Civil Engineering Reference
In-Depth Information
words implies that a large mesh (domain) is assembled, in principle by the global assembly
techniques described in the previous Chapters, and that the resulting global equations are
“decomposed” or torn apart into smaller pieces. Indeed some early implementations of
this process were called
diakoptics
, implying a cutting up procedure. “Substructuring,” and
“block” and “frontal” methods are similar variants, the main application being the solution,
by elimination techniques, of the very large sets of linear algebraic equations which govern
the static equilibrium of linear and non-linear FE systems.
All of this suggests the parallelisation of the “global” strategy used in earlier Chapters of
this topic. When Gaussian elimination methods are used, elimination can proceed indepen-
dently on equations relating to a particular subdomain, stored on a particular processor, until
the boundaries of the subdomain are reached. Then communication is necessary between
processors. This can be a rather complicated procedure, and a large literature has devel-
oped around the theme of optimising subdomain distributions. Often equations are solved
directly within subdomains but iteratively at the boundaries.
In contrast, in this topic a simple approach is used, based on the element-by-element
methods used in previous Chapters. In these, no global equation system matrices are ever
constructed and therefore it is more meaningful to speak of “domain composition” rather
than “decomposition.” But of course subdomains have to be identified.
In the element-by-element technique, see Section 3.5, the essential operations that
involve inter-processor communication are the “gather, matrix multiply, scatter” procedure
typified by equations (3.23) and the dot products typified by equations (3.22), (3.28) and
(3.29). Clearly different subdomain distributions will affect the amount of communcation
Element 1
Element 1
Element 1
Element 1
Figure 12.7
Alternative domain compositions
