Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting

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

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

“…The following result concerning the global strong solvability of SDAEs of the type (1.1) is a slight generalization of [23,Theorem 3.1]. Compared to [23] we consider weaker assumptions on the drift coefficient a, a more general class of constraint functions g, M is not assumed to be the identity matrix, and finite moments of all orders are established. Theorem 2.6 (Existence and uniqueness).…”

“…The proof of Theorem 2.6 is mostly analogous to that of [23,Theorem 3.1]. A sketch of the key steps of the proof and the differences to [23] can be found in Appendix A.…”

“…The precise assumptions are stated in Section 2.1. By a slight generalization of the existence result in [23], we know that for all initial conditions r 0 ∈ M, v 0 ∈ T r0 M there exists a unique global strong solution (r, v, µ) to the SDAE (1.1), see Section 2.2 for details.…”

“…The process µ = (µ(t)) t 0 is an m-dimensional semimartingale, implicitly given as the Lagrange multiplier to the holonomic constraint g(r(t)) = 0. It thus determines the constraint force, which, according to d'Alembert's principle, acts orthogonal to the tangent space T x M = {y ∈ R n : Dg(x)y = 0} at the current position x ∈ M. Constrained SDEs of the form (1.1) occur in various applications, ranging from molecular dynamics [19-22, 40, 41] to models for the dynamics of fibers in turbulent airflows in the context industrial production processes of non-woven textiles [23,24].…”

“…We assume that a and B are locally Lipschitz continuous, of polynomial and linear growth, respectively, and that the mapping (x, y) → (y, a(x, y)) satisfies a one-sided linear growth condition. Note that the diffusion coefficient B is allowed to depend on both position and velocity, which is motivated by [23,24]. The derivatives of g are assumed to satisfy suitable boundedness and non-degeneracy conditions in a neighborhood of the manifold.…”