Alternative Formulations to PML for an Infinitely Long Beam on Elastic Foundation

Abstract

Following the major evolution of computers that provided the possibility of using numerical methods such as the finite element method (FEM) in the solution of complex problems, the solution of problems of infinite domain became a plausibility. One common problem of infinite domain in civil engineering has been in railroads that use continuously welded rails. Because of the need for an extensive number of elements and computer time, special procedures would have to be devised for the solution of the latter class of problems to become practical. The main problem with application of standard FEM to problems of infinite domain is the error generated because of the proximity of the boundaries. In dealing with dynamic response of railroads, for example, especially under seismic loading, the required analysis time can be prohibitive, demanding the use of effective modeling techniques. Because of possible variation in properties of the rail system along the line, the use of finite element method is required. For some simple cases, a model with a large number of elements can be used. In most cases, however, the number of elements and the computer time become prohibitive or else the effects of the boundaries would not be adequately eliminated to avoid erroneous results. The perfectly matched layer (PML) method, somewhat recently introduced for electromagnetic problems, seems to provide one of the best means of dealing with problems of infinite domain. In the standard PML approach, the differential equation of a dynamic problem is first transformed to frequency domain. In that domain, a stretching of the axial coordinate is performed leading to an exponential decay of the displacement in the time domain, that is, producing the effect of attenuation of the waves away from the segment of interest. Unfamiliarity of this procedure to structural engineers and the difficulty of its physical interpretation make it difficult to use. In this study, a procedure is introduced for overcoming the problems mentioned. Thus, the PML method is applied in time domain directly and with two slightly different procedures. The results show the applicability of these procedures despite their simplicity.