On a nonlinear partial differential algebraic system arising in technical textile industry: Analysis and numerics
For researchers in textile engineering and applied mathematics, this provides a validated numerical method for simulating inextensible elastic fibers, though the problem is domain-specific.
The paper develops a numerical scheme for a nonlinear fourth-order system of partial differential algebraic equations modeling slender inextensible elastica in textile manufacturing, proving stability and convergence with finite element simulations confirming theoretical properties.
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.