L1 ⚛ L1-498 ⊙ Testnet

Topology Optimization (SIMP)

Optimization · Structural topology

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⬇ L1-498.md

# ⚛ L1 Principle — Topology Optimization (SIMP) **ID:** `L1-498` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Optimization — *Structural topology* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 2D density field > **📡 Carrier:** N/A · **🌫 Noise:** deterministic > **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555317 --- ## 🧠 1. Introduction **Topology Optimization (SIMP)** is a **nonlinear inverse problem** whose unknown lives in **2D density field** space, within the **Structural topology** sub-domain of **Optimization**. Measurements consist of N/A via a **compliance minimization** sensing mechanism. The forward operator applies, in order: F · filter · density operator; S · simp · penalty operator; an unspecified linear measurement operator; O · compliance · gradient operator. Observations are corrupted by no stochastic noise (deterministic measurement). Existence of the recovered 2D_density_field is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 200); load_direction_error dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor. ## ⚙ 2. Forward Model Physical chain: **x** → S · simp · penalty → Generic linear operator → O · compliance · gradient → **y** (detector). ``` y = `O.compliance.gradient` A `S.simp.penalty` x (deterministic) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `S.simp.penalty` | S · simp · penalty operator | | `L.linear_op` | An unspecified linear measurement operator | | `O.compliance.gradient` | O · compliance · gradient operator | **🛰 Estimator components** _(used inside the solver, not in the forward equation)_: | Primitive | What it does | |---|---| | `F.filter.density` | F · filter · density operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Optimization | | Sub domain | Structural topology | | Carrier | N/A | | Problem class | nonlinear_inverse | | Solution space | 2D_density_field | | Noise model | deterministic | | Integration axis | spatial_domain | | Difficulty delta | 3 | | L dag | 3.5 | ## 📡 4. Measurement Model Existence of the recovered 2D_density_field is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 200); load_direction_error dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor. | Metric | Value | |---|---| | Metric | compliance_ratio | | Secondary | volume_fraction_error | ## 📏 5. Operating Range (Ω) **Center problem class:** `nonlinear_inverse` · **Forward operator:** `compliance_minimization` **Center point:** | Parameter | Unit | Value | |---|---|---| | N e | — | 2500 | | V f | — | 0.5 | | Filter r | — | 2 | | Penalty p | — | 3 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | N e | — | 100 – 10000 | | V f | — | 0.3 – 0.7 | | Filter r | — | 1.5 – 5.0 | | Penalty p | — | 1 – 5 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 0.45 compliance_ratio | Metric | Range | |---|---| | Compliance ratio | 0.2 – 0.8 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **compliance_ratio**, with κ = `10000.0` and δ = `3`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xca78b708c444f5e5390117508c96c169c4927e40fa5e7ecb4425c93ddb40fde0` - **Chain tx hash:** `0x73b7e7539603f461ddbf2820c5f257a6a1f9adbafbdd7726e819292f1f29b656` - **Chain block:** `41555317` --- ## File Mapping This bundle consists of: `L1-498.md`, `L1-498.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-498.md` | Source of truth — edit this | Human or LLM | | `L1-498.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 `L1-498`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-498.md).
Regenerate L1-498.json so every field matches the Markdown.
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L2-498-001 Topology Optimization (SIMP) — Nominal + Mismatch Specs

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📋 JSON metadata

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  "chain_block": 41555317,
  "chain_hash": "0xca78b708c444f5e5390117508c96c169c4927e40fa5e7ecb4425c93ddb40fde0",
  "chain_tx_hash": "0x73b7e7539603f461ddbf2820c5f257a6a1f9adbafbdd7726e819292f1f29b656",
  "domain": "Optimization",
  "hardness_fn": {
    "delta": 3,
    "kappa": 10000.0,
    "metric": "compliance_ratio",
    "type": "epsilon_fn"
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      "license_hash": null,
      "name": "primary",
      "weight": 1.0
    }
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    "metric": "compliance_ratio",
    "regime": "Existence of the recovered 2D_density_field is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 200); load_direction_error dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Deterministic sets the irreducible data-fidelity floor.",
    "secondary": "volume_fraction_error"
  },
  "physics_fingerprint": {
    "L_DAG": 3.5,
    "carrier": "N/A",
    "difficulty_delta": 3,
    "domain": "Optimization",
    "integration_axis": "spatial_domain",
    "noise_model": "deterministic",
    "primitives": [
      "F.filter.density",
      "S.simp.penalty",
      "L.linear_op",
      "O.compliance.gradient"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "compliance_minimization",
    "solution_space": "2D_density_field",
    "sub_domain": "Structural topology",
    "title": "Topology Optimization (SIMP)"
  },
  "size_tiers": {
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      "compliance_minimization"
    ],
    "allowed_omega_dimensions": [
      "N_e",
      "V_f",
      "penalty_p",
      "filter_r"
    ],
    "allowed_problem_classes": [
      "nonlinear_inverse"
    ],
    "center_spec": {
      "epsilon_fn_center": "0.45 compliance_ratio",
      "forward_operator": "compliance_minimization",
      "input_format": "measurement_only",
      "omega": {
        "N_e": 2500,
        "V_f": 0.5,
        "filter_r": 2.0,
        "penalty_p": 3.0
      },
      "problem_class": "nonlinear_inverse"
    },
    "epsilon_bounds": {
      "compliance_ratio": [
        0.2,
        0.8
      ]
    },
    "omega_bounds": {
      "N_e": [
        100,
        10000
      ],
      "V_f": [
        0.3,
        0.7
      ],
      "filter_r": [
        1.5,
        5.0
      ],
      "penalty_p": [
        1,
        5
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Structural topology",
  "title": "Topology Optimization (SIMP)"
}

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