Soil Behavior: Dynamic Soil-Structure Interactions - Dynamic Behavior of Concrete and Seismic Engineering - page 156

4.3.4.3. Geometrical modeling of the medium
The finite element formulation is standard; nevertheless, some conditions have to
be respected. Transmitting high frequencies imposes a maximum dimension on the
elements, at most equal to a fraction of the corresponding wavelength. Typically, we
use a value between 1/8 and 1/5 of the wavelength:

11 V

s

h d to 58f

[4.41]

max

max

where f max represents the highest frequency to be transmitted, and V s the propagation
rate of the shear waves. This criterion is generally applied to the vertical dimension
of the mesh because, considering the generally admitted assumption of wave vertical
propagation, the displacement field varies faster vertically than horizontally,
especially some distance away from the structure.

The extension of the finite element mesh constitutes one of the most critical
problems in the resolution of a dynamic problem involving propagation phenomena
using the finite element method. As a matter of fact, without any special conditions,
the side and lower limits of the model are open surfaces that completely reverberate
the wave fronts that hit them. The energy carried by these waves is reverberated
back to the structure instead of being carried ad infinitum inside the soil. As the only
energy dissipation takes place through material damping, the model has to be
extended so that the waves reverberated at the limits do not reach the structure while
its response is being estimated. The procedure soon makes the calculation cost
prohibitive.

To free us from such reverberations, some special devices called absorbing
boundaries have been developed. Located at the ends of the model, these boundaries
are supposed to represent the exact stress conditions existing at that limit, due to the
presence of soil outside the model.

Generally speaking, the side boundaries of the model can be divided into local
boundaries or consistent boundaries.

The local boundaries generally consist of localized dashpots, the characteristics
of which depend on the mechanical properties of the medium around them. These
boundaries do not couple the different degrees of freedom of the nodes along the
boundary and perfectly absorb only the waves with a normal incidence [LYS 69].
They can advantageously be implanted into time or frequency calculations.