A true PML approach for steady-state vibration analysis of an elastically supported beam under moving load by a DLSFEM formulation Diego Froio a , Egidio Rizzi a,⇑ , Fernando M.F. Simões b , António Pinto da Costa b a Dipartimento di Ingegneria e Scienze Applicate, Università degli studi di Bergamo, viale G. Marconi 5, I-24044 Dalmine (BG), Italy b CERIS, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal article info Article history: Received 2 November 2018 Accepted 15 May 2020 Keywords: Perfectly Matched Layer (PML) Discontinuous Least-Squares Finite Element Method (DLSFEM) Infinite beam Pasternak visco-elastic support Moving Load (ML) Steady-state vibration abstract This paper concerns a computational implementation for solving a Moving Load (ML) problem on an infi- nite Euler–Bernoulli elastic beam on a Pasternak visco-elastic support. A steady-state dynamic response in convected coordinate is sought, by a numerical approach with discretization over a finite domain, implying spurious boundary reflections of non-evanescent waves. This is effectively solved by: (a) analyt- ically formulating a new, true Perfectly Matched Layer (PML) approach, toward handling the underlying fourth-order differential problem and the corresponding far-field conditions, without adopting special boundary conditions; (b) outlining a local Discontinuous Least-Squares Finite Element Method (DLSFEM) formulation, apt to provide a robust approach for the present non self-adjoint problem and to conveniently handle the jump condition in the shear force at the concentrated ML position. Consistent numerical results are illustrated and compared to an available analytical solution, showing a perfect match, with a complete removal of spurious boundary effects and a proof of theoretical a priori error estimates. Further results are produced for a case with multiple MLs. The paper shows that the pre- sent innovative DLSFEM-PML formulation is effectively suitable to numerically solve a steady-state ML problem on an infinite beam, setting up a new computational tool in such a challenging mechanical context. Ó 2020 Elsevier Ltd. All rights reserved. 1. Introduction 1.1. Physical problem framework Moving Load (ML) problems relative to beams or more complex structures resting on elastic supports constitute dynamical sub- jects of a considerable interest within both the scientific and the technical literature, referring to several and diversified important engineering application contexts, a main one being that of railway engineering (see e.g. the discussion produced in Froio et al. [6]). As a characteristic feature of this type of dynamic problems, especially for fast moving loads, the dynamic response of a beam may become very different when the load velocity is smaller (subcritical or subsonic case) or greater (supercritical or supersonic case) than the so-called critical velocity, corresponding to the minimum among the velocities of the waves propagating within the beam (see e.g. Kenney [1], Dimitrovová and Rodrigues [2], Eftekhari [3] for a Winkler type foundation and Mallik et al. [4], Basu and Kameswara Rao [5], Froio et al. [6] for a Pasternak type foundation, and Dimitrovová [7] for a foundation described as a continuum of a finite depth). Such critical velocity may reach rather low values in case of a track resting on a soft soil site, as experimentally observed by Madshus and Kaynia [8]. Since current technological trends in transportation engineering are keeping raising the velocities of moving vehicles, for instance in the order of 1000 km/h for rail- ways [9], the critical velocity value of the mechanical system may now be approached, requiring dedicated modelization approaches and effective solutions, toward consistent design and verification purposes. Comprehensive literature reviews about the problem of moving loads acting on beams on elastic supports may be found in Fry ´ ba [10], Kerr [11], Ouyang [12] and Beskou and Theodorakopoulos [13]. Moreover, by the continuous advances in computer technology and the raising demand of effectively analyzing more detailed structural modelizations, accounting also for various sources of possible nonlinearities, ad hoc numerical methods, such as the Finite Element Method (FEM), have become increasingly attractive for the treatment of moving load problems (see e.g. Castro et https://doi.org/10.1016/j.compstruc.2020.106295 0045-7949/Ó 2020 Elsevier Ltd. All rights reserved. ⇑ Corresponding author at: University of Bergamo, Department of Engineering and Applied Sciences (Dalmine), viale G. Marconi 5, I-24044 Dalmine (BG), Italy. E-mail addresses: diego.froio@unibg.it (D. Froio), egidio.rizzi@unibg.it (E. Rizzi), fernando.simoes@tecnico.ulisboa.pt (F.M.F. Simões), antonio.pinto.da.costa@ tecnico.ulisboa.pt (A. Pinto da Costa). Computers and Structures 239 (2020) 106295 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc