An optimization-based strategy for peridynamic-FEM coupling and for the prescription of nonlocal boundary conditions

D’Elia, Marta; Littlewood, David John; Trageser, Jeremy; Perego, Mauro; Bochev, Pavel B.

doi:10.48550/arxiv.2110.04420

Cited by 1 publication

(1 citation statement)

References 56 publications

(75 reference statements)

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“…Solving such a continuum model by well-established numerical methods would reduce the computational cost. Following this domain-decomposition strategy, several works that combine continuum and microscopic models have been proposed [18,37,38,47,58,59,74,76,85] to model multiscale systems. Another approach focuses on developing a fast surrogate as the fine-scale model represented by homogenization [7,9,22,82,90,96].…”

Section: Introductionmentioning

confidence: 99%

Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems

Yin¹,

Zhang²,

Yu³

et al. 2022

Preprint

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Multiscale modeling is an effective approach for investigating multiphysics systems with largely disparate size features, where models with different resolutions or heterogeneous descriptions are coupled together for predicting the system's response. The solver with lower fidelity (coarse) is responsible for simulating domains with homogeneous features, whereas the expensive high-fidelity (fine) model describes microscopic features with refined discretization, often making the overall cost prohibitively high, especially for timedependent problems. In this work, we explore the idea of multiscale modeling with machine learning and employ DeepONet, a neural operator, as an efficient surrogate of the expensive solver. DeepONet is trained offline using data acquired from the fine solver for learning the underlying and possibly unknown finescale dynamics. It is then coupled with standard PDE solvers for predicting the multiscale systems with new boundary/initial conditions in the coupling stage. The proposed framework significantly reduces the computational cost of multiscale simulations since the DeepONet inference cost is negligible, facilitating readily the incorporation of a plurality of interface conditions and coupling schemes. We present various benchmarks to assess accuracy and speedup, and in particular we develop a coupling algorithm for a timedependent problem, and we also demonstrate coupling of a continuum model (finite element methods, FEM) with a neural operator representation of a particle system (Smoothed Particle Hydrodynamics, SPH) for a uniaxial tension problem with hyperelastic material. What makes this approach unique is that a well-trained over-parametrized DeepONet can generalize well and make predictions at a negligible cost.

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