# ⚛  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`

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## 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._