29 research outputs found
On a nonlinear partial differential algebraic system arising in technical textile industry: Analysis and numerics
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.Comment: Abstract and introduction are partially rewritten. The numerical
study in Section 4 is completely rewritte
Stochastic fiber dynamics in a spatially semi-discrete setting
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.Comment: 20 pages; typos removed, references adde
Adjoint-based optimal control using meshfree discretizations
AbstractThe paper at hand presents a combination of optimal control approaches for PDEs with meshless discretizations. Applying a classical Lagrangian type particle method to optimization problems with hyperbolic constraints, several adjoint-based strategies differing in the sequential order of optimization and discretization of the Lagrangian or Eulerian problem formulation are proposed and compared. The numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior. Moreover, they exemplify the superiority of meshless methods over the conventional mesh-based approaches for the numerical handling and optimization of problems with time-dependent geometries and freely moving boundaries