{
  "abstract": "We study free-material optimization for nonlinear bond-based nonlocal diffusion and its localization as the interaction horizon $δ$ vanishes. Bond conductivities are subject to pointwise bounds and a global material budget constraint. After normalization, the asymptotic behavior of these problems is governed by the fraction $θ_δ$ of potentially active bonds that can be assigned the maximal conductivity. We identify three distinct limiting regimes. If $θ_δ\\to0$, the normalized optimal values converge to those of the classical vanishing-material optimal design problem, equivalently formulated as a Lipschitz-constrained state problem or a measure-valued flux problem. For $θ_δ\\toθ\\in(0,1)$, we obtain a directional local free-material problem governed by the upper tail of the directional $p$-strain over position-direction space. Thus the state variable localizes, while the limiting constitutive law retains an internal directional variable inherited from the bonds. If $θ_δ\\toθ\\geq1$, the limit is the isotropic local $p$-Dirichlet problem. A convex minimax argument eliminates the normalized conductivity variable exactly, after which the analysis combines convex duality, nonlocal compactness, and $Γ$-convergence for convex integrands. We prove a unified $Γ$-convergence result for the reduced state energies whenever $θ_δ\\toθ\\in[0,+\\infty)$, yielding convergence of normalized optimal values and compactness and identification of optimal states throughout the full parameter range. At the vanishing-material endpoint, we additionally recover optimal measure-valued local fluxes from optimal nonlocal two-point fluxes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19776",
  "author_total": 1,
  "authors": [
    {
      "name": "Anton Evgrafov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4jbc9v8dy278g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19776",
  "fulltext_url": null,
  "id": "4jbc9v8dy278g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19776",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2205|Topology Optimization in Engineering"
    ]
  ],
  "openalex": "W7222229339",
  "pdf_url": "https://arxiv.org/pdf/2610.19776",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4jbc9v8dy278g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19776",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19776"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Free-Material Optimization in a Peridynamic World",
  "title_text": "Free-Material Optimization in a Peridynamic World",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/4jbc9v8dy278g",
  "venue": "arXiv",
  "versions": []
}