Static analysis
The solution in the far field is assumed to be linear, so only linear behavior is provided in the infinite elements. The static behavior of the infinite elements is based on modeling the basic solution variable, u (in stress analysis u is a displacement component) with respect to spatial distance r measured from a “pole” of the solution, so that as , and as . The interpolation provides terms of order , , and, when the solution variable is a stress-like variable (such as the pore liquid pressure in the analysis of flow through a porous medium), as . The far-field behavior of many common cases, such as a point load on a half-space, is thereby included. This modeling is achieved by using standard quadratic or cubic interpolation for in , where s is a mapped coordinate that is chosen such that the mapping causes as . We obtain two- and three-dimensional models of domains that reach to infinity by combining this interpolation in the s-direction in a product form with standard linear or quadratic interpolation in orthogonal directions in the mapped space.
In using infinite elements for static analysis, the pole must be located so as to provide a reasonable far-field solution for the particular problem being modeled. The infinite elements in Abaqus are written with nodes on the interface between the finite and infinite elements and, on each edge that stretches to infinity, a node that must be placed in the infinite direction such that the straight line from that node through the corresponding interface node passes through the pole for that ray at a distance on the other side of the interface from the infinite element equal to the distance between these nodes (Figure 1).
The one-dimensional concept is, thus, based on a node (node 1) on the interface between the finite and infinite elements, distance from the pole and at in the mapped space, and node 2, at from the pole (the pole is at ) and at is the mapped space. The mapping is chosen as
so that
which inverts to give
When an element with and behavior is required, we combine this geometric mapping with standard quadratic interpolation of u with respect to s, written in terms of its values at node 1 and at node 2:
(this gives at , where ). Using the inverted geometric mapping to define then gives
which provides the desired behavior. Likewise, when behavior is also required, we use cubic interpolation of u with respect to s, written in terms of its values at nodes 1 and 2 and at a third node, which we choose to place at :
The inverted geometric mapping then provides
The infinite elements in Abaqus consist of two- and three-dimensional elements for uncoupled stress analysis that use quadratic interpolation for displacement components and two- and three-dimensional elements for coupled stress-pore liquid pressure elements, in which the displacements use quadratic interpolation and the pore liquid pressure uses cubic interpolation in the infinite direction. This higher-order interpolation is used for the pore liquid pressure for compatibility: since the displacement varies as , the strain (and, therefore, the stress) may vary as .