This paper describes the implementation of a PCG equation solver in a Finite Element Methods (FEM) program, FEAP, on a parallel platform. FEAP is a general purpose FEM program, developed by Professor R.L.Taylor of the Civil Engineering department at U.C.Berkeley, and is used by many researchers in Civil Engineering and related disciplines.
Finite Element Methods are popular for the
analysis of structural system in Civil Engineering, as well as
the approximation of partial differential equations in many other
fields. FEM consists of a
discretization of a domain of interest, into elements with discrete endpoints,
or nodes, in space. These elements calculate an approximate solution to the
governing equations, associated with the problem at hand
(e.g. heat or stress), in the physical space which they implicitly
occupy. For structural mechanics this process requires discretizing
the structure, selecting elements, applying external or
internal loads (e.g. a car or an
earthquake acceleration field, respectively). The elements are then
required to form a stiffness, or derivative of their resistance to the
load with respect to the displacement of their end points; this stiffness
is then assembled into a global matrix 
. The elements also form
their current resisting force (this includes body forces such as acceleration
fields), which are added to the global force, or residual, vector (
)
- to form 
. The displacements (
) are then solved for,
and used to evaluate the state of the structure - usually
requiring the elements to calculate
their stresses, so as to determine the condition of the structure. These
displacements may also be used by the elements to calculate their resisting
force again, and the process repeated. In the case of a linear analysis, the
reformed residual vector () will be the zero vector
(within the machine's precision), in this second iteration.
For nonlinear analysis this process is repeated, with solutions () used to
increment the nodal displacements, until convergence is achieved.
The solution of this system of simultaneous equations, requires the bulk of time in most structural analysis, though the element state determination phase (i.e. forming an element stiffness and residual), can require anywhere between 1% and 50% of the total cost of the solution. Thus the equation solver, and the element state determination, are the primary processes which are of interest in the performance of FEAP.
This project will address some issues of the increasing the performance of the equation solver by performing some of the computations simultaneously on a parallel computer. The execution of the element state determination in parallel is an important issue as well, not only to utilize the hardware as much as possible (to derive the benefit of computing in parallel) - but also to reduce the costs in the distribution of the data (and thus reduce the overhead of computing in parallel). This paper will not address the issue of the element state determination, but will instead concentrate on the equation solver.