Dynamic and Transient Infinite Elements

Effective and efficient modelling of infinite media is important for the production of accurate and useful solutions for many scientific and engineering problems involving infinite domains (Bettess 1977, 1980; Chow and Smith 1981; Medina and Taylor 1983; Zhang and Zhao 1987; Zhao et al. 1989; Zhao and Valliappan 1993a, b, c, d; Astley 1996, 1998; Yang et al. 1996; Yang and Huang 2001; Yun et al. 2000, 2007; Wang et al. 2006). Some typical examples involving infinite domains are as follows: (1) earthquake wave propagation within the upper crust of the Earth in the fields of geophysics and seismology; (2) dynamic structure–foundation interaction in the fields of geotechnical, civil and dam engineering; and (3) transient pore-fluid flow, heat transfer and mass transport within the interior of the Earth in the fields of geoscience and geoenvironmental engineering.

Numerical simulation of wave propagation problems in infinite media has attracted significant attention in many scientific and engineering fields such as geophysics, seismology, civil engineering and earthquake engineering. From a wave motion point of view, structural vibration problems can be divided into two categories. One is a wave radiation problem, or an interior domain problem, in which wave energy is produced within the near field and then propagated into the far field of the problem in various wave forms. Typical examples of this category are foundation vibration problems as a result of trains passing on railways, machine vibration problems on the foundations of buildings, impacting vibration problems on the ground surface of an airport during airplanes landing, to name just a few.

Extensive studies on the dynamic response of concrete gravity dams due to earthquake loadings have demonstrated that their dynamic response is mainly affected by the following factors: (1) the interaction between the dam and impounded reservoir water (Chopra 1968; Chakarbarti and Chopra 1974; Liam-Finn et al. 1977); (2) the compressibility of the impounded water (Chopra and Gupta 1982); (3) the interaction between the dam and the foundation rock (Liam-Finn et al. 1977; Liam-Finn and Varoglu 1972a, b, 1975); and (4) the materials at the reservoir bottom (Hall and Copra 1982; Fenves and Chopra 1983, 1984, 1985; Lotfi et al. 1987; Medina et al. 1990). By means of a substructure method, Chopra and his colleagues considered the above factors and made some interesting conclusions on the dynamic response of concrete gravity dams due to earthquake loadings (Chopra 1968; Chakarbarti and Chopra 1974; Hall and Copra 1982; Fenves and Chopra 1983, 1984, 1985).

Previous studies related to the surface motion of wave propagation demonstrated that the topographical and geological features of a canyon have a significant influence on the motions of the ground (Bouchon 1973; Rogers et al. 1974; Datta and El-Akily 1978; Bard 1982; Shah et al. 1982, 1983; Dravinski 1983; Ohtsuki and Harumi 1984; Zahradnik and Urban 1984; Sanchez-Sesma et al. 1985; Wong et al. 1985; Geli et al. 1988; Zhao and Valliappan 1993a, b). This fact may have an important impact on the dynamic response of such large-scale structures as dams and bridges, because free-field motions have obviously different amplitudes and phases along an abutment on which a structure is founded. One of the key questions associated with the selection of a dam or bridge site is to investigate which kind of canyon is more beneficial to the cost and safety of a structure, from a seismic resistance point of view.

Numerical simulation of infinite media is an important topic in dynamic soil–structure interaction problems. This topic arose from numerous practical problems, such as numerical simulation of building structural foundations, offshore structural foundations, dam foundations, nuclear power station foundations, just to name a few. The study of this topic becomes more important when the structure is large and the effects of earthquake waves are considered. Owing to the importance of dynamic soil–structure interaction effects, a large amount of research has been carried out in the past few decades (Elorduy et al. 1967; Lysmer and Kuhlemyer 1969; Kausel 1974; Zienkiewicz and Bettess 1975; Wong and Luco 1976; White et al. 1977; Cundall et al. 1978; Chow and Smith 1981; Hamidzadeh-Eraghi and Grootenhuis 1981; Medina and Taylor 1983; Liao et al. 1984; Wolf 1985, 1988; Zhao et al. 1987, 1989; Zhang and Zhao 1987; Zhao and Liu 2002, 2003). The general methodology of dealing with a dynamic soil–structure interaction problem is to divide the whole infinite foundation of the problem into a near field, which is comprised of a limited region of the infinite foundation, and a far field, which is comprised of the remaining part of the infinite foundation. As the near field is usually simulated by using the finite element method, both the geometrical irregularity and the non-homogeneity of an infinite foundation can be considered to determine the boundary of the near field. Since the far field is usually simplified as an isotropic, homogeneous, elastic medium, its effect on the near field can be represented either by some special artificial boundaries (Lysmer and Kuhlemyer 1969; Kausel 1974; White et al. 1977; Cundall et al. 1978; Liao et al. 1984; Zhao and Liu 2002, 2003) or by some special elements (Ungless 1973; Zienkiewicz and Bettess 1975; Bettess 1977, 1980; Chow and Smith 1981; Medina and Taylor 1983; Zhao et al. 1987, 1989). Through applying these special artificial boundaries or elements on the interface between the near field and the far field, the effect of the far field on the near field can be considered in the corresponding computational models.

Numerical simulation of dynamic structure–foundation interaction problems has been an important research topic in many scientific and engineering fields (Zhao et al. 1991, 1992; Zhao and Valliappan 1991, 1993c, e). In terms of a structure–foundation interaction system, the foundation is referred to as the natural or built-up formation of soil, subsoil or rock upon which a building or structure is supported. If a structure is founded on a soft soil foundation, the dynamic structure–foundation interaction problem is also called the dynamic soil–structure interaction problem (Wolf 1985, 1988). For example, the dynamic analysis of an embankment dam–foundation system is a typical dynamic soil–structure interaction problem. Owing to the vast diversity of structural configurations and foundation materials, it is impossible to deal with all the dynamic structure–foundation interaction problems in one chapter.

Pore-fluid flow and heat transfer in fluid-saturated porous media of infinite domains are important phenomena in many scientific and engineering fields. For example, in the field of exploration geoscience, pore-fluid flow and heat transfer from the interior of the Earth to the surface of the Earth are two important physical processes to control ore body formation and mineralization within the upper crust of the Earth. Owing to the increasing demand of natural minerals and the possible exhaustion of existing mineral resources in the foreseeable future, there has been an ever-increasing interest in the study of key controlling processes associated with ore body formation and mineralization within the upper crust of the Earth (Phillips 1991; Yeh and Tripathi 1991; Nield and Bejan 1992; Steefel and Lasaga 1994; Raffensperger and Garven 1995; Schafer et al. 1998a, b; Xu et al. 1999; Schaubs and Zhao 2002; Ord et al. 2002; Gow et al. 2002; Zhao et al. 1997–2008).

Numerical simulation of contaminant transport in fractured porous media of infinite domains is a complex problem in various aspects. The solution for this kind of problem becomes more difficult once practical considerations, such as the infinite extension of the problem domain, the leakage effect between the porous medium and the fissured network, the characterization of the fissured network and other physical and chemical parameters, are appropriately included in the analysis (Rowe and Booker 1989, 1990a, 1991). On the other hand, the practical problems involving contaminant transport in fractured porous media have received rapidly increasing attention as a result of the treatment of industrial and agricultural wastes, the evaluation of potential contamination from nuclear power plants and the disposal activities of wastes from our daily lives.