260 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
Analysis and computations for a model of quasi-static deformation of a thinning sheet arising in superplastic forming
We consider a mathematical model for the quasi-static deformation of a thinning sheet. The model couples a first-order equation for the thickness of the sheet to a prescribed curvature equation for the displacement of the sheet. We prove a local in time existence and uniqueness theorem for this system when the sheet can be written as a graph. A contact problem is formulated for a sheet constrained to be above a mould. Finally we present some computational results
Sufficient conditions for unique global solutions in optimal control of semilinear equations with nonlinearity
We consider a semilinear elliptic optimal control problem possibly
subject to control and/or state constraints. Generalizing previous work we
provide a condition which guarantees that a solution of the necessary first
order conditions is a global minimum. A similiar result also holds at the
discrete level where the corresponding condition can be evaluated explicitly.
Our investigations are motivated by G\"unter Leugering, who raised the question
whether our previous results can be extended to the nonlinearity
. We develop a corresponding analysis and present several
numerical test examples demonstrating its usefulness in practice
Finite element methods for surface PDEs
In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples
Optimal control of the propagation of a graph in inhomogeneous media
We study an optimal control problem for viscosity solutions of a Hamilton–Jacobi equation describing the propagation of a one-dimensional graph with the control being the speed function. The existence of an optimal control is proved together with an approximate controllability result in the -norm. We prove convergence of a discrete optimal control problem based on a monotone finite difference scheme and describe some numerical results
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