OPTIMIZATION 367If the difference in the function values ΔE≤ 0, the probability is one and the point Xt+1 is always
accepted. When ΔE > 0, Xt+1 is worse than Xt and is consequently rejected by most traditional
gradient based searches. In simulated annealing however, there is a finite probability that the point
is accepted. If the temperature parameter T is large, the probability is high for Xt+1 to be accepted
even for largely disparate function values (high ΔE). Thus, at each stage, the system can move either
to a state for which the energy is higher than its present state, or to a state of lower energy. Eq. (12.37)
allows the system to move consistently towards lower energy states, yet still jump out of local
minima due to the probabilistic acceptance of some upward moves.
The initial temperature T, the number of function evaluations nperformed at a particular temperature
and the cooling schedule are three essentials governing the performance of simulated annealing. If a
large initial T is chosen, it takes long to converge. For small initial T, the search is not adequate to
thoroughly investigate the problem space before converging to a true (global) optimum. An estimate
of the initial temperature parameter can be obtained by computing the average of the function values
at randomly generated points in the search space. A large value of n is recommended to achieve the
quasi-equilibrium state for which the computation time is more. For most problems, n is usually
chosen between 20 and 100 depending on the computation resources and time.
This chapter discussed some of the many optimization methods in use in engineering applications.
First, one-variable optimization methods were discussed which are of the bracketing and open type,
the former being of zeroth order requiring only function values while the latter being of the first or
second order requiring the first or second function derivatives. The classical necessary and sufficient
optimality conditions for multi-variable optimization methods with and without constraints were
derived followed by a discussion on the Karush-Kuhn-Tucker or KKT conditions. Linear programming
which involves linear function and constraints was discussed in some detail with simplex implementation
following which sequential linear programming (SLP) was briefed. Numerical implementation of the
sequential quadratic programming (SQP) which employs the KKT conditions iteratively was discussed
in brief followed by a mention on some stochastic methods like genetic algorithms and simulated
Annealing. The scope of this chapter is restricted only to the aforementioned noting that numerous
texts and codes are available exclusively on optimization. We close this chapter by citing an example
on topology optimization of compliant mechanisms with local stress constraints which involves
linear frame finite element analysis discussed in Chapter 11.
Example 12.11. Compliant mechanisms are single-piece devices designed for prescribed motion,
force and/or energy transduction through elastic deformation. Consider a design region discretized
using linear frame elements shown in Figure 12.14(a) with length 150 mm and width 50 mm for a
compliant crimper. The left vertical edge is fixed while the bottom edge is on a roller support. A load
of 20 N is applied as input at the top right corner and it is desired to maximize the deformation at
pointP along the direction shown. To compute the deformation at P, the virtual work principle is used
where a unit dummy load is applied at P along the direction of desired deformation and the nodal
displacementsV are computed as a response only to the dummy load. If U is the displacement vector
due to the input load only then
Δout (P) = VTKUwhereK is the global stiffness matrix. The axial stress in each frame element i can be computed as
σi = EiBiuiwhereEi is the Young’s modulus of the ith frame element, Bi the strain displacement matrix and ui
the local nodal displacements due to the input load. The optimization problem can be formulated as