–Bernoulli free–free beam, β₁L = 4.7300; Timoshenko, Young and Weaver, Vibration Problems in Engineering, 4th ed., Wiley 1974, §5.3. Tolerances — rigid_6: a free body has six zero frequencies; numerical rigid-body frequencies below 10⁻³ of the first elastic one (eigenvalues below 10⁻⁶ of it) are rounding of the shifted solve; f7: Euler–Bernoulli neglects shear and rotary inertia, about 0.8% at L/h = 20 for the free–free mode; mass: the mass matrix integrates the density exactly
|
||||
- **[j2-bauschinger](../tests/oracle/j2-bauschinger/)** (theory): Bauschinger effect of linear kinematic hardening: tension, then back to zero strain. Source: J2 plasticity with Prager's linear kinematic hardening in uniaxial stress (W. Prager, Ing.-Arch. 1955; Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3): from σ_a = σ_y + E_t (ε_a − σ_y/E) the material unloads elastically by 2σ_y and then hardens with E_t, so at zero strain σ = σ_a − 2σ_y − E_t (ε_a − 2σ_y/E). Tolerances — residual_stress: a homogeneous small-strain state the element represents exactly, with increments that end at the load path's turn; isotropic hardening would give −281.835 (it yields again only at −σ_a); plastic_strain: (σ_a − σ_y)/H forward plus E/(E + H) (ε_a − 2σ_y/E) in reverse
|
||||
- **[j2-heated-bar](../tests/oracle/j2-heated-bar/)** (theory): A steel bar held at its ends and heated past yield (small strain). Source: Uniaxial J2 plasticity with linear hardening (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) with the additive thermal strain: the bar's total axial strain is held at zero, so its mechanical strain is −α ΔT and σ = −(σ_y + E_t (α ΔT − σ_y/E)) once α ΔT exceeds σ_y/E. Tolerances — sigma: a homogeneous uniaxial state the elements represent exactly; plastic_strain: (\|σ\| − σ_y)/H, as for the stress
|
||||
- **[j2-uniaxial-explicit](../tests/oracle/j2-uniaxial-explicit/)** (theory): Uniaxial tension of a J2 bilinear-hardening cube to large strain (explicit). Source: von Mises plasticity with linear isotropic hardening under uniaxial stress (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3); logarithmic strain for a hypoelastic–plastic law under coaxial stretching. Tolerances — sigma: uniaxial J2 with linear hardening: σ = σ_y + E_t (ε − σ_y/E) in logarithmic strain; the stretch takes 10⁴ wave transits, so inertia and the ringing of the start (ρcv ≈ 0.1% of σ) stay well under 1%; plastic_strain: ε_p = ε − σ/E, as for the stress
|
||||
- **[kirsch-plate](../tests/oracle/kirsch-plate/)** (theory): Plate with a circular hole in tension (Kirsch). Source: Kirsch, Zeitschrift des VDI 42 (1898) 797–807; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §35. Tolerances — kt: the reference is the infinite plate; the finite plate (W = 60a, Heywood's width correction about +0.1%, and the loaded ends 30a away) raises the peak by a few tenths of a percent, and the stress extrapolated to the hole surface adds up to about 1%; side: infinite-plate value −S; finite width and extrapolation as for Kt, on a smaller value
|
||||
- **[kirsch-plate-plane-stress](../tests/oracle/kirsch-plate-plane-stress/)** (theory): Plate with a circular hole in tension, 2D plane stress (Kirsch). Source: Kirsch, Zeitschrift des VDI 42 (1898) 797–807; Timoshenko and Goodier, Theory of Elasticity, 3rd ed., 1970, §35. Tolerances — kt: the reference is the infinite plate; the finite plate (W = 60a, loaded ends 30a away) raises the peak by a few tenths of a percent, and the stress extrapolated to the hole surface adds up to about 1%; side: infinite-plate value −S; finite width and extrapolation as for Kt, on a smaller value
|
||||
@@ -198,9 +213,13 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
|
||||
- **[nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/)** (theory): Inflation of a thick neo-Hookean cylinder by a follower pressure (plane strain). Source: Finite inflation of an incompressible neo-Hookean thick-walled tube in plane strain: r² = R² + a² − A², σ_θ − σ_r = μ (λ² − λ⁻²) with λ = r/R, integrated through the wall: p = (μ/2)[(ln λ_a² − λ_a⁻²) − (ln λ_b² − λ_b⁻²)] (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3.2). Tolerances — ua: the closed form is incompressible and our material nearly so (κ/μ = 1000, a volume change of order p/κ ≈ 4·10⁻⁴); hybrid hex20, 6 × 16, follow the wall's strain gradient well inside; ub: as at the bore
|
||||
- **[nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/)** (theory): Large-deflection cantilever under a dead tip load against the elastica (solids). Source: The elastica of a cantilever under a vertical end load: K. E. Bisshopp, D. C. Drucker, Large deflection of cantilever beams, Quarterly of Applied Mathematics 3(3):272–275, 1945 (elliptic-integral solution); values computed by integrating θ″ = −(P/EI) cos θ by shooting (our own RK4, 20000 steps), agreeing with the table of K. Mattiasson, Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals, IJNME 17(1):145–153, 1981, to all six digits. Tolerances — v1: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u1: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v2: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u2: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v5: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u5: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's; v10: hex20, 40 × 2 through the depth, L/h = 20: shear deformation (0.6 (h/L)² ≈ 0.15 % of the deflection) and the clamp's local restraint, both absent from the elastica; u10: as δ_v; the shortening is the smaller, second-order quantity, so its relative error is about twice the deflection's
|
||||
- **[nl-j2-uniaxial](../tests/oracle/nl-j2-uniaxial/)** (theory): Uniaxial tension of a J2 bilinear-hardening cube to large strain (implicit, finite strain). Source: J2 plasticity with linear isotropic hardening in uniaxial stress (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) in the logarithmic strain, which Simo's exponential return map (CMAME 99:61–112, 1992) integrates exactly on a proportional path: Kirchhoff stress τ = σ_y + E_t (ε − σ_y/E), Cauchy stress σ = τ/J with ln J = τ/(3K) (Hencky's volume law). Tolerances — sigma: a homogeneous deformation the element represents exactly, reached on a proportional path: the tolerance only admits the Newton tolerance and rounding; plastic_strain: ε_p = (τ − σ_y)/H, as for the stress
|
||||
- **[nl-mooney-rivlin-modes](../tests/oracle/nl-mooney-rivlin-modes/)** (theory): Mooney–Rivlin block in uniaxial, equibiaxial and pure-shear tension. Source: Treloar's homogeneous modes of rubber testing (L. R. G. Treloar, The Physics of Rubber Elasticity, 3rd ed., Oxford 1975, ch. 10) for the decoupled compressible energy W = C₁₀ (Ī₁ − 3) + C₀₁ (Ī₂ − 3) + κ/2 (J − 1)²: principal Cauchy stresses σᵢ = (λᵢ/J) ∂W/∂λᵢ with the free stretch found so the free faces carry no stress (bisection; E = 3, ν = 0.49, C₀₁ = 0.2, C₁₀ = μ/2 − C₀₁). Tolerances — uniaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; uniaxial_lateral: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; equibiaxial: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_xx: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; shear_yy: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding
|
||||
- **[nl-mooney-rivlin-tube](../tests/oracle/nl-mooney-rivlin-tube/)** (theory): Inflation of a stretched Mooney–Rivlin tube (follower pressure). Source: Finite inflation and extension of an incompressible Mooney–Rivlin tube (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3): at axial stretch λ_z, r² = a² + (R² − A²)/λ_z, σ_θ − σ_r = 2 (λ_θ² − λ_r²)(C₁₀ + C₀₁ λ_z²) with λ_θ = r/R, λ_r = 1/(λ_θ λ_z), integrated through the wall: p = ∫ (σ_θ − σ_r) dr/r (Simpson, 2·10⁵ intervals). Tolerances — ua: 16 × 6 hybrid hex20 around the quarter; ν = 0.4995 stands in for incompressibility (a volume change of order p/κ ≈ 10⁻³ shifts the radii by about 0.1 %), and the tube is stretched axially by its ends, which a nearly incompressible solid takes with larger pressures than the plane-strain neo-Hookean tube; ub: 0.2 % of the outer radius b = 20.616: the same volume change of order 10⁻³ moves b by about 0.03, which is 4.5 % of b − B; the incompressible kinematics r² = a² + (R² − A²)/λ_z amplify it there
|
||||
- **[nl-neo-hookean-uniaxial](../tests/oracle/nl-neo-hookean-uniaxial/)** (theory): Uniaxial stretch of a neo-Hookean block to twice its length. Source: Closed form of the compressible neo-Hookean model W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)² in uniaxial stress (G. A. Holzapfel, Nonlinear Solid Mechanics, Wiley 2000, §6.5, eq. 6.148 with the volumetric term of §6.4): σ_yy = μ J^(−5/3)(l² − Ī/3) + κ(J − 1) = 0 fixes the lateral stretch l, then σ_xx = μ J^(−5/3)(λ² − l²). Tolerances — sigma: a homogeneous deformation every element represents exactly: the tolerance only admits rounding and the Newton tolerance; lateral: a homogeneous deformation every element represents exactly: the tolerance only admits rounding and the Newton tolerance
|
||||
- **[nl-patch-hex8](../tests/oracle/nl-patch-hex8/)** (theory): Nonlinear patch test, hex8: a homogeneous finite deformation of a distorted mesh. Source: The patch test (B. M. Irons, A. Razzaque, in A. K. Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, 557–587) at finite strain: a boundary moved by x = F X makes F the solution everywhere (J. Bonet, R. D. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd ed., Cambridge 2008, §8); the neo-Hookean Cauchy stress σ = μ J^(−5/3) dev b + κ (J − 1) I of that F. Tolerances — ux: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; uy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; uz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxx: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; syy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; szz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxy: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; sxz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance; syz: a patch test is passed exactly: the tolerance admits rounding and the Newton tolerance
|
||||
- **[nl-rigid-rotation](../tests/oracle/nl-rigid-rotation/)** (theory): A quarter turn strains and stresses nothing (objectivity). Source: Objectivity of the total Lagrangian formulation and of the hyperelastic stress: a rigid motion x = R X + c gives C = I and σ = 0 whatever the rotation (Bonet & Wood 2008, §3.4 and §5.2; T. Belytschko, W. K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, 2nd ed., Wiley 2014, §5.5). Tolerances — ux: the inner nodes follow the rigid rotation exactly; uy: the inner nodes follow the rigid rotation exactly; uz: the inner nodes follow the rigid rotation exactly; von_mises: zero up to rounding of stresses of order E = 1000: a 90° turn leaves no strain
|
||||
- **[nl-thermal-expansion](../tests/oracle/nl-thermal-expansion/)** (theory): Free and fully constrained thermal expansion at finite strain. Source: The multiplicative split F = F_m F_θ with F_θ = (1 + α ΔT) I (V. A. Lubarda, Appl. Mech. Rev. 57(2):95–108, 2004): a free body follows F = F_θ stress-free; held on every face, F = I and F_m = ϑ⁻¹ I, so the compressible neo-Hookean material carries the pressure of its volume ratio, σ = κ (ϑ⁻³ − 1). Tolerances — free_u: a homogeneous stretch every element represents exactly; free_stress: rounding only; held_stress: a homogeneous state; E = 10, ν = 0.3 (κ = 8.3333), ϑ = 1.2
|
||||
- **[nl-yeoh-uniaxial](../tests/oracle/nl-yeoh-uniaxial/)** (theory): Yeoh block in uniaxial tension. Source: Uniaxial tension of the decoupled compressible Yeoh energy W = C₁₀ (Ī₁ − 3) + C₂₀ (Ī₁ − 3)² + C₃₀ (Ī₁ − 3)³ + κ/2 (J − 1)² (O. H. Yeoh, Rubber Chem. Technol. 66(5):754–771, 1993): σ = (λ/J) ∂W/∂λ with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.49, C₁₀ = μ/2, C₂₀ = −0.02, C₃₀ = 0.004). Tolerances — sigma: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding; lateral: a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding
|
||||
- **[patch-hex20](../tests/oracle/patch-hex20/)** (theory): Constant-stress patch test, hex20. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
|
||||
- **[patch-hex20-reduced](../tests/oracle/patch-hex20-reduced/)** (theory): Constant-stress patch test, hex20-reduced. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
|
||||
- **[patch-hex8](../tests/oracle/patch-hex8/)** (theory): Constant-stress patch test, hex8. Source: Irons and Razzaque, 'Experience with the patch test for convergence of finite elements', in Aziz (ed.), The Mathematical Foundations of the Finite Element Method, Academic Press 1972, pp. 557–587; MacNeal and Harder, Finite Elements in Analysis and Design 1 (1985) 3–20, constant-stress loading, p. 14 Table 7. Tolerances — ux: a patch test is passed exactly: the tolerance only admits rounding; uy: a patch test is passed exactly: the tolerance only admits rounding; uz: a patch test is passed exactly: the tolerance only admits rounding; sxx: a patch test is passed exactly: the tolerance only admits rounding; syy: a patch test is passed exactly: the tolerance only admits rounding; szz: a patch test is passed exactly: the tolerance only admits rounding; sxy: a patch test is passed exactly: the tolerance only admits rounding (stresses of order 1)
|
||||
|
||||
+2
-2
@@ -44,7 +44,7 @@ products are run, no documentation text is copied). Status: **done**, **started*
|
||||
| Mesh | Mesh editing (move, merge, split, collapse, remesh region), imported orphan meshes | M6 | yes | started (node/element add/remove) |
|
||||
| Model | Sets and surfaces bound to geometry names | M3 | yes | done |
|
||||
| Model | Materials: elastic (isotropic, orthotropic), density, unit-system check | M3 | | done |
|
||||
| Model | Materials: plasticity, hyperelasticity, damage, thermal | M8 | | started (von Mises plasticity: bilinear, table, Johnson–Cook; Johnson–Cook damage, erosion; plasticity in explicit and nonlinear static steps; written to Abaqus `*PLASTIC`, Nastran `MATS1`, LS-DYNA plastic materials where expressible; compressible neo-Hookean for nonlinear steps, Abaqus `*HYPERELASTIC, NEO HOOKE`) |
|
||||
| Model | Materials: plasticity, hyperelasticity, damage, thermal | M8 | | started (von Mises plasticity: bilinear, table, Johnson–Cook; Johnson–Cook damage, erosion; plasticity in explicit and nonlinear static steps; written to Abaqus `*PLASTIC`, Nastran `MATS1`, LS-DYNA plastic materials where expressible; compressible neo-Hookean, Mooney–Rivlin and Yeoh for nonlinear steps, Abaqus `*HYPERELASTIC, NEO HOOKE`, `MOONEY-RIVLIN`, `YEOH`; thermal strain in nonlinear steps) |
|
||||
| Model | Sections: solid, shell, beam profiles; assignment by set | M3 | | done |
|
||||
| Model | 2D sections: plane stress, plane strain, axisymmetric (XY plane, Y the axis); element edges in sets; centrifugal load | M4 | | done |
|
||||
| Model | Composite layups, material orientations, continuum shells | M8 | yes | planned |
|
||||
@@ -63,7 +63,7 @@ products are run, no documentation text is copied). Status: **done**, **started*
|
||||
| Decks | LS-DYNA keyword writer (implicit; explicit in M8) | M3 | | done (implicit static and eigenvalue; beam sections planned) |
|
||||
| Decks | Readers keeping unknown keywords | M9 | | planned |
|
||||
| Solve | Own linear static and modal solver, verified against analytic solutions | M4 | yes | started (solid elements: tet4, tet10, wedge6, hex8, hex20; 2D plane stress, plane strain and axisymmetric tri3, tri6, quad4 (full, incompatible modes), quad8 (full, reduced); quad4/tri3 MITC shells and Timoshenko beams with profiles, mixed with solids; reference points, kinematic and distributing couplings, moments; centrifugal loads) |
|
||||
| Solve | Own nonlinear implicit solver: Newton–Raphson with line search and automatic increments, total Lagrangian solids and 2D elements, mixed u–p at finite strain, follower pressure | M15 | yes | started (solids, plane strain, axisymmetric; St Venant–Kirchhoff, neo-Hookean and J2 plasticity (Simo's exponential return map at finite strain, radial return at small strain, consistent tangents, the return shared with explicit dynamics); verified by the elastica, tube inflation, patch and objectivity tests, uniaxial tension to large strain, Hill's thick cylinder, a cantilever's collapse and the Bauschinger effect; Mooney–Rivlin, thermal strain, shells, contact, arc-length next) |
|
||||
| Solve | Own nonlinear implicit solver: Newton–Raphson with line search and automatic increments, total Lagrangian solids and 2D elements, mixed u–p at finite strain, follower pressure | M15 | yes | started (solids, plane strain, axisymmetric; St Venant–Kirchhoff, neo-Hookean and J2 plasticity (Simo's exponential return map at finite strain, radial return at small strain, consistent tangents, the return shared with explicit dynamics); verified by the elastica, tube inflation, patch and objectivity tests, uniaxial tension to large strain, Hill's thick cylinder, a cantilever's collapse and the Bauschinger effect; Mooney–Rivlin and Yeoh; thermal strain at finite and small strain; shells, contact, arc-length next) |
|
||||
| Solve | Own explicit dynamics solver (central differences, one-point hex8 with hourglass control, nodal-pressure tet4, bulk viscosity, energy balance), verified against analytic wave and impact solutions | M10 | yes | started (X1 core and X2: large deformation, objective rates, J2 and Johnson–Cook plasticity, damage, erosion; contact planned) |
|
||||
| Results | Deformed shape, contours, mode shapes, probes, reaction sums | M4 | | started (results in the session, `solve.probe`, reaction sums; display planned) |
|
||||
| Results | XY and path plots, cuts, display groups, free-body forces, animation | M9 | | planned |
|
||||
|
||||
@@ -1211,3 +1211,20 @@ update are written from (cited again in the module docs):
|
||||
B. G. Neal, *The Plastic Methods of Structural Analysis*, 3rd ed., Chapman & Hall 1977, and the
|
||||
continuum's velocity fields, W. Prager, P. G. Hodge, *Theory of Perfectly Plastic Solids*,
|
||||
Wiley 1951.
|
||||
|
||||
## Part O: Polynomial hyperelasticity and thermal strain in nonlinear steps, as implemented
|
||||
|
||||
- Decoupled invariant-based hyperelasticity, the fictitious stress and elasticity and their
|
||||
projection: G. A. Holzapfel, *Nonlinear Solid Mechanics*, Wiley 2000, §6.4–6.6 (eqs. 6.168,
|
||||
6.193).
|
||||
- Mooney–Rivlin: M. Mooney, *A theory of large elastic deformation*, J. Appl. Phys.
|
||||
11(9):582–592, 1940; R. S. Rivlin, *Large elastic deformations of isotropic materials IV*,
|
||||
Phil. Trans. R. Soc. A 241:379–397, 1948. Yeoh: O. H. Yeoh, *Some forms of the strain energy
|
||||
function for rubber*, Rubber Chem. Technol. 66(5):754–771, 1993.
|
||||
- The homogeneous modes of rubber testing: L. R. G. Treloar, *The Physics of Rubber
|
||||
Elasticity*, 3rd ed., Oxford 1975, ch. 10; the inflation and extension of a tube: R. S. Rivlin,
|
||||
Phil. Trans. R. Soc. A 242:173–195, 1949, and R. W. Ogden, *Non-Linear Elastic Deformations*,
|
||||
Dover 1997, §5.3.
|
||||
- Thermal strain by the multiplicative split F = F_m F_θ: V. A. Lubarda, *Constitutive
|
||||
theories based on the multiplicative decomposition of deformation gradient: thermoelasticity,
|
||||
elastoplasticity, and biomechanics*, Appl. Mech. Rev. 57(2):95–108, 2004.
|
||||
@@ -0,0 +1,8 @@
|
||||
# A heated bar held at its ends past yield
|
||||
|
||||
A steel bar 4 × 1 × 1 of 4 hex8 elements (E = 200000, ν = 0.3, σ_y = 250, H = 2000 isotropic,
|
||||
α = 1.2·10⁻⁵ from 20 degrees) on symmetry planes at x = 0, y = 0, z = 0 and held at x = 4 (its
|
||||
axial strain is zero; the lateral faces are free), heated uniformly to 320 by a temperature load
|
||||
in a nonlinear static step without nonlinear geometry. The mechanical strain is −α ΔT = −0.0036,
|
||||
2.9 times the yield strain, so the bar yields in compression: σ = −(σ_y + E_t (α ΔT − σ_y/E)) =
|
||||
−254.653 with E_t = E H/(E + H), and ε_p = (|σ| − σ_y)/H.
|
||||
@@ -0,0 +1,45 @@
|
||||
{
|
||||
"title": "A steel bar held at its ends and heated past yield (small strain)",
|
||||
"kind": "theory",
|
||||
"source": "Uniaxial J2 plasticity with linear hardening (Simo and Hughes, Computational Inelasticity, Springer 1998, §2.3) with the additive thermal strain: the bar's total axial strain is held at zero, so its mechanical strain is −α ΔT and σ = −(σ_y + E_t (α ΔT − σ_y/E)) once α ΔT exceeds σ_y/E",
|
||||
"quantities": [
|
||||
{
|
||||
"id": "sigma",
|
||||
"quantity": "axial stress, ΔT = 300 (α ΔT = 3.6·10⁻³, 2.9 times the yield strain)",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Heat",
|
||||
"elements": [
|
||||
2
|
||||
]
|
||||
},
|
||||
"pointer": "/elements/0/stress/0/0"
|
||||
},
|
||||
"value": -254.65346534653466,
|
||||
"tolerance": {
|
||||
"relative": 1e-08
|
||||
},
|
||||
"reason": "a homogeneous uniaxial state the elements represent exactly"
|
||||
},
|
||||
{
|
||||
"id": "plastic_strain",
|
||||
"quantity": "equivalent plastic strain",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Heat",
|
||||
"elements": [
|
||||
2
|
||||
]
|
||||
},
|
||||
"pointer": "/elements/0/plastic_strain/0"
|
||||
},
|
||||
"value": 0.0023267326732673297,
|
||||
"tolerance": {
|
||||
"relative": 1e-07
|
||||
},
|
||||
"reason": "(|σ| − σ_y)/H, as for the stress"
|
||||
}
|
||||
]
|
||||
}
|
||||
@@ -0,0 +1,13 @@
|
||||
{
|
||||
"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
|
||||
"values": {
|
||||
"plastic_strain": {
|
||||
"band": 2.3267326732675587e-12,
|
||||
"value": 0.0023267326732675587
|
||||
},
|
||||
"sigma": {
|
||||
"band": 2.5465346534653467e-7,
|
||||
"value": -254.6534653464418
|
||||
}
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,17 @@
|
||||
[
|
||||
{"command": "mesh.block", "params": {"body": "Bar", "kind": "hex8", "corners": [[0, 0, 0], [4, 0, 0], [4, 1, 0], [0, 1, 0], [0, 0, 1], [4, 0, 1], [4, 1, 1], [0, 1, 1]], "divisions": [4, 1, 1], "faces": {"6": "x0", "3": "y0", "1": "z0", "4": "x1"}}},
|
||||
{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Bar"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Bar", "face": "x0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "y0", "geometry": [{"body": "Bar", "face": "y0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "z0", "geometry": [{"body": "Bar", "face": "z0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x1", "geometry": [{"body": "Bar", "face": "x1"}]}},
|
||||
{"command": "model.material.create", "params": {"name": "Steel", "elastic": {"type": "isotropic", "young": 200000.0, "poisson": 0.3}, "plastic": {"type": "bilinear", "yield_stress": 250.0, "hardening_modulus": 2000.0}, "thermal": {"expansion": 1.2e-05, "reference_temperature": 20.0}}},
|
||||
{"command": "model.section.create", "params": {"name": "bar", "set": "all", "material": "Steel", "kind": {"type": "solid"}}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sx0", "type": "symmetry", "target": {"set": "x0"}, "plane": "x"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sy0", "type": "symmetry", "target": {"set": "y0"}, "plane": "y"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sz0", "type": "symmetry", "target": {"set": "z0"}, "plane": "z"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sx1", "type": "symmetry", "target": {"set": "x1"}, "plane": "x"}},
|
||||
{"command": "model.load.create", "params": {"name": "heat", "type": "temperature", "target": {"set": "all"}, "value": 320.0}},
|
||||
{"command": "model.step.create", "params": {"name": "Heat", "procedure": {"type": "nonlinear_static", "nonlinear_geometry": false}, "boundary_conditions": ["sx0", "sy0", "sz0", "sx1"], "loads": ["heat"]}},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Heat"}, "id": "solve"}
|
||||
]
|
||||
@@ -0,0 +1,14 @@
|
||||
# Mooney–Rivlin block in Treloar's three modes
|
||||
|
||||
One hybrid hex20 unit cube on three symmetry planes, E = 3, ν = 0.49 (μ = 1.006711,
|
||||
κ = 50), Mooney–Rivlin with C₀₁ = 0.2 (C₁₀ = μ/2 − C₀₁ = 0.303356), in three nonlinear
|
||||
static steps:
|
||||
|
||||
- **Uniaxial**: the face x = 1 pulled to λ = 2, the others free.
|
||||
- **Equibiaxial**: x = 1 and y = 1 pulled to λ = 1.5, z free.
|
||||
- **Pure shear**: x = 1 pulled to λ = 2, y = 1 held (λ_y = 1), z free.
|
||||
|
||||
The references are the compressible decoupled energy's principal Cauchy stresses
|
||||
σᵢ = (λᵢ/J) ∂W/∂λᵢ, by differences of W alone, with the free stretch found by bisection so the
|
||||
free faces carry no stress. The deformation is homogeneous, so the element represents it
|
||||
exactly.
|
||||
@@ -0,0 +1,122 @@
|
||||
{
|
||||
"title": "Mooney–Rivlin block in uniaxial, equibiaxial and pure-shear tension",
|
||||
"kind": "theory",
|
||||
"source": "Treloar's homogeneous modes of rubber testing (L. R. G. Treloar, The Physics of Rubber Elasticity, 3rd ed., Oxford 1975, ch. 10) for the decoupled compressible energy W = C₁₀ (Ī₁ − 3) + C₀₁ (Ī₂ − 3) + κ/2 (J − 1)²: principal Cauchy stresses σᵢ = (λᵢ/J) ∂W/∂λᵢ with the free stretch found so the free faces carry no stress (bisection; E = 3, ν = 0.49, C₀₁ = 0.2, C₁₀ = μ/2 − C₀₁)",
|
||||
"quantities": [
|
||||
{
|
||||
"id": "uniaxial",
|
||||
"quantity": "axial Cauchy stress, uniaxial stretch λ = 2",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Uniaxial",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/stress/0"
|
||||
},
|
||||
"value": 2.736633402084229,
|
||||
"tolerance": {
|
||||
"relative": 1e-06
|
||||
},
|
||||
"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
|
||||
},
|
||||
{
|
||||
"id": "uniaxial_lateral",
|
||||
"quantity": "lateral displacement at (1, 1, 1), uniaxial: l − 1",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Uniaxial",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/displacement/1"
|
||||
},
|
||||
"value": -0.28647206688129256,
|
||||
"tolerance": {
|
||||
"relative": 1e-06
|
||||
},
|
||||
"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
|
||||
},
|
||||
{
|
||||
"id": "equibiaxial",
|
||||
"quantity": "in-plane Cauchy stress, equibiaxial stretch λ = 1.5",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Equibiaxial",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/stress/0"
|
||||
},
|
||||
"value": 2.842734818108988,
|
||||
"tolerance": {
|
||||
"relative": 1e-06
|
||||
},
|
||||
"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
|
||||
},
|
||||
{
|
||||
"id": "shear_xx",
|
||||
"quantity": "Cauchy stress along the stretch, pure shear λ = 2",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Shear",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/stress/0"
|
||||
},
|
||||
"value": 3.526124960232999,
|
||||
"tolerance": {
|
||||
"relative": 1e-06
|
||||
},
|
||||
"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
|
||||
},
|
||||
{
|
||||
"id": "shear_yy",
|
||||
"quantity": "Cauchy stress across, in the held direction, pure shear λ = 2",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Shear",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/stress/1"
|
||||
},
|
||||
"value": 1.506681177907714,
|
||||
"tolerance": {
|
||||
"relative": 1e-06
|
||||
},
|
||||
"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
|
||||
}
|
||||
]
|
||||
}
|
||||
@@ -0,0 +1,25 @@
|
||||
{
|
||||
"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
|
||||
"values": {
|
||||
"equibiaxial": {
|
||||
"band": 2.8427348183579755e-9,
|
||||
"value": 2.8427348183579753
|
||||
},
|
||||
"shear_xx": {
|
||||
"band": 3.5261249603786212e-9,
|
||||
"value": 3.526124960378621
|
||||
},
|
||||
"shear_yy": {
|
||||
"band": 1.5066811779077142e-9,
|
||||
"value": 1.5066811778673217
|
||||
},
|
||||
"uniaxial": {
|
||||
"band": 2.736633402109491e-9,
|
||||
"value": 2.736633402109491
|
||||
},
|
||||
"uniaxial_lateral": {
|
||||
"band": 2.8647206688145424e-10,
|
||||
"value": -0.2864720668814542
|
||||
}
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,25 @@
|
||||
[
|
||||
{"command": "mesh.block", "params": {"body": "Cube", "kind": "hex20", "corners": [[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1]], "divisions": [1, 1, 1], "faces": {"6": "x0", "3": "y0", "1": "z0", "4": "x1", "5": "y1"}}},
|
||||
{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Cube"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Cube", "face": "x0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "y0", "geometry": [{"body": "Cube", "face": "y0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "z0", "geometry": [{"body": "Cube", "face": "z0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x1", "geometry": [{"body": "Cube", "face": "x1"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "y1", "geometry": [{"body": "Cube", "face": "y1"}]}},
|
||||
{"command": "model.material.create", "params": {"name": "Rubber", "elastic": {"type": "isotropic", "young": 3.0, "poisson": 0.49}, "hyperelastic": {"type": "mooney_rivlin", "c01": 0.2}}},
|
||||
{"command": "model.section.create", "params": {"name": "cube", "set": "all", "material": "Rubber", "kind": {"type": "solid"}, "formulation": {"hybrid": true}}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sx", "type": "symmetry", "target": {"set": "x0"}, "plane": "x"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sy", "type": "symmetry", "target": {"set": "y0"}, "plane": "y"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sz", "type": "symmetry", "target": {"set": "z0"}, "plane": "z"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "pull_u", "type": "displacement", "target": {"set": "x1"}, "values": [1.0, null, null, null, null, null]}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "pull_e", "type": "displacement", "target": {"set": "x1"}, "values": [0.5, null, null, null, null, null]}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "pull_ey", "type": "displacement", "target": {"set": "y1"}, "values": [null, 0.5, null, null, null, null]}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "pull_s", "type": "displacement", "target": {"set": "x1"}, "values": [1.0, null, null, null, null, null]}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "hold_y", "type": "displacement", "target": {"set": "y1"}, "values": [null, 0.0, null, null, null, null]}},
|
||||
{"command": "model.step.create", "params": {"name": "Uniaxial", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz", "pull_u"], "loads": []}},
|
||||
{"command": "model.step.create", "params": {"name": "Equibiaxial", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz", "pull_e", "pull_ey"], "loads": []}},
|
||||
{"command": "model.step.create", "params": {"name": "Shear", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz", "pull_s", "hold_y"], "loads": []}},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Uniaxial"}, "id": "uniaxial"},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Equibiaxial"}, "id": "equibiaxial"},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Shear"}, "id": "shear"}
|
||||
]
|
||||
@@ -0,0 +1,10 @@
|
||||
# Inflation of a stretched Mooney–Rivlin tube
|
||||
|
||||
The quarter tube of `nl-cylinder-inflation` (A = 10, B = 20, one unit long, 6 × 16 hybrid
|
||||
hex20), now Mooney–Rivlin (E = 3, ν = 0.4995, so μ = 1.000333; C₀₁ = 0.2, C₁₀ = μ/2 − C₀₁) and
|
||||
stretched axially by its end face (u_z = 0.5 at z = 1: λ_z = 1.5) while a follower pressure
|
||||
p = 0.540850 inflates the bore. In plane strain (λ_z = 1) Mooney–Rivlin is neo-Hookean with
|
||||
μ = 2(C₁₀ + C₀₁), since Ī₁ = Ī₂ there; the axial stretch makes the second invariant's share
|
||||
count: σ_θ − σ_r = 2(λ_θ² − λ_r²)(C₁₀ + C₀₁ λ_z²). Integrated through the incompressible wall,
|
||||
r² = a² + (R² − A²)/λ_z, that pressure puts the bore at a = 15 and the outer wall at
|
||||
b = 20.615528.
|
||||
@@ -0,0 +1,53 @@
|
||||
{
|
||||
"title": "Inflation of a stretched Mooney–Rivlin tube (follower pressure)",
|
||||
"kind": "theory",
|
||||
"source": "Finite inflation and extension of an incompressible Mooney–Rivlin tube (R. S. Rivlin, Large elastic deformations of isotropic materials VI, Phil. Trans. R. Soc. A 242:173–195, 1949; R. W. Ogden, Non-Linear Elastic Deformations, Dover 1997, §5.3): at axial stretch λ_z, r² = a² + (R² − A²)/λ_z, σ_θ − σ_r = 2 (λ_θ² − λ_r²)(C₁₀ + C₀₁ λ_z²) with λ_θ = r/R, λ_r = 1/(λ_θ λ_z), integrated through the wall: p = ∫ (σ_θ − σ_r) dr/r (Simpson, 2·10⁵ intervals)",
|
||||
"quantities": [
|
||||
{
|
||||
"id": "ua",
|
||||
"quantity": "radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 at λ_z = 1.5",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Inflate",
|
||||
"at": [
|
||||
[
|
||||
10.0,
|
||||
0,
|
||||
0
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/displacement/0"
|
||||
},
|
||||
"value": 5.0,
|
||||
"tolerance": {
|
||||
"relative": 0.01
|
||||
},
|
||||
"reason": "16 × 6 hybrid hex20 around the quarter; ν = 0.4995 stands in for incompressibility (a volume change of order p/κ ≈ 10⁻³ shifts the radii by about 0.1 %), and the tube is stretched axially by its ends, which a nearly incompressible solid takes with larger pressures than the plane-strain neo-Hookean tube"
|
||||
},
|
||||
{
|
||||
"id": "ub",
|
||||
"quantity": "radial displacement of the outer wall (20, 0, 0): b − B, a small difference of radii",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Inflate",
|
||||
"at": [
|
||||
[
|
||||
20.0,
|
||||
0,
|
||||
0
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/displacement/0"
|
||||
},
|
||||
"value": 0.6155281280883038,
|
||||
"tolerance": {
|
||||
"absolute": 0.04
|
||||
},
|
||||
"reason": "0.2 % of the outer radius b = 20.616: the same volume change of order 10⁻³ moves b by about 0.03, which is 4.5 % of b − B; the incompressible kinematics r² = a² + (R² − A²)/λ_z amplify it there"
|
||||
}
|
||||
]
|
||||
}
|
||||
@@ -0,0 +1,13 @@
|
||||
{
|
||||
"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
|
||||
"values": {
|
||||
"ua": {
|
||||
"band": 5.031118164303424e-9,
|
||||
"value": 5.0311181643034235
|
||||
},
|
||||
"ub": {
|
||||
"band": 0.00004000064339064634,
|
||||
"value": 0.64339064633861
|
||||
}
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,18 @@
|
||||
[
|
||||
{"command": "mesh.block", "params": {"body": "Pipe", "kind": "hex20", "corners": [[10.0, 0, 0], [20.0, 0, 0], [0, 20.0, 0], [0, 10.0, 0], [10.0, 0, 1], [20.0, 0, 1], [0, 20.0, 1], [0, 10.0, 1]], "divisions": [6, 16, 1], "edges": [{"corners": [2, 3], "arc": {"center": [0, 0, 0]}}, {"corners": [1, 4], "arc": {"center": [0, 0, 0]}}, {"corners": [6, 7], "arc": {"center": [0, 0, 1]}}, {"corners": [5, 8], "arc": {"center": [0, 0, 1]}}], "faces": {"6": "bore", "3": "y0", "5": "x0", "1": "z0", "2": "z1"}}},
|
||||
{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Pipe"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "bore", "geometry": [{"body": "Pipe", "face": "bore"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Pipe", "face": "x0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "y0", "geometry": [{"body": "Pipe", "face": "y0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "z0", "geometry": [{"body": "Pipe", "face": "z0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "z1", "geometry": [{"body": "Pipe", "face": "z1"}]}},
|
||||
{"command": "model.material.create", "params": {"name": "Rubber", "elastic": {"type": "isotropic", "young": 3.0, "poisson": 0.4995}, "hyperelastic": {"type": "mooney_rivlin", "c01": 0.2}}},
|
||||
{"command": "model.section.create", "params": {"name": "pipe", "set": "all", "material": "Rubber", "kind": {"type": "solid"}, "formulation": {"hybrid": true}}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sx", "type": "symmetry", "target": {"set": "x0"}, "plane": "x"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sy", "type": "symmetry", "target": {"set": "y0"}, "plane": "y"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sz0", "type": "symmetry", "target": {"set": "z0"}, "plane": "z"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "stretch", "type": "displacement", "target": {"set": "z1"}, "values": [null, null, 0.5, null, null, null]}},
|
||||
{"command": "model.load.create", "params": {"name": "p", "type": "pressure", "set": "bore", "magnitude": 0.5408503415684837}},
|
||||
{"command": "model.step.create", "params": {"name": "Inflate", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz0", "stretch"], "loads": ["p"]}},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Inflate"}, "id": "solve"}
|
||||
]
|
||||
@@ -0,0 +1,8 @@
|
||||
# Free and constrained thermal expansion at finite strain
|
||||
|
||||
A unit cube of 2 × 2 × 2 hex20, compressible neo-Hookean (E = 10, ν = 0.3) with an expansion
|
||||
coefficient α = 10⁻³ from 20 degrees, heated to 220 by a temperature load in two nonlinear static
|
||||
steps: **Free** (three symmetry planes) and **Held** (symmetry planes on all six faces). The
|
||||
thermal stretch is ϑ = 1 + α ΔT = 1.2. Free, the cube grows to 1.2 along every edge without
|
||||
stress; held, F = I, the material's mechanical part is F_m = ϑ⁻¹ I and its stress the neo-Hookean
|
||||
pressure κ (J_m − 1) with J_m = ϑ⁻³: −3.510802.
|
||||
@@ -0,0 +1,76 @@
|
||||
{
|
||||
"title": "Free and fully constrained thermal expansion at finite strain",
|
||||
"kind": "theory",
|
||||
"source": "The multiplicative split F = F_m F_θ with F_θ = (1 + α ΔT) I (V. A. Lubarda, Appl. Mech. Rev. 57(2):95–108, 2004): a free body follows F = F_θ stress-free; held on every face, F = I and F_m = ϑ⁻¹ I, so the compressible neo-Hookean material carries the pressure of its volume ratio, σ = κ (ϑ⁻³ − 1)",
|
||||
"quantities": [
|
||||
{
|
||||
"id": "free_u",
|
||||
"quantity": "displacement of the corner (1, 1, 1) along x, free, α ΔT = 0.2",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Free",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/displacement/0"
|
||||
},
|
||||
"value": 0.2,
|
||||
"tolerance": {
|
||||
"relative": 1e-08
|
||||
},
|
||||
"reason": "a homogeneous stretch every element represents exactly"
|
||||
},
|
||||
{
|
||||
"id": "free_stress",
|
||||
"quantity": "stress of the free body (σ_xx at the corner)",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Free",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/stress/0"
|
||||
},
|
||||
"value": 0.0,
|
||||
"tolerance": {
|
||||
"absolute": 1e-08
|
||||
},
|
||||
"reason": "rounding only"
|
||||
},
|
||||
{
|
||||
"id": "held_stress",
|
||||
"quantity": "stress of the held block, σ_xx (hydrostatic)",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Held",
|
||||
"at": [
|
||||
[
|
||||
0.5,
|
||||
0.5,
|
||||
0.5
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/stress/0"
|
||||
},
|
||||
"value": -3.5108024691358017,
|
||||
"tolerance": {
|
||||
"relative": 1e-08
|
||||
},
|
||||
"reason": "a homogeneous state; E = 10, ν = 0.3 (κ = 8.3333), ϑ = 1.2"
|
||||
}
|
||||
]
|
||||
}
|
||||
@@ -0,0 +1,17 @@
|
||||
{
|
||||
"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
|
||||
"values": {
|
||||
"free_stress": {
|
||||
"band": 1e-8,
|
||||
"value": -1.0882345665298319e-14
|
||||
},
|
||||
"free_u": {
|
||||
"band": 2.0000000000000037e-10,
|
||||
"value": 0.20000000000000034
|
||||
},
|
||||
"held_stress": {
|
||||
"band": 3.510802469135803e-9,
|
||||
"value": -3.5108024691358026
|
||||
}
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,23 @@
|
||||
[
|
||||
{"command": "mesh.block", "params": {"body": "Cube", "kind": "hex20", "corners": [[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1]], "divisions": [2, 2, 2], "faces": {"6": "x0", "3": "y0", "1": "z0", "4": "x1", "5": "y1", "2": "z1"}}},
|
||||
{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Cube"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Cube", "face": "x0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "y0", "geometry": [{"body": "Cube", "face": "y0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "z0", "geometry": [{"body": "Cube", "face": "z0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x1", "geometry": [{"body": "Cube", "face": "x1"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "y1", "geometry": [{"body": "Cube", "face": "y1"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "z1", "geometry": [{"body": "Cube", "face": "z1"}]}},
|
||||
{"command": "model.material.create", "params": {"name": "Rubber", "elastic": {"type": "isotropic", "young": 10.0, "poisson": 0.3}, "hyperelastic": {"type": "neo_hookean"}, "thermal": {"expansion": 0.001, "reference_temperature": 20.0}}},
|
||||
{"command": "model.section.create", "params": {"name": "cube", "set": "all", "material": "Rubber", "kind": {"type": "solid"}}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sx0", "type": "symmetry", "target": {"set": "x0"}, "plane": "x"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sy0", "type": "symmetry", "target": {"set": "y0"}, "plane": "y"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sz0", "type": "symmetry", "target": {"set": "z0"}, "plane": "z"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sx1", "type": "symmetry", "target": {"set": "x1"}, "plane": "x"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sy1", "type": "symmetry", "target": {"set": "y1"}, "plane": "y"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sz1", "type": "symmetry", "target": {"set": "z1"}, "plane": "z"}},
|
||||
{"command": "model.load.create", "params": {"name": "heat", "type": "temperature", "target": {"set": "all"}, "value": 220.0}},
|
||||
{"command": "model.step.create", "params": {"name": "Free", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx0", "sy0", "sz0"], "loads": ["heat"]}},
|
||||
{"command": "model.step.create", "params": {"name": "Held", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx0", "sy0", "sz0", "sx1", "sy1", "sz1"], "loads": ["heat"]}},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Free"}, "id": "free"},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Held"}, "id": "held"}
|
||||
]
|
||||
@@ -0,0 +1,6 @@
|
||||
# Yeoh block in uniaxial tension
|
||||
|
||||
One hybrid hex20 unit cube on three symmetry planes, E = 3, ν = 0.49, Yeoh with C₁₀ = μ/2,
|
||||
C₂₀ = −0.02 and C₃₀ = 0.004, pulled to λ = 2 in a nonlinear static step. The reference is the
|
||||
compressible decoupled energy's axial Cauchy stress σ = (λ/J) ∂W/∂λ with the lateral stretch
|
||||
found by bisection so σ_yy = 0.
|
||||
@@ -0,0 +1,53 @@
|
||||
{
|
||||
"title": "Yeoh block in uniaxial tension",
|
||||
"kind": "theory",
|
||||
"source": "Uniaxial tension of the decoupled compressible Yeoh energy W = C₁₀ (Ī₁ − 3) + C₂₀ (Ī₁ − 3)² + C₃₀ (Ī₁ − 3)³ + κ/2 (J − 1)² (O. H. Yeoh, Rubber Chem. Technol. 66(5):754–771, 1993): σ = (λ/J) ∂W/∂λ with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.49, C₁₀ = μ/2, C₂₀ = −0.02, C₃₀ = 0.004)",
|
||||
"quantities": [
|
||||
{
|
||||
"id": "sigma",
|
||||
"quantity": "axial Cauchy stress, stretch λ = 2",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Uniaxial",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/stress/0"
|
||||
},
|
||||
"value": 3.174263919393508,
|
||||
"tolerance": {
|
||||
"relative": 1e-06
|
||||
},
|
||||
"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
|
||||
},
|
||||
{
|
||||
"id": "lateral",
|
||||
"quantity": "lateral displacement at (1, 1, 1): l − 1",
|
||||
"measure": {
|
||||
"command": "solve.probe",
|
||||
"params": {
|
||||
"step": "Uniaxial",
|
||||
"at": [
|
||||
[
|
||||
1,
|
||||
1,
|
||||
1
|
||||
]
|
||||
]
|
||||
},
|
||||
"pointer": "/nodes/0/displacement/1"
|
||||
},
|
||||
"value": -0.2854505757255106,
|
||||
"tolerance": {
|
||||
"relative": 1e-06
|
||||
},
|
||||
"reason": "a homogeneous deformation the hybrid hex20 represents exactly: the tolerance admits the Newton tolerance and rounding"
|
||||
}
|
||||
]
|
||||
}
|
||||
@@ -0,0 +1,13 @@
|
||||
{
|
||||
"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
|
||||
"values": {
|
||||
"lateral": {
|
||||
"band": 2.854505757255106e-10,
|
||||
"value": -0.285450575725269
|
||||
},
|
||||
"sigma": {
|
||||
"band": 3.174263919404631e-9,
|
||||
"value": 3.1742639194046305
|
||||
}
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,16 @@
|
||||
[
|
||||
{"command": "mesh.block", "params": {"body": "Cube", "kind": "hex20", "corners": [[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1]], "divisions": [1, 1, 1], "faces": {"6": "x0", "3": "y0", "1": "z0", "4": "x1", "5": "y1"}}},
|
||||
{"command": "model.set.create", "params": {"name": "all", "geometry": [{"body": "Cube"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Cube", "face": "x0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "y0", "geometry": [{"body": "Cube", "face": "y0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "z0", "geometry": [{"body": "Cube", "face": "z0"}]}},
|
||||
{"command": "model.set.create", "params": {"name": "x1", "geometry": [{"body": "Cube", "face": "x1"}]}},
|
||||
{"command": "model.material.create", "params": {"name": "Rubber", "elastic": {"type": "isotropic", "young": 3.0, "poisson": 0.49}, "hyperelastic": {"type": "yeoh", "c20": -0.02, "c30": 0.004}}},
|
||||
{"command": "model.section.create", "params": {"name": "cube", "set": "all", "material": "Rubber", "kind": {"type": "solid"}, "formulation": {"hybrid": true}}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sx", "type": "symmetry", "target": {"set": "x0"}, "plane": "x"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sy", "type": "symmetry", "target": {"set": "y0"}, "plane": "y"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "sz", "type": "symmetry", "target": {"set": "z0"}, "plane": "z"}},
|
||||
{"command": "model.boundary_condition.create", "params": {"name": "pull", "type": "displacement", "target": {"set": "x1"}, "values": [1.0, null, null, null, null, null]}},
|
||||
{"command": "model.step.create", "params": {"name": "Uniaxial", "procedure": {"type": "nonlinear_static"}, "boundary_conditions": ["sx", "sy", "sz", "pull"], "loads": []}},
|
||||
{"command": "solve.nonlinear", "params": {"step": "Uniaxial"}, "id": "solve"}
|
||||
]
|
||||
Reference in new issue
Block a user