# Using Semi-Independent Variables to Generate More Feasible Designs and Improve Search Process

Sometimes we intend for a set of design variables to be independent, but then realize that these variables need to satisfy a given relationship. How can a variable be both independent and dependent at the same time? We call these semi-independent variables.

Let’s illustrate this concept with two separate examples.

In our first example, the goal is to optimize the thickness of each layer in a three-layer laminated composite plate, as shown below. The thickness of the ith layer is ti. But the total thickness of the laminated plate must remain equal to a specified value T, so we have three design variables: t1, t2, t3 and a required relationship:

t1 + t2 + t= T                                                                                (1)

Figure 1. A three-layer laminated composite plate

There are two possible approaches for representing this problem in a design optimization study. One way is introduce a constraint that enforces the relationship in Equation (1). Within the range of allowable values of t1, t2, t3, very few designs will actually satisfy this constraint. So this approach practically guarantees that a large percentage of designs will be infeasible, causing the search to be very inefficient. For this reason, this approach is not recommended unless it is the only option available.

The preferred approach is to redefine the problem so that all designs produced during the optimization study automatically satisfy the constraint in Equation (1). We can do this using semi-independent variables. Here are the detailed implementation steps for the current problem:

1. Define the total thickness T as a constant parameter and assign its specified value as the baseline.
2. Define two continuous variables p1 and p2 with a range (0.01, 0.99) for each one. Think of these as percentages. These will be our only two independent variables in the problem.
3. Define three dependent thickness variables t1, t2, t3 as follows:

t1 = p1 * T              So p1 is a percentage of the total thickness

t2 = p2 * (Tt1)     So p2 is a percentage of the remaining thickness after t1 is determined

t3 = Tt1t2        So t3 is the remaining thickness after t1 and t2 are determined

The ranges used for p1 and p2 are somewhat arbitrary, but we should not use (0.00, 1.00) to avoid layers with zero thickness.

The resulting variable definitions in HEEDS MDO look like this:

Figure 2. Implementation of the variable representation for Example 1 in HEEDS MDO

Note that the implementation above has reduced the number of independent variables from three to two. This is expected when an equality constraint is imposed. As demonstrated in the next example, an inequality constraint can be implemented in a similar way, but there will not be a reduction in the number of independent variables.

For our second example, consider a shape optimization problem for an axisymmetric pipe or nozzle. The shape will be determined by a spline with four control points, as shown in Figure 3. The design variables are: r1, r2, r3 and r4. For performance and manufacturability reasons, the design variables must satisfy the relationships:

r1  ≤  r2  ≤  r≤  r4  ≤  R                                                                           (2)

Figure 3. A spline curve with four control points

In this case, we could represent this problem with four constraints that enforce the relationships in Equation (2). As in the first example, this would again cause a very large percentage of designs to be infeasible, so the search would be inefficient.

A better approach is to represent the problem in a way that all designs produced during the study automatically satisfy the set of constraints in Equation (2). Semi-independent variables can be used to achieve this goal, as described below.

1. Define the maximum radius R as a constant parameter and assign its specified value as the baseline.
2. Define four continuous variables p1, p2, p3, and p4 with a range (0.0, 1.0) for each one. Think of these as percentages. These will be our four independent variables in the problem.
3. Define four dependent thickness variables r1, r2, r3, r4 as follows:

r1 = p1 * R                      So p1 is a percentage of the total radius

r2 = r1 + p2 * (Rr1)     So p2 is a percentage of the remaining radius after r1 is determined

r3 = r2 + p3 * (R r2)     So p3 is a percentage of the remaining radius after r2 is determined

r4 = r3 + p4 * (Rr3)    So p4 is a percentage of the remaining radius after r3 is determined

Because the constraint in Equation (2) allows for the radii of two adjacent control points to be equal, the range used for p1, p2, p3, and p4 is set to (0.0, 1.0). Further, because the constraints are inequality constraints, we are not able to eliminate one of the variables as we did in the previous example.

The resulting variable definitions in HEEDS MDO look like this:

Figure 4. Implementation of the variable representation for Example 2 in HEEDS MDO

Generally speaking, a problem representation that automatically generates mostly feasible designs enables a more productive search than one that generates mostly infeasible designs. So it is usually worth the small amount of extra effort needed to construct a problem definition that produces mostly feasible designs. Strategies similar to those described above can often be used to achieve this.

We hope this tip helps you to Discover Better Designs, faster.