Direct And Inverse Catenary Problems With Inhomogeneously Distributed Mass: Mathematical Modelling And Numerical Simulation

Contrary to inhomogeneously distributed mass, the analytical solution for cate-nary problem or the hanging chain problem in a uniformly distributed case was found in 1690. But in this dissertation, given inhomogeneously distributed mass, this direct problem is to find the shape of cable fixed on two suspended endpoints. Existence and uniqueness of the solution to direct catenary problem are established, which we prove through exploiting several convex optimization theories. A constraint on 2nd-order derivative is introduced for both the theoretical proof and fast convergence of its numerical algorithm. In terms of inverse catenary problem, we intend to recover inhomogeneously distributed mass from the given shape of the cable. We present a trick to decomposing this nonlinear problem as two (linear) stable numerical differ-entiation problems. Furthermore, existence, uniqueness and conditional stability for this regularized inverse problem are also set up. Its numerical solution is provided by a series of reconstruction algorithms on the basis of a well-developed numerical differentiation algorithm. Numerical tests using interior point method for convex quadratic programming with constraints are presented and discussed.