Seminario del 2025

Febbraio
26
2025
Sabrina Pellegrino
Spectral collocation methods for nonlocal perydynamic problems and applications
nell'ambito della serie: SCUBE
analisi numerica
Peridynamics is a nonlocal version of continuum mechanics theory able to incorporate singularities since it does not take into account spatial partial derivatives. As a consequence, it assumes long-range interactions among material particles and is able to describe the formation and the evolution of fractures. The discretization of such nonlocal model requires the use of raffinate numerical tools for approximating the solutions to the model. Due to the presence of a convolution product in the definition of the nonlocal operator, we propose a spectral collocation method based on the implementation of Fourier and Chebyshev polynomials to discretize the model. The choice can benefit of the FFT algorithm and allow us to deal efficiently with the imposition of non-periodic boundary conditions by a volume penalization technique. We prove the convergence of such methods in the framework of fractional Sobolev space and discuss numerically the stability of the scheme. We also investigate the qualitative aspects of the convolution kernel and of the nonlocality parameters by solving an inverse peridynamic problem by using a Physics-Informed Neural Network activated by suitable Radial Basis functions. Additionally, we propose a virtual element approach to obtain the solution of a nonlocal diffusion problem. The main feature of the proposed technique is that we are able to construct a nonlocal counterpart for the divergence operator in order to obtain a weak formulation of the peridynamic model and exploit the analogies with the known results in the context of Galerkin approximation. We prove the convergence of the proposed method and provide several simulations to validate our results. References: [1] Lopez, L., Pellegrino, S. F. (2021). A spectral method with volume penalization for a nonlinear peridynamic model International Journal for Numerical Methods in Engineering 122(3): 707–725. https://doi.org/10.1002/nme.6555 [2] Lopez, L., Pellegrino, S. F. (2022). A space-time discretization of a nonlinear peridynamic model on a 2D lamina Computers and Mathematics with Applications 116: 161–175. https://doi.org/10.1016/j.camwa.2021.07.0041 [3] Lopez, L., Pellegrino, S. F. (2022). A non-periodic Chebyshev spectral method avoiding penalization techniques for a class of nonlinear peridynamic models International Journal for Numerical Methods in Engineering 123(20): 4859–4876. https://doi.org/10.1002/nme.7058 [4] Difonzo, F. V., Lopez, L., Pellegrino, S. F. (2024). Physics informed neural networks for an inverse problem in peridynamic models Engineering with Computers. https://doi.org/10.1007/s00366-024-01957-5 [5] Difonzo, F. V., Lopez, L., Pellegrino, S. F. (2024). Physics informed neural networks for learning the horizon size in bond-based peridynamic models Computer Methods in Applied Mechanics and Engineering. https://doi.org/10.1016/j.cma.2024.117727
organizzato da: Davide Palitta e Valeria Simoncini

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