14.10 Multiobjective Optimization 761
If a multilevel (decomposition) approach is used, the optimization of various subsystems
(at different levels) can be performed on parallel processors while the solution of the
coordinating optimization problem can be accomplished on the main processor. If the
optimization problem involves an extensive analysis, such as a finite-element analysis,
the problem can be decomposed into subsystems (substructures) and the analyses of
subsystems can be conducted on parallel processors with a main processor performing
the system-level computations. Such an approach was used by El-Sayed and Hsiung
[14.17, 14.20]. The procedure can be summarized as follows:
1.Initialize the optimization process. The current (related) design variables are
sent to the various processors.
2.The finite-element analyses of the substructures are performed on different
(associated) processors.
3.The main processor collects the stiffness and force contribution matrices from
the various processors, solves for the displacements at the shared (common)
boundary nodes of substructures, and sends the data to various processors.
4.The associated processors perform the detailed calculations to find the displace-
ments and stresses needed for the evaluation of the constraints.
5.The main processor collects the constraint-related data from the associate pro-
cessors and checks the convergence of the optimization process. If convergence
is not achieved, it performs the computations of the optimization algorithm and
the procedure is repeated from step 1 onward.
Numerical examples were solved on a Cray X-MP four-processor supercomputer
[14.17]. For a 200-member planar truss, the weight was minimized with constraints on
stresses using four substructures. It was reported [14.17] that the parallel computations
required 10.585 s of CPU time, while the sequential computations required a CPU time
of 13.518 s (with a speedup factor of 1.28)
For most mechanical and structural problems, parallel computers with MIMD (mul-
tiple instruction multiple data) architecture are better suited. Atiqullah and Rao [14.21]
presented a procedure for the parallel implementation of the simulated annealing algo-
rithm. In this method, certain design variables assigned to each processor perform the
variable specific optimization. This information is later combined to complete one cycle
of optimization. Since the entire (variable-specific) optimization process is repeated on
each processor, all processors will be equally busy most of the time, except for any
input/output done by the specific processors. Thus the “divide and conquer” strategy
of optimization needs a “communicate and combine” process, which should be kept to
a minimum. The detailed procedure is shown as a flow diagram in Fig. 14.8.
The minimum-weight design of a 128-bar planar truss was considered with
stress and buckling constraints. A speedup factor of 10.2569 was achieved using the
eight-node configuration of an iPSC/860 computer.
14.10 Multiobjective Optimization
A multiobjective optimization problem with inequality constraints can be stated as
(equality constraints, if they exist, can also be included in the formulation of the
problem)
