Application of PML to Analysis of Nonlinear Soil-Structure-Fluid Problem Using Mixed Element

Soil-structure-fluid interaction may have significant effects on seismic responses of structures. Mixed element may be conveniently used to express non-linear constitutive equation of fluid and to avoid volumetric locking. X-FEM may be well suited to model discontinuity of displacements between solid and fluid. In the X-FEM analysis, as well as FEM and FDM analyses, appropriate boundary conditions should be set at the boundaries of numerical models not to reflect outgoing waves. Several methods are proposed (Wolf 1988). The first is the extensive mesh models using a finite element method or a finite difference method with approximate energy transmitting boundaries. The second is the substructure method using, for example, finite element and time domain boundary element method. In the former, the degrees of freedom of the models are often very large. The latter method may be more efficient, but the nonlinearity must be restricted within the nearby portion of structures modeled by finite element method, i.e. ,constitutive equations are assumed to be linear at and outer domain of the boundary. The third is FEM with PML or convoltion PML(Berenger 1994, Collino 2001,Basu 2003 2004,Drossaert 2007). PML and convolution PML are proved to have efficient wave absorbing capability for linear elasto-dynamic problem, and, the nonlinearity must be restricted within finite element domain. In the severe earthquakes, however, soil may become nonlinear to a large extent so that the second and the third methods may be inadequate. Convolutional PML is extended to cope with non-linear problem, so that nonlinear soil can be analyzed with a limited number of meshes without loss of accuracy (Shiojiri 2010, Reheman 2011). But, it is restricted to displacement based FEM. Here, complex frequency shifted convolution-PML without splitting of variables is developed for mixed finite element and for X-FEM, and the performances of PML are confirmed. The formulation of PML is completely consistent with corresponding FEM or X-FEM. It can be easily extended to any type of element and any nonlinear constitutive equations of the corresponding FEM or X-FEM. The resulting mass and stiffness matrices for PML are symmetric for linear models.