DLSFEM–PML formulation for the steady-state response of a taut string on visco-elastic support under moving load

Abstract

The numerical solution of the steady-state response of a uniform taut string on visco-elastic support under a concentrated transverse moving load is addressed. By recasting the governing second-order differential equation as a first-order system in convected coordinate, a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation is developed within a complex-valued function space, to overcome numerical instabilities linked to high-velocity loads and handle far-field conditions through an effective Perfectly Matched Layer (PML) implementation. As an original advancement of the present DLSFEM–PML formulation, a coercivity theorem is proven for any first-order ordinary differential system and uniform error estimates are established for the finite element approximation for both$$L^2$$- and$$H^1$$-norms. Thus, the formulation newly joins a DLSFEM approach and a PML implementation, for solving the above-mentioned moving load problem. Numerical examples illustrate feasibility and accuracy of the method in reproducing the expected trends of solution and a priori error estimates.