A State-Space Framework for trivial and Topological Metamaterial Stochastic Analysis
Abstract
Topological phononic crystals and elastic metamaterials support edge states defined by global topological invariants, offering a route toward vibration-control and wave-guiding devices that remain functional in the presence of defects. However, manufacturing-induced spatial variability can perturb these invariants and compromise their robustness, making its quantification essential during design. In this work, we first demonstrate that a previously proposed linear time-varying (LTV) formulation is mathematically equivalent to the spectral element method based on transfer matrices for elementary rod, Saint-Venant shaft, and Euler-Bernoulli beam theories. The deterministic formulation is then extended to stochastic analyses through a stochastic linear time-varying (SLTV) framework. The proposed methodology combines Monte Carlo simulations with stochastic Fourier series and an analytical Karhunen--Loève expansion, providing closed-form stochastic fields and their derivatives required by the LTV formulation. The SLTV approach enables the computation of stochastic dispersion diagrams and forced responses of one-dimensional waveguides with arbitrarily varying geometry and mechanical properties. Because the LTV-based transition matrix isolates individual wavemodes without requiring the mode tracking needed by conventional eigenproblem-based formulations, the framework is particularly suited for evaluating topological invariants, specifically the Zak phase, and assessing the robustness of topological bands under spatial variability. Numerical results for rods, shafts, and Euler-Bernoulli beams demonstrate the applicability of the proposed framework as a unified methodology for deterministic and stochastic analyses of trivial and topological periodic waveguides under continuous spatial uncertainty.
Related
- Dual Topological Channels in a Programmable Mechanical Metamaterial
- Topological chiral edge modes in the continuum of a trivial bulk
- Classification of topological phases of matter in stochastic systems
- Tensor network representations for intrinsically mixed-state topological orders
- Emergent gravity from topological quantum field theory: Stochastic gradient flow perspective away from the quantum gravity problem
BibTeX
@misc{ribeiro2026statespace,
title = {{A State-Space Framework for trivial and Topological Metamaterial Stochastic Analysis}},
author = {Luiz Henrique Marra da Silva Ribeiro and Abhilash Sreekumar and Vinícius Fonseca Dal Poggetto and Vicent Romero-García and Dimitrios Chronopoulos and Carlos De Marqui Jr. and José Roberto de França Arruda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24808},
eprint = {2609.24808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24808},
}
RIS
TY - UNPB
TI - A State-Space Framework for trivial and Topological Metamaterial Stochastic Analysis
AU - Luiz Henrique Marra da Silva Ribeiro
AU - Abhilash Sreekumar
AU - Vinícius Fonseca Dal Poggetto
AU - Vicent Romero-García
AU - Dimitrios Chronopoulos
AU - Carlos De Marqui Jr.
AU - José Roberto de França Arruda
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24808
UR - https://arxiv.org/abs/2609.24808
AB - Topological phononic crystals and elastic metamaterials support edge states defined by global topological invariants, offering a route toward vibration-control and wave-guiding devices that remain functional in the presence of defects. However, manufacturing-induced spatial variability can perturb these invariants and compromise their robustness, making its quantification essential during design. In this work, we first demonstrate that a previously proposed linear time-varying (LTV) formulation is mathematically equivalent to the spectral element method based on transfer matrices for elementary rod, Saint-Venant shaft, and Euler-Bernoulli beam theories. The deterministic formulation is then extended to stochastic analyses through a stochastic linear time-varying (SLTV) framework. The proposed methodology combines Monte Carlo simulations with stochastic Fourier series and an analytical Karhunen--Loève expansion, providing closed-form stochastic fields and their derivatives required by the LTV formulation. The SLTV approach enables the computation of stochastic dispersion diagrams and forced responses of one-dimensional waveguides with arbitrarily varying geometry and mechanical properties. Because the LTV-based transition matrix isolates individual wavemodes without requiring the mode tracking needed by conventional eigenproblem-based formulations, the framework is particularly suited for evaluating topological invariants, specifically the Zak phase, and assessing the robustness of topological bands under spatial variability. Numerical results for rods, shafts, and Euler-Bernoulli beams demonstrate the applicability of the proposed framework as a unified methodology for deterministic and stochastic analyses of trivial and topological periodic waveguides under continuous spatial uncertainty.
ER -