# Optimization of Weightless Design Variables

As seen in Equation (2.1), the weight of the pile foundation is only governed by the length and area of the piles. Thus, optimizing the size of the piles, and assuming an infinite selection of pile sizes, is rather straightforward because 3WT/3di = dWT/dAi = pLi. The weight gradient, located in the denominator of the optimality criteria, Equation (2.6), is non-zero. However, for other design variables, such as batter (when the length is constant) or pile spacing, SW^ddi = 0.

Even though the weight gradient of these topological design variables is zero, they still have an indirect effect on the weight. Unfortunately, this effect is immeasurable. To solve this problem, Hoback and Truman created a new optimality criterion that can model the behavior of topological variables (Hoback et al., 1993). First the optimality criteria for the weightless gradient is rewritten from Equation (2.4) with 3WT/5di = 0:

Because the efficiency is one when the design variable is optimal, the weightless optimality criteria from Equation (2.6) is rewritten as:

The weightless scaling factor, w, replaces the weight gradient in the optimality criteria, effectively acting as a pseudo-weight gradient. Because the weightless optimality criteria is formulated to approach one at optimum, Ti can still be used in the same recurrence relationship, Equation (2.7).

The difficult step is finding an acceptable value for w. Hoback’s method involved two steps. First, he scaled w so that the coefficients of the weightless Lagrange multipliers, Aj, along the main diagonal of the linear equations were at least as large as the corresponding weighted coefficients. The second step involved increasing w until an estimated weight change converged. Although Hoback found this method successful, his work showed inconsistencies at finding a better, lower-weight design, and a new method has been developed.

After testing random optimization problems for the effects of the weightless scaling factor, w, on the optimization process, a pattern was discovered. First, a single value for w would not work for several reasons. If w is too large, the weightless variable would remain unchanged, for Ti will always be too close to one. The final solution would still optimize the weighted variables, but the final weight would be less than optimal without the optimization of the weightless variables. If w is too small, Ti will take on very large and very small values, throwing the weightless variable to their extremes, never reaching an optimum. This creates a very unstable optimization process that may never converge to a single design.

Figure 3.1 shows the typical results of simultaneously optimizing weighted variables and weightless variables. Weightless scaling factors of 1012, 108, 106, and 103 are each tried from the same initial design point. When w = 1012, the weightless variables were untouched because w was too high, and when w = 103, the optimization became unstable, resulting in sudden large increases and decreases in weight.

 -w – 10л12 ■ w – 10Л8 — w – 10Л6 w – 10Л3

Figure 3.1. A typical optimization, including weightless variables, with different w values tested. Note that a gradually decreasing weightless scaling factor, beginning at 1012 and gradually reducing to 103, would end with a near-optimal design.

Because several random examples displayed similar results as figure 3.1, a new method for determining w was created. The weightless scaling factor, w, will begin at a very large value (such as 1012 in figure 3.1). Optimization will continue at that value of w until the weight converges (at iteration 4 in figure 3.1). For the next iteration, the square root, or another appropriate reduction factor, of w will be taken (thus, reducing w to 106). Then, this process will be repeated, continuing to reduce the weightless scaling factor and further decreasing the overall weight of the pile foundation design. Figure 3.1 shows how gradually decreasing w would result in a continuously decreasing weight if w were to decrease from 1012 to 106 to 103. Eventually, the weightless scaling factor will become too small, and the design will become unstable. At this point, the optimization is complete, and the lowest weight design is chosen.