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.

中文翻译：

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弹性基础上无限长梁的 PML 替代公式

随着计算机的重大发展，提供了在解决复杂问题时使用诸如有限元法 (FEM) 等数值方法的可能性，无限域问题的解决方案变得合理。土木工程中无限域的一个常见问题是使用连续焊接钢轨的铁路。由于需要大量元素和计算机时间，必须设计特殊程序才能使后一类问题的解决方案变得可行。将标准 FEM 应用于无限域问题的主要问题是由于边界接近而产生的误差。例如，在处理铁路的动力响应时，特别是在地震荷载下，所需的分析时间可能会令人望而却步，要求使用有效的建模技术。由于沿线轨道系统的特性可能会发生变化，因此需要使用有限元方法。对于一些简单的情况，可以使用具有大量元素的模型。然而，在大多数情况下，元素的数量和计算机时间变得令人望而却步，否则边界的影响将无法充分消除以避免错误结果。最近针对电磁问题引入的完美匹配层 (PML) 方法似乎提供了处理无限域问题的最佳方法之一。在标准 PML 方法中，动态问题的微分方程首先转换到频域。在那个领域，执行轴向坐标的拉伸导致时域中位移的指数衰减，即产生远离感兴趣段的波衰减效应。结构工程师对这个程序不熟悉，而且其物理解释的困难使其难以使用。在这项研究中，介绍了一个程序来克服提到的问题。因此，PML 方法直接应用于时域，并采用两个略有不同的程序。结果显示了这些程序的适用性，尽管它们很简单。在这项研究中，介绍了一个程序来克服提到的问题。因此，PML 方法直接应用于时域，并采用两个略有不同的程序。结果显示了这些程序的适用性，尽管它们很简单。在这项研究中，介绍了一个程序来克服提到的问题。因此，PML 方法直接应用于时域，并采用两个略有不同的程序。结果显示了这些程序的适用性，尽管它们很简单。