Spectrally stable dispersion matching for peridynamic approximation of periodic atomistic chains

Peridynamics and atomistic lattice models both describe motion through nonlocal interactions, but a generic peridynamic kernel does not necessarily reproduce atomistic dispersion. This article constructs a spectrally stable, dispersion-matched peridynamic approximation for one-dimensional periodic chains with finite-range harmonic interactions. The finite-horizon kernel is treated as a design variable rather than prescribed as a constant micromodulus. Using the analytical Fourier symbol of the prescribed finite-range atomistic chain as the target, exact moment constraints fix the low-frequency behavior. The remaining kernel coefficients are fitted on a selected mesoscopic wavenumber band. Spectral nonnegativity is imposed independently over the full atomistic Brillouin zone, permitting sign-changing kernels while retaining linear energy stability. The analysis establishes consistency and energy estimates, a grid-based sufficient condition for continuous spectral positivity, and an L 2 dynamic error bound controlled by the resolved symbol mismatch. For a representative two-neighbor chain, the fitted kernel reduces both relative symbol error and wave-packet error by more than one order of magnitude compared with constant-kernel calibration. Parameter and spectrum sweeps quantify the effects of the fitted band, horizon, atomistic stiffnesses, and number of radial shells. A positivity ablation further shows that a small in-band fitting error can coexist with unstable modes outside the fitted band. Thus, within the linear periodic setting, a finite-horizon continuum operator can reproduce selected atomistic wave dynamics without copying the individual atomistic bonds.