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solver: mortar contact in 2D models
Plane strain, plane stress and axisymmetric models take contact pairs and general (self) contact on the linear edges of tri3 and quad4 elements: edges oriented out of their elements, the edge mortar of fem with the thickness or 2 pi r in the weights, the element's area over the edge's length as the thickness under it, and the in-plane rows of the tangent blocks. Verified by the contact patch test in plane strain and axisymmetric (to 1e-9 small sliding, 1e-4 finite), Hertz's line contact on quad4 edges (Johnson 1985, 4.3; equal to the hex8 model's to 1e-5), and a hybrid rubber C-ring closing on itself that moves node for node as the one-layer solid ring. Co-Authored-By: Claude Opus 5.5 <[email protected]> Claude-Session: https://claude.ai/code/session_01UhhZMBTviDQxZvwnyP89fo
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jyvikoandClaude Opus 5.5 authored and Irakli Kourtis committed
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@@ -468,8 +468,20 @@ contact patch test. General (single-surface) contact uses one face list, a set's
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exterior face of the solids, as both sides: each face against every other one facing it that
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shares no node with its element (its neighbours) and that it does not reach through more than
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half an element (the far side of a thin wall), paired both ways, so the folds of one body press
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on each other symmetrically. Linear faces (tri3, quad4: tet4, wedge6, hex8); quadratic faces, 2D
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models and arc-length steps are refused by name.
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on each other symmetrically. Linear faces (tri3, quad4: tet4, wedge6, hex8), and in 2D models
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(plane strain, plane stress, axisymmetric) the linear edges of tri3 and quad4 elements (sets'
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edges; general contact defaults to every exterior edge); quadratic faces and edges and
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arc-length steps are refused by name.
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**2D models**: each edge is oriented with its normal (t_y, −t_x) out of its element; the main
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edge is projected along the secondary edge's normal onto its line, which maps main points to the
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secondary parameter linearly, so the overlap is an interval integrated with three-point
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Gauss–Legendre (exact for the degree-3 Φ_a N_b r of the axisymmetric patch test). The
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out-of-plane measure (the thickness, or 2πr at the secondary point) is in the weights, so A_A,
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g̃_A and the forces are totals over the slab or the ring, as the 2D elements' are (B. Yang,
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T. A. Laursen, X. Meng, IJNME 62:1183–1225, 2005). Points keep z = 0 and the same 24 variables
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as faces, and the tangent blocks keep their in-plane rows. The element's thickness under an edge
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is its area over the edge's length.
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**Segmentation** (`fieldcraft_fem::mortar`): both faces are projected along the secondary
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face's unit normal onto its plane, the main polygon is clipped by the secondary one
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@@ -540,7 +552,9 @@ entries, and the results display draws `contact_pressure`, `contact_gap`, `conta
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`contact_shear` and `contact_slip`.
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**Verification** (`crates/solver/tests/contact.rs`, `tests/oracle/nl-contact-patch`,
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`tests/oracle/nl-hertz-cylinder`): the contact patch test on non-matching hex8 and tet4 meshes
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`tests/oracle/nl-hertz-cylinder`, and their 2D forms `nl-contact-patch-plane-strain`,
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`nl-contact-patch-axisymmetric`, `nl-hertz-cylinder-plane-strain`): the contact patch test on
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non-matching hex8, tet4 and (plane strain and axisymmetric) quad4 meshes
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(uniform σ_zz, pressure equal to it, no lateral stress: to 10⁻⁹ with small sliding, 10⁻⁴ with
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finite sliding, where the gaps the augmentation leaves enter through their geometric terms);
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Hertz's line contact, plane strain (peak pressure within 0.3%, the pressure at two nodes within
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@@ -550,7 +564,10 @@ Newton's quadratic rate near the solution with finite sliding; Coulomb friction
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blocks (stick well below the cone with the forces of glued blocks, slip on it carrying μ N to 0.1%
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with small sliding and 3% with finite sliding, where the block turns a little, and μ_k N once
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slipping; `tests/oracle/nl-friction-sliding-blocks`); initial overclosure adjusted and
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refused; bit-for-bit results on one and four threads; the refusals.
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refused; bit-for-bit results on one and four threads; the refusals. In 2D: the patch tests pass
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to 10⁻⁹ (small sliding) and 10⁻⁴ (finite), Hertz's line contact on quad4 edges matches the
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hex8 model to 10⁻⁵, and a hybrid Mooney–Rivlin C-ring closing on itself in plane strain moves as
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the one-layer solid ring with z held, node for node to 10⁻⁵.
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## Ties (non-matching meshes)
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@@ -2,7 +2,7 @@
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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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**232 of 238 quantities pass** (6 known failure(s)) in 66 active cases: 57 theory, 9 published benchmark, 0 experiment; 10 pending cases wait for capabilities still being built.
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**270 of 276 quantities pass** (6 known failure(s)) in 70 active cases: 61 theory, 9 published benchmark, 0 experiment; 10 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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@@ -95,6 +95,36 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
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| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: the contact pressure at the secondary node (0.5, 0.5, 1) over −σ_zz of element 19 | 1 | 0.99998 | -0.00172% | 0.02% | pass | matches |
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| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Finite: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 0.02% | pass | matches |
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| [nl-contact-patch](../tests/oracle/nl-contact-patch/) | Small: σ_zz of element 19 (uniaxial stress E ε = −10 for the 1% squeeze) | -10 | -9.9996 | 0.00426% | 0.01% | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Small: σ_yy of element 4 over that of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Small: σ_yy of element 6 over that of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Small: σ_yy of element 1 over that of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Small: σ_yy of element 10 over that of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Small: σ_xx of element 6 over its σ_yy (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-6 | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Small: the contact pressure at the secondary node (1.5, 1) over −σ_yy of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Small: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Finite: σ_yy of element 4 over that of element 7 | 1 | 1 | 0.0000126% | 0.02% | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Finite: σ_yy of element 6 over that of element 7 | 1 | 0.99997 | -0.00325% | 0.02% | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Finite: σ_yy of element 1 over that of element 7 | 1 | 0.99999 | -0.00149% | 0.02% | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Finite: σ_yy of element 10 over that of element 7 | 1 | 0.99998 | -0.00185% | 0.02% | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Finite: σ_xx of element 6 over its σ_yy (uniaxial stress: 0) | 0 | -6.2023e-6 | -6.2e-6 | ± 2.0000e-4 | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Finite: the contact pressure at the secondary node (1.5, 1) over −σ_yy of element 7 | 1 | 0.99997 | -0.00285% | 0.02% | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Finite: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 0.02% | pass | matches |
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| [nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/) | Small: σ_yy of element 7 (uniaxial stress E ε = −10 for the 1% squeeze) | -10 | -9.9996 | 0.00426% | 0.01% | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Small: σ_yy of element 4 over that of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Small: σ_yy of element 6 over that of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Small: σ_yy of element 1 over that of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Small: σ_yy of element 10 over that of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Small: σ_xx of element 6 over its σ_yy (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 1.0000e-6 | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Small: the contact pressure at the secondary node (0.5, 1) over −σ_yy of element 7 | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Small: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 1e-6 rel. | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Finite: σ_yy of element 4 over that of element 7 | 1 | 1 | 0.0000708% | 0.02% | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Finite: σ_yy of element 6 over that of element 7 | 1 | 0.99999 | -0.00087% | 0.02% | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Finite: σ_yy of element 1 over that of element 7 | 1 | 1 | -0.000312% | 0.02% | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Finite: σ_yy of element 10 over that of element 7 | 1 | 1 | -0.000428% | 0.02% | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Finite: σ_xx of element 6 over its σ_yy (uniaxial stress: 0) | 0 | ≈ 0 | ≈ 0 | ± 2.0000e-4 | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Finite: the contact pressure at the secondary node (0.5, 1) over −σ_yy of element 7 | 1 | 0.99999 | -0.00075% | 0.02% | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Finite: the integrated contact pressure over the reaction at the pushed top | 1 | 1 | < 1e-7% | 0.02% | pass | matches |
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| [nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/) | Small: σ_yy of element 7 (plane strain uniaxial stress E ε/(1 − ν²) = −10.989 for the 1% squeeze) | -10.989 | -10.988 | 0.00516% | 0.01% | pass | matches |
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| [nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/) | radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 | 5 | 5.004 | 0.0793% | 0.5% | pass | matches |
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| [nl-cylinder-inflation](../tests/oracle/nl-cylinder-inflation/) | radial displacement of the outer wall (20, 0, 0) | 2.9129 | 2.9173 | 0.152% | 0.5% | pass | matches |
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| [nl-elastica-cantilever](../tests/oracle/nl-elastica-cantilever/) | tip deflection δ_v/L at P L²/(E I) = 1 | 0.30172 | 0.30226 | 0.179% | 1% | pass | matches |
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@@ -114,6 +144,12 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
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| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.36701 (node 8) | 1959 | 2078.4 | 6.09% | 10% | pass | matches |
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| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the contact pressure at the secondary node at x = 0.63325 (node 11, beyond the half-width b = 0.5) | 0 | ≈ 0 | ≈ 0 | ± 1 | pass | matches |
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| [nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/) | the vertical stress σ_yy under the centre of contact at depth 0.47127 (node 713) | -2099.2 | -2059.8 | 1.87% | 4% | pass | matches |
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| [nl-hertz-cylinder-plane-strain](../tests/oracle/nl-hertz-cylinder-plane-strain/) | the largest contact pressure p₀ (at x = 0) | 2884.6 | 2892 | 0.255% | 2% | pass | matches |
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| [nl-hertz-cylinder-plane-strain](../tests/oracle/nl-hertz-cylinder-plane-strain/) | the force across the contact (half model, 1 mm thick) | 1132.8 | 1133 | 0.0228% | 0.2% | pass | matches |
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| [nl-hertz-cylinder-plane-strain](../tests/oracle/nl-hertz-cylinder-plane-strain/) | the contact pressure at the secondary node at x = 0.29594 | 2325.1 | 2257.9 | -2.89% | 5% | pass | matches |
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| [nl-hertz-cylinder-plane-strain](../tests/oracle/nl-hertz-cylinder-plane-strain/) | the contact pressure at the secondary node at x = 0.36701 | 1959 | 2078.4 | 6.09% | 10% | pass | matches |
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| [nl-hertz-cylinder-plane-strain](../tests/oracle/nl-hertz-cylinder-plane-strain/) | the contact pressure at the secondary node at x = 0.63325 (beyond the half-width b = 0.5) | 0 | ≈ 0 | ≈ 0 | ± 1 | pass | matches |
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| [nl-hertz-cylinder-plane-strain](../tests/oracle/nl-hertz-cylinder-plane-strain/) | the vertical stress σ_yy under the centre of contact at depth 0.47127 | -2099.2 | -2059.8 | 1.87% | 4% | pass | matches |
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| [nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/) | axial Cauchy stress, stretch λ = 2, ν = 0.4999 | 3.4988 | 3.4988 | 0.000000135% | 1e-6 rel. | pass | matches |
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| [nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/) | lateral displacement at (1, 1, 1): l − 1 | -0.29281 | -0.29281 | < 1e-7% | 1e-6 rel. | 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 | matches |
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@@ -125,6 +161,8 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
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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 | matches |
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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 | matches |
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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 | matches |
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| [nl-mooney-rivlin-tube-axisymmetric](../tests/oracle/nl-mooney-rivlin-tube-axisymmetric/) | radial displacement of the bore (10, 0, 0) at the pressure that inflates it to a = 15 at λ_z = 1.5 | 5 | 5.0329 | 0.658% | 1% | pass | matches |
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| [nl-mooney-rivlin-tube-axisymmetric](../tests/oracle/nl-mooney-rivlin-tube-axisymmetric/) | radial displacement of the outer wall (20, 0, 0): b − B, a small difference of radii | 0.61553 | 0.64473 | 2.9e-2 | ± 0.04 | pass | matches |
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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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@@ -291,14 +329,18 @@ Generated by `cargo xtask oracle` from [`tests/oracle/`](../tests/oracle/); the
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- **[nafems-le1-plane-stress](../tests/oracle/nafems-le1-plane-stress/)** (benchmark): NAFEMS LE1 elliptic membrane with 2D plane stress elements (quad8). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE1 (target σ_yy = 92.7 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE1: Plane stress elements—elliptic membrane'. Tolerances — syy_D: the target is published to three figures; the stress extrapolated to the edge on a converged mesh adds a few tenths of a percent
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- **[nafems-le10](../tests/oracle/nafems-le10/)** (benchmark): NAFEMS LE10 thick plate under pressure (hex20). Source: NAFEMS, The Standard NAFEMS Benchmarks, TNSB Rev. 3, October 1990, test LE10 (target σ_yy = −5.38 MPa at D); as reproduced in the Abaqus 2017 Benchmarks Guide, 'LE10: Thick plate under pressure'. Tolerances — syy_D: the target is published to three figures; at D the stress is extrapolated to a corner where the hole edge meets the loaded face, and published converged solutions scatter by 1–2% (Abaqus Benchmarks Guide fine meshes: −5.51 to −5.89)
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- **[nl-contact-patch](../tests/oracle/nl-contact-patch/)** (theory): Contact patch test: two blocks with non-matching meshes in uniaxial compression (implicit mortar contact). Source: The contact patch test: a contact discretisation must transmit a uniform pressure across a non-matching interface exactly; Taylor & Papadopoulos, On a patch test for contact problems in two dimensions, Nonlinear Computational Mechanics, Springer 1991, 690–702; Crisfield, Re-visiting the contact patch test, IJNME 48:435–449, 2000; passed by mortar methods: Puso & Laursen, CMAME 193:601–629, 2004. Uniaxial stress between frictionless symmetry planes is uniform in both blocks. Tolerances — small_lower_corner: small sliding is linear and its mortar integrals exact on these rectangular faces: a uniform stress is the discrete solution up to the convergence tolerance (10⁻¹⁰); small_lower_centre: as for the corner element; small_lower_base: as for the corner element; small_upper_top: as for the corner element; small_lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them; small_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; small_force: the pressure carries the whole load across the interface; finite_lower_corner: finite sliding is exact up to the geometric terms of the gaps the augmentation leaves (10⁻⁵ of the element thickness) and the default convergence tolerances; finite_lower_centre: as for the corner element; finite_lower_base: as for the corner element; finite_upper_top: as for the corner element; finite_lateral: frictionless symmetry planes and a frictionless interface leave the blocks free to expand sideways; the non-matching interface must not hold them; finite_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; finite_force: the pressure carries the whole load across the interface; small_stress: the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness, a few 10⁻⁵ of the squeeze)
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- **[nl-contact-patch-axisymmetric](../tests/oracle/nl-contact-patch-axisymmetric/)** (theory): Contact patch test, axisymmetric: two rings with non-matching edges in axial compression (implicit mortar contact). Source: The contact patch test: a contact discretisation must transmit a uniform pressure across a non-matching interface exactly; Taylor & Papadopoulos, On a patch test for contact problems in two dimensions, Nonlinear Computational Mechanics, Springer 1991, 690–702; passed by 2D mortar methods: Yang, Laursen & Meng, IJNME 62:1183–1225, 2005. Uniaxial stress between frictionless supports is uniform in both blocks. Tolerances — small_lower_corner: small sliding is linear and its edge mortar integrals exact (degree-3 Gauss on the overlaps): the uniform stress is the discrete solution up to the convergence tolerance; small_lower_far: as for the corner element; small_lower_base: as for the corner element; small_upper_top: as for the corner element; small_lateral: a frictionless interface (and symmetry plane) leaves the blocks free to expand sideways; the non-matching interface must not hold them; small_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; small_force: the pressure carries the whole load across the interface (both are totals over the circumference); finite_lower_corner: finite sliding is exact up to the geometric terms of the gaps the augmentation leaves (10⁻⁵ of the element thickness) and the default convergence tolerances; finite_lower_far: as for the corner element; finite_lower_base: as for the corner element; finite_upper_top: as for the corner element; finite_lateral: a frictionless interface (and symmetry plane) leaves the blocks free to expand sideways; the non-matching interface must not hold them; finite_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; finite_force: the pressure carries the whole load across the interface (both are totals over the circumference); small_stress: the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness)
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- **[nl-contact-patch-plane-strain](../tests/oracle/nl-contact-patch-plane-strain/)** (theory): Contact patch test in plane strain: two blocks with non-matching edges in uniaxial compression (implicit mortar contact). Source: The contact patch test: a contact discretisation must transmit a uniform pressure across a non-matching interface exactly; Taylor & Papadopoulos, On a patch test for contact problems in two dimensions, Nonlinear Computational Mechanics, Springer 1991, 690–702; passed by 2D mortar methods: Yang, Laursen & Meng, IJNME 62:1183–1225, 2005. Uniaxial stress between frictionless supports is uniform in both blocks. Tolerances — small_lower_corner: small sliding is linear and its edge mortar integrals exact (degree-3 Gauss on the overlaps): the uniform stress is the discrete solution up to the convergence tolerance; small_lower_far: as for the corner element; small_lower_base: as for the corner element; small_upper_top: as for the corner element; small_lateral: a frictionless interface (and symmetry plane) leaves the blocks free to expand sideways; the non-matching interface must not hold them; small_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; small_force: the pressure carries the whole load across the interface (both are totals over the thickness); finite_lower_corner: finite sliding is exact up to the geometric terms of the gaps the augmentation leaves (10⁻⁵ of the element thickness) and the default convergence tolerances; finite_lower_far: as for the corner element; finite_lower_base: as for the corner element; finite_upper_top: as for the corner element; finite_lateral: a frictionless interface (and symmetry plane) leaves the blocks free to expand sideways; the non-matching interface must not hold them; finite_pressure: the mortar multipliers are the nodal values of the contact pressure, uniform and equal to the stress across the interface; finite_force: the pressure carries the whole load across the interface (both are totals over the thickness); small_stress: the squeeze less the penetration the augmented Lagrangian leaves (up to 10⁻⁵ of the element thickness)
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- **[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
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- **[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
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- **[nl-friction-sliding-blocks](../tests/oracle/nl-friction-sliding-blocks/)** (theory): Coulomb friction: a block pressed on another and slid (implicit mortar contact). Source: Coulomb's law of dry friction: a sliding interface carries μ_k times the normal force against the slip (C. A. Coulomb, Théorie des machines simples, Mém. Math. Phys. Acad. Sci. 10:161–332, 1785; P. Wriggers, Computational Contact Mechanics, 2nd ed., Springer 2006, §5.2). Tolerances — kinetic: once the interface slips its friction is μ_k N exactly (the return map puts every slipping node's traction on the cone); the force opposes the upper block's motion; balance: the friction carries the whole sideways load; slipping: a sideways demand far beyond μ N slips the whole interface
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- **[nl-hertz-cylinder](../tests/oracle/nl-hertz-cylinder/)** (theory): Hertz line contact: a cylinder pressed on a flat (implicit mortar contact, plane strain). Source: H. Hertz, Über die Berührung fester elastischer Körper, J. reine angew. Math. 92:156–171, 1882; K. L. Johnson, Contact Mechanics, Cambridge University Press 1985, §4.3 (two-dimensional contact of cylindrical bodies, eqs. 4.45–4.49): b = √(4PR/(πE*)), p₀ = 2P/(πb), with 1/E* = 2(1 − ν²)/E for two bodies of one material and R the cylinder's radius on a flat. Tolerances — peak_pressure: p₀ = 2P/(πb) for the load per length P = 2265.57 N/mm, b = 0.5 mm; the contact width spans seven elements of the graded mesh; normal_force: the pressure carries the whole load; the soft springs that hold the indenter before contact take a millionth; pressure_x0296: p₀ √(1 − x²/b²) at the node; the nodal pressure (the node's normal force over the area it carries) averages the distribution over the node's elements; pressure_x0367: as at node 7, nearer the edge, where the elements round off the square-root fall of the pressure to zero at b; pressure_x0633: outside the contact the surfaces are apart: no pressure (an element beyond b, since the edge element carries part of the last closed node's pressure); syy_depth: σ_z = −p₀ / √(1 + z²/b²) on the axis of a half-plane (Johnson 1985, eq. 4.49); the foundation is ten half-widths wide and deep, not a half-plane, and the stress is the nodal average of the elements around the node
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- **[nl-hertz-cylinder-plane-strain](../tests/oracle/nl-hertz-cylinder-plane-strain/)** (theory): Hertz line contact: a cylinder pressed on a flat (implicit mortar contact of 2D edges, plane strain). Source: H. Hertz, Über die Berührung fester elastischer Körper, J. reine angew. Math. 92:156–171, 1882; K. L. Johnson, Contact Mechanics, Cambridge University Press 1985, §4.3 (two-dimensional contact of cylindrical bodies, eqs. 4.45–4.49): b = √(4PR/(πE*)), p₀ = 2P/(πb), with 1/E* = 2(1 − ν²)/E for two bodies of one material and R the cylinder's radius on a flat. Tolerances — peak_pressure: p₀ = 2P/(πb) for the load per length P = 2265.57 N/mm, b = 0.5 mm; the contact width spans seven elements of the graded mesh; normal_force: the pressure carries the whole load; the soft springs that hold the indenter before contact take a millionth; pressure_x0296: p₀ √(1 − x²/b²) at the node; the nodal pressure (the node's normal force over the area it carries) averages the distribution over the node's elements; pressure_x0367: as at node 7, nearer the edge, where the elements round off the square-root fall of the pressure to zero at b; pressure_x0633: outside the contact the surfaces are apart: no pressure (an element beyond b, since the edge element carries part of the last closed node's pressure); syy_depth: σ_z = −p₀ / √(1 + z²/b²) on the axis of a half-plane (Johnson 1985, eq. 4.49); the foundation is ten half-widths wide and deep, not a half-plane, and the stress is the nodal average of the elements around the node
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- **[nl-hybrid-incompressible](../tests/oracle/nl-hybrid-incompressible/)** (theory): Nearly incompressible hybrid block under prescribed stretch. Source: Uniaxial stress of the decoupled compressible neo-Hookean energy W = μ/2 (Ī₁ − 3) + κ/2 (J − 1)² (P. J. Flory, Trans. Faraday Soc. 57:829–838, 1961): σ = μ J^(−5/3) (λ² − I₁/3) + κ (J − 1) with the lateral stretch found so σ_yy = 0 (bisection; E = 3, ν = 0.4999: μ = 1.0000667, κ = 5000). Tolerances — sigma: a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding; lateral: a homogeneous deformation every hybrid hex8 represents exactly; the 27-element mesh makes the global mixed tangent indefinite in the first iterations (LDLᵀ): the tolerance admits the Newton tolerance and rounding
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- **[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
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- **[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
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- **[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
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- **[nl-mooney-rivlin-tube-axisymmetric](../tests/oracle/nl-mooney-rivlin-tube-axisymmetric/)** (theory): Inflation of a stretched Mooney–Rivlin tube (axisymmetric hybrid quad4). 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
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- **[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
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- **[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
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- **[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
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