Free-Material Optimization in a Peridynamic World
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 θδ of potentially active bonds that can be assigned the maximal conductivity. We identify three distinct limiting regimes. If θδ→0, 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 θδ→θ∈(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 θδ→θ≥1, 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 θδ→θ∈[0,+∞), 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.
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BibTeX
@misc{evgrafov2026freematerial,
title = {{Free-Material Optimization in a Peridynamic World}},
author = {Anton Evgrafov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19776},
eprint = {2610.19776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19776},
}
RIS
TY - UNPB
TI - Free-Material Optimization in a Peridynamic World
AU - Anton Evgrafov
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19776
UR - https://arxiv.org/abs/2610.19776
AB - 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→0, 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→θ∈(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→θ≥1, 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→θ∈[0,+∞), 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.
ER -