Stochastic fiber dynamics in a spatially semi-discrete setting

TL;DR

This paper develops a rigorous mathematical framework for a semi-discrete stochastic model of fiber dynamics in turbulent flows, proving existence and uniqueness of solutions and enhancing numerical methods for simulation.

Contribution

It introduces a novel semi-discrete stochastic fiber model derived from a continuous beam model, with a proof of well-posedness and improved numerical solution techniques.

Findings

Proved existence and uniqueness of solutions for the model.

Developed an explicit Lagrange multiplier representation.

Enhanced numerical methods for time discretization.

Abstract

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.