5. Reliability-based Structural Optimization
5.5 Practical Aspects of Structural Optimization
5.5.1 Design Variable Linking
Having an independent design variable for each free parameter or finite element gives additional degrees of freedom for solving the mathematical optimization problem. But sometimes this results in impractical or difficult to manufacture components. In addition, a problem solved with hundreds or thousands of design variables may not be tractable. Hence there are several practical advantages to reducing the number of design variables. One way to do this is to link the local design variables with global variables. The global variables b are the ones that are directly involved in the design process. The local variables are linked to the global values through a matrix relationship of the form
b T
t (5.46)
where t is a vector of local optimization variables, b is a vector of global design variables, and T is the linking matrix. There are various forms of linking options possible, based on the physics of the problem. The idea is to significantly reduce the number of optimization variables using the T matrix. Linking of design variables imposes additional constraints on the problem and may not lead to the lowest possible objective function.
5.5.2 Reduction of Number of Constraints
A multidisciplinary design problem often involves a large number of inequality constraints—both behavioral and side constraints. The large number of constraints arises because it is usually necessary to guard against a wide variety of failure modes in each of several distinct maneuver and load conditions. During each stage of an iterative process, only critical and potentially critical constraints play significant roles in deriving the solution. Non-critical and redundant constraints that are not currently influencing the iterative design process are temporarily ignored. Two commonly used techniques to accomplish this are regionalization and “throw-away.” In regionalization, for example under multiple static loading conditions, if the region contains various types of finite elements (e.g. bars, shear panels, quadrilaterals, beams), it may be desirable to retain the one most critical stress constraint for each load condition and element type. Reduction of constraints using the regionalization concept hinges upon the assumption that the design changes made during the redesign step are not so drastic as to result in a shift of the constraint location within a region. In the “throw away” approach, unimportant (redundant or very inactive) constraints are temporarily ignored in a particular iteration.
5.5.3 Approximation Concepts
The basic objective in the approximate structural analysis approach is to obtain high quality algebraically explicit expressions for the objective function and behavior constraints. These explicit approximations are used in place of the detailed analysis during different parts of the iterative process. The function approximations play a very significant role in reliability-based optimization, and they were extensively discussed in Chapters 3 and 4, where they were successfully applied in computing the safety index, E , and the failure probability, Pf. As shown in Chapters 3 and 4, these concepts have been used extensively in mathematical optimization and are also popular in reliability analysis and design.
5.5.4 Move Limits
The approximations constructed at a specific point are accurate within certain bounds of n-dimensional space. Once the exact problem is replaced with a surrogate problem, in order to maintain the validity of the approximations, bounds
are placed on how much a design variable can change during a single design cycle.
Move limits artificially restrict the design space. Due to the nonlinearity of functions, a certain percentage of change in design variables may not proportionately translate to the same percentage change in the response function.
Proper selection of move limits is important for convergence to the optimum.
These artificial bounds always fall within the bounds of the original design problem.
Figure 5.13. Surrogate Models and Sequential Optimization
Figure 5.13 shows a sketch of how an approximate problem converges to the true optimum. For a 2-D problem, the move limits are shown as a box around the current design point. Design freedom decreases as we move closer to the optimum where the constraint violations are not acceptable. For the first iteration, the solution to the approximate problem is found to be at a corner of the box, since only the objective is minimized in this iteration where the constraints are not active.
Typically, an approximate problem is reconstructed at this solution using an exact analysis (simulation), and the optimization-seeking process is continued. By the end of the second iteration, one of the constraints is violated due to the typical nature of approximations. At the end of the third iteration, a near optimal solution is reached. Multiple types of move-limit strategies are implemented in the design optimization community.
5.6 Convergence to Local Optimum
Since numerical optimization is an iterative process, one of the steps is to define termination criteria. The convergence parameters vary from problem to problem;
all of the following conditions may not be needed for every problem.
The first criterion requires that the relative change in the objective between iterations is less than a specific tolerance H1. Thus, the criterion is satisfied if
1 1
1
) (
) ( ) (
H d
q q q
b F
b F b F
(5.47)
The second criterion is that the absolute change in the objective between the iterations is less than a specified tolerance H2. This criterion is satisfied if
2 1) ( )
(bq F bq dH
F (5.48)
The third criterion is that the absolute change in the design variables is less than a specified tolerance, H3. The criterion s satisfied if
3 1 dH q
q b
b (5.49)