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首页> 外文期刊>Finite Elements in Analysis and Design >Time-domain PML formulation for modeling viscoelastic waves with Rayleigh-type damping in an unbounded domain: Theory and application in ABAQUS
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Time-domain PML formulation for modeling viscoelastic waves with Rayleigh-type damping in an unbounded domain: Theory and application in ABAQUS

机译：时域PML公式，用于在无界域中用瑞利型阻尼建模粘弹性波：理论和在ABAQUS中的应用

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摘要

The Perfectly Matched Layer (PML) method is a powerful approach to absorb outgoing waves from a finite computational domain in various media, such as elastic, poroelastic, anisotropic, and viscoelastic, by means of layers of artificial material placed at the finite domain boundaries. However, its use in most finite-element method (FEM) codes, which take into account viscous damping by employing only the Rayleigh and Lindsday damping formulation, is inadequate. This paper introduces a viscoelastic Perfectly Matched Layer (PML) formulation with Rayleigh-type damping for finite-element time-domain analyses. The PML formulations in the frequency- and time-domain are derived using a two-parameter complex coordinate stretching function. The displacement field is the only unknown variable in the formulation, so that the approach can be readily implemented in general finite-element codes. The weak form formulation in the time-domain is implemented in ABAQUS/Standard by merging it with a user-defined element (UEL) subroutine in Fortran90. The UEL is developed for 2D plane-strain (PE4ML) and plane-stress (PS4ML) four-node, linear, isoparametric, quadrilateral elements. The Hilber-Hughes-Taylor implicit time integration scheme is used to evaluate the unknown displacements, velocities, and accelerations. The validity and efficiency of the viscoelastic PML formulation and the UEL subroutine are examined with two numerical examples, namely a single-layer and a multi-layered soil profile with different damping ratios and properties. The results also highlight the poor performance of elastic PML layers, when they are placed at the boundaries of a viscoelastic interior domain, because, as the damping ratio increases, the reflection at the interface of the interior domain and the elastic PML domain increases. Long-time stability of the model is also examined, and no instabilities are observed.
机译：完全匹配层（PML）方法是一种强大的方法，可以通过在有限域边界处放置人造材料层来吸收各种介质（例如弹性，多孔弹性，各向异性和粘弹性）中来自有限计算域的输出波。但是，它在大多数有限元方法（FEM）代码中的使用是不够的，这些方法仅通过使用Rayleigh和Lindsday阻尼公式来考虑粘性阻尼。本文介绍了具有瑞利型阻尼的粘弹性完全匹配层（PML）公式，用于有限元时域分析。频域和时域中的PML公式是使用两参数复数坐标拉伸函数得出的。位移场是公式中唯一未知的变量，因此该方法可以很容易地在通用有限元代码中实现。通过将ABAQUS / Standard中的弱形式公式与Fortran90中的用户定义元素（UEL）子例程合并，可以实现该形式。 UEL是为二维平面应变（PE4ML）和平面应力（PS4ML）四节点，线性，等参，四边形单元开发的。 Hilber-Hughes-Taylor隐式时间积分方案用于评估未知的位移，速度和加速度。通过两个数值示例检验了粘弹性PML配方和UEL子程序的有效性和效率，即具有不同阻尼比和特性的单层和多层土壤剖面。该结果还突出显示了当将弹性PML层放置在粘弹性内部区域的边界时的性能不佳，因为随着阻尼比的增加，内部区域和弹性PML区域的界面处的反射会增加。还检查了模型的长期稳定性，未观察到任何不稳定性。

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