On a nonlinear partial differential algebraic system arising in technical textile industry: Analysis and numerics

TL;DR

This paper develops and analyzes a numerical scheme for a complex nonlinear PDE-DAE system modeling slender elastica in the textile industry, ensuring stability, convergence, and practical simulation accuracy.

Contribution

It introduces a semi-discretization approach transforming the PDE-DAE system into constrained optimization problems with proven stability and convergence.

Findings

Numerical scheme is stable and convergent.

Finite element discretization effectively simulates elastica dynamics.

Simulation results match analytical predictions.

Abstract

In this paper we explore a numerical scheme for a nonlinear fourth order system of partial differential algebraic equations that describes the dynamics of slender inextensible elastica as they arise in the technical textile industry. Applying a semi-discretization in time, the resulting sequence of nonlinear elliptic systems with the algebraic constraint for the local length preservation is reformulated as constrained optimization problems in a Hilbert space setting that admit a solution at each time level. Stability and convergence of the scheme are proved. The numerical realization is based on a finite element discretization in space. The simulation results confirm the analytically predicted properties of the scheme.