L1 ⚛ L1-222 ⊙ Testnet

XFEM — extended finite element for cracks

Structural Mechanics · Enriched FE with discontinuous shape functions

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⬇ L1-222.md

# ⚛ L1 Principle — XFEM — extended finite element for cracks **ID:** `L1-222` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Structural Mechanics — *Enriched FE with discontinuous shape functions* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** crack field enriched > **📡 Carrier:** mechanical · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554012 --- ## 🧠 1. Introduction **XFEM — extended finite element for cracks** is a **linear inverse problem** whose unknown lives in **crack field enriched** space, within the **Enriched FE with discontinuous shape functions** sub-domain of **Structural Mechanics**. Measurements consist of mechanical vibrations or strain signals via a **dic** sensing mechanism. The forward operator applies, in order: L · level set operator; L · enrich shape operator; L · heaviside branch operator; pixel-level spatial averaging on the detector. Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds. ## ⚙ 2. Forward Model Physical chain: **x** → L · level set → L · enrich shape → L · heaviside branch → Spatial integration → **y** (detector). ``` y = ∫_A dA `L.heaviside_branch` `L.enrich_shape` `L.level_set` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.level_set` | L · level set operator | | `L.enrich_shape` | L · enrich shape operator | | `L.heaviside_branch` | L · heaviside branch operator | | `int.spatial` | Pixel-level spatial averaging on the detector | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Structural Mechanics | | Sub domain | Enriched FE with discontinuous shape functions | | Carrier | mechanical | | Problem class | linear_inverse | | Solution space | crack_field_enriched | | Noise model | gaussian | | Integration axis | spatial | | Difficulty delta | 5 | | L dag | 3.3 | ## 📡 4. Measurement Model Conditional stability; mismatch parameters dominate at Omega bounds. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `xfem` · **Forward operator:** `xfem_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | N dof | — | 50000 | | Snr db | dB | 30 | | N enrich | — | 100 | | Level set error | — | 0 | | Enrichment error | — | 0 | | Quadrature error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N dof | — | 1000 – 10000000 | | Snr db | dB | 0 – 40 | | N enrich | — | 1 – 10000 | | Level set error | — | 0.0 – 0.1 | | Enrichment error | — | 0.0 – 0.2 | | Quadrature error | — | 0.0 – 0.1 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 23.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `360` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xb8e9a27c8f2bf2d4002c0cb713f85d51e49ff2e64997a5ffa93d299e0399ca74` - **Chain tx hash:** `0xd50fd6f56d7e0d2049c2912fe935462b2bce8b0125d3288d2cc4b8456cc31487` - **Chain block:** `41554012` --- ## File Mapping This bundle consists of: `L1-222.md`, `L1-222.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-222.md` | Source of truth — edit this | Human or LLM | | `L1-222.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-222`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-222.md).
Regenerate L1-222.json so every field matches the Markdown.
Preserve the schema documented in the File Mapping footer at the bottom of the MD.
Output each file in its own fenced code block tagged with the filename.

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L2-222-001 XFEM — extended finite element for cracks Mismatch-Only Spec

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📋 JSON metadata

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  "artifact_id": "L1-222",
  "chain_block": 41554012,
  "chain_hash": "0xb8e9a27c8f2bf2d4002c0cb713f85d51e49ff2e64997a5ffa93d299e0399ca74",
  "chain_tx_hash": "0xd50fd6f56d7e0d2049c2912fe935462b2bce8b0125d3288d2cc4b8456cc31487",
  "domain": "Structural Mechanics",
  "hardness_fn": {
    "delta": 5,
    "kappa": 360,
    "metric": "PSNR_dB",
    "type": "epsilon_fn"
  },
  "initiator_dataset": [
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      "ipfs_cid": null,
      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
  ],
  "layer": "L1",
  "observable_profile": {
    "metric": "PSNR_dB",
    "regime": "Conditional stability; mismatch parameters dominate at Omega bounds.",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 3.3,
    "carrier": "mechanical",
    "difficulty_delta": 5,
    "domain": "Structural Mechanics",
    "integration_axis": "spatial",
    "noise_model": "gaussian",
    "primitives": [
      "L.level_set",
      "L.enrich_shape",
      "L.heaviside_branch",
      "int.spatial"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "dic",
    "solution_space": "crack_field_enriched",
    "sub_domain": "Enriched FE with discontinuous shape functions",
    "title": "XFEM \u2014 extended finite element for cracks"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "xfem_forward"
    ],
    "allowed_omega_dimensions": [
      "N_DOF",
      "N_enrich",
      "SNR_dB",
      "level_set_error",
      "enrichment_error",
      "quadrature_error",
      "level_set_error",
      "enrichment_error",
      "quadrature_error"
    ],
    "allowed_problem_classes": [
      "xfem"
    ],
    "center_spec": {
      "epsilon_fn_center": "23.0",
      "forward_operator": "xfem_forward",
      "input_format": "measurement_only",
      "omega": {
        "N_DOF": 50000,
        "N_enrich": 100,
        "SNR_dB": 30,
        "enrichment_error": 0.0,
        "level_set_error": 0.0,
        "quadrature_error": 0.0
      },
      "problem_class": "xfem"
    },
    "epsilon_bounds": {
      "psnr_db": [
        10.0,
        45.0
      ]
    },
    "omega_bounds": {
      "N_DOF": [
        1000,
        10000000
      ],
      "N_enrich": [
        1,
        10000
      ],
      "SNR_dB": [
        0,
        40
      ],
      "enrichment_error": [
        0.0,
        0.2
      ],
      "level_set_error": [
        0.0,
        0.1
      ],
      "quadrature_error": [
        0.0,
        0.1
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Enriched FE with discontinuous shape functions",
  "title": "XFEM \u2014 extended finite element for cracks"
}

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