Nonlinear Problems - The Boundary Element Method with Programming - page 406

15.3.3 Method of solution

For the solution of problems in plasticity we use a similar method as in Finite Elements
known as the “initial stress” method. In this method we compute initial stresses as
outlined in the previous section and apply this as loading. For this we have to amend the
discretisation of the problem. In addition to surface elements we require the specification
of volume cells in the parts of the domain that are likely to yield, for the integration of
initial stresses. These volume cells have been discussed in Chapter 3. Figure 15.5 shows
examples of discretisations for a cantilever beam and a circular hole in an infinite
domain. The discretisations actually look almost like finite element meshes and it could
be argued that one might as well use finite elements for this problem.
However, there are subtle differences:

x There is no requirement of continuity, i.e. elements do not need to connect to each

other as finite elements need.

x There are no additional unknown associated with the mesh of volume cells.

Therefore the system of equations does not increase in size.

x The representation of stress is still more accurate than with the FEM.

x The mesh of cells only needs to cover zones where plastic behaviour is expected.

Figure 15.5 Volume cells for the example of a cantilever beam and a circular hole

The iterative process is described in the structure chart in Fig 15.6. First we may
divide the total applied load into increments to optimise the number of iterations. Then
we solve for the unknown displacements/tractions with the applied loading. With the
boundary results we compute the stresses at each cell node and check the yield
condition. If F >0 is detected then the “initial stress” is computed as explained
previously. The residual vector ^ R

is computed as will be explained later and a new