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verification: the tie patch test
An oracle case of uniaxial tension across a tie of non-matching meshes; the tie formulation in docs/fem.md and the literature survey; ROADMAP M7 and parity updated. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> 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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# Tie patch test (non-matching meshes)
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Two unit cubes stacked along z, meshed independently: 3 × 3 × 2 hex8 below (graded in x and y),
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2 × 2 × 2 above, so the nodes at z = 1 do not match. A tie bonds the upper block's bottom
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(secondary) to the lower block's top (main). The faces x = 0 and y = 0 are symmetry planes, the
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base z = 0 is held in z, and the top z = 2 is pulled up by 0.02 (E = 1000, ν = 0.3) in a linear
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static step. The exact state is the uniaxial stress σ_zz = 10, uniform in both blocks.
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{
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"title": "Tie patch test: two blocks with non-matching meshes tied together in uniaxial tension",
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"kind": "theory",
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"source": "The patch test for mesh tying: a tie across non-matching meshes must transmit a constant stress exactly; B. I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal. 38(3):989–1012, 2000; M. A. Puso, A 3D mortar method for solid mechanics, IJNME 59:315–336, 2004. Uniaxial stress between frictionless symmetry planes is uniform in both blocks",
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"quantities": [
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{
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"id": "szz_10",
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"quantity": "σ_zz of the lower block's corner element under the tie (element 10)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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10
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]
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},
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"pointer": "/elements/0/stress/0/2"
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},
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"value": 10.0,
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"tolerance": {
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"relative": 1e-09
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},
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"reason": "uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes"
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},
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{
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"id": "szz_14",
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"quantity": "σ_zz of the lower block's centre element under the tie (element 14)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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14
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]
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},
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"pointer": "/elements/0/stress/0/2"
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},
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"value": 10.0,
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"tolerance": {
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"relative": 1e-09
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},
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"reason": "uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes"
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},
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{
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"id": "szz_1",
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"quantity": "σ_zz of the lower block's bottom corner element (element 1)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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1
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]
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},
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"pointer": "/elements/0/stress/0/2"
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},
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"value": 10.0,
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"tolerance": {
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"relative": 1e-09
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},
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"reason": "uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes"
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},
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{
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"id": "szz_19",
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"quantity": "σ_zz of the upper block's corner element over the tie (element 19)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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19
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]
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},
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"pointer": "/elements/0/stress/0/2"
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},
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"value": 10.0,
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"tolerance": {
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"relative": 1e-09
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},
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"reason": "uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes"
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},
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{
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"id": "szz_26",
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"quantity": "σ_zz of the upper block's top far corner element (element 26)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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26
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]
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},
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"pointer": "/elements/0/stress/0/2"
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},
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"value": 10.0,
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"tolerance": {
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"relative": 1e-09
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},
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"reason": "uniaxial stress E ε = 1000 × 0.01 is the exact discrete solution: the dual mortar conditions hold for the linear field on both meshes"
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},
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{
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"id": "sxx_14",
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"quantity": "σ_xx of element 14 (uniaxial stress: 0)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"elements": [
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14
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]
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},
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"pointer": "/elements/0/stress/0/0"
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},
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"value": 0.0,
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"tolerance": {
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"absolute": 1e-09
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},
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"reason": "the tie must not hold the lateral contraction of either mesh"
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},
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{
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"id": "uz_interface",
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"quantity": "u_z at the tied secondary node (0.5, 0.5, 1)",
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"measure": {
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"command": "solve.probe",
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"params": {
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"step": "Pull",
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"at": [
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[
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0.5,
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0.5,
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1.0
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]
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]
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},
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"pointer": "/nodes/0/displacement/2"
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},
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"value": 0.01,
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"tolerance": {
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"relative": 1e-09
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},
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"reason": "the tied node follows the uniform strain field"
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}
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]
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}
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{
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"note": "FieldCraft's accepted results for this case, written by `cargo xtask oracle --bless`. A result outside its band fails CI.",
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"values": {
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"sxx_14": {
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"band": 1e-9,
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"value": -5.317400101732486e-15
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},
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"szz_1": {
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"band": 1.0000000000000018e-8,
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"value": 10.000000000000018
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},
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"szz_10": {
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"band": 1e-8,
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"value": 9.999999999999993
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},
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"szz_14": {
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"band": 1.0000000000000002e-8,
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"value": 10.000000000000002
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},
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"szz_19": {
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"band": 1.0000000000000017e-8,
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"value": 10.000000000000016
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},
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"szz_26": {
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"band": 1.000000000000001e-8,
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"value": 10.000000000000009
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},
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"uz_interface": {
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"band": 1.0000000000000001e-11,
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"value": 0.009999999999999997
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}
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}
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}
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[
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{"command": "mesh.block", "params": {"body": "Lower", "kind": "hex8", "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": [3, 3, 2], "faces": {"1": "bottom", "2": "top", "3": "y0", "6": "x0"}, "bias": [2.0, 0.6, 1.0]}},
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{"command": "mesh.block", "params": {"body": "Upper", "kind": "hex8", "corners": [[0, 0, 1], [1, 0, 1], [1, 1, 1], [0, 1, 1], [0, 0, 2], [1, 0, 2], [1, 1, 2], [0, 1, 2]], "divisions": [2, 2, 2], "faces": {"1": "bottom", "2": "top", "3": "y0", "6": "x0"}, "merge": 0}},
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{"command": "model.set.create", "params": {"name": "lower", "geometry": [{"body": "Lower"}]}},
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{"command": "model.set.create", "params": {"name": "upper", "geometry": [{"body": "Upper"}]}},
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{"command": "model.set.create", "params": {"name": "lower_top", "geometry": [{"body": "Lower", "face": "top"}]}},
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{"command": "model.set.create", "params": {"name": "upper_bottom", "geometry": [{"body": "Upper", "face": "bottom"}]}},
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{"command": "model.set.create", "params": {"name": "x0", "geometry": [{"body": "Lower", "face": "x0"}, {"body": "Upper", "face": "x0"}]}},
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{"command": "model.set.create", "params": {"name": "y0", "geometry": [{"body": "Lower", "face": "y0"}, {"body": "Upper", "face": "y0"}]}},
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{"command": "model.set.create", "params": {"name": "z0", "geometry": [{"body": "Lower", "face": "bottom"}]}},
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{"command": "model.set.create", "params": {"name": "top", "geometry": [{"body": "Upper", "face": "top"}]}},
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{"command": "model.material.create", "params": {"name": "Mat", "elastic": {"type": "isotropic", "young": 1000.0, "poisson": 0.3}}},
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{"command": "model.section.create", "params": {"name": "lower", "set": "lower", "material": "Mat", "kind": {"type": "solid"}}},
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{"command": "model.section.create", "params": {"name": "upper", "set": "upper", "material": "Mat", "kind": {"type": "solid"}}},
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{"command": "interaction.tie", "params": {"name": "glue", "main": "lower_top", "secondary": "upper_bottom"}},
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{"command": "model.boundary_condition.create", "params": {"name": "x0", "type": "displacement", "target": {"set": "x0"}, "values": [0, null, null, null, null, null]}},
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{"command": "model.boundary_condition.create", "params": {"name": "y0", "type": "displacement", "target": {"set": "y0"}, "values": [null, 0, null, null, null, null]}},
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{"command": "model.boundary_condition.create", "params": {"name": "z0", "type": "displacement", "target": {"set": "z0"}, "values": [null, null, 0, null, null, null]}},
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{"command": "model.boundary_condition.create", "params": {"name": "push", "type": "displacement", "target": {"set": "top"}, "values": [null, null, 0.02, null, null, null]}},
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{"command": "model.step.create", "params": {"name": "Pull", "procedure": {"type": "static"}, "boundary_conditions": ["x0", "y0", "z0", "push"], "loads": []}},
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{"command": "solve.static", "params": {"step": "Pull"}, "id": "pull"}
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]
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