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 *i ^{th}* layer is

*t*. But the total thickness of the laminated plate must remain equal to a specified value

_{i}*T*, so we have three design variables:

*t*,

_{1}*t*,

_{2}*t*and a required relationship:

_{3}*t _{1 }+ t_{2 }+ t_{3 }= T *(1)

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 *t _{1}*,

*t*,

_{2}*t*, 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.

_{3}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:

- Define the total thickness
*T*as a*constant parameter*and assign its specified value as the baseline. - Define two
*continuous*variables*p*and_{1}*p*with a range (0.01, 0.99) for each one. Think of these as_{2}*percentages*. These will be our only two independent variables in the problem. - Define three
*dependent*thickness variables*t*,_{1}*t*,_{2}*t*as follows:_{3}

* t _{1}* =

*p**

_{1}*T*So

*p*is a percentage of the total thickness

_{1}*t _{2}* =

*p** (

_{2}*T*–

*t*) So

_{1}*p*is a percentage of the remaining thickness after

_{2}*t*

_{1}is determined

* t _{3}* =

*T*–

*t*–

_{1}*t*So

_{2}*t*is the remaining thickness after

_{3}*t*

_{1}and

*t*

_{2}are determined

The ranges used for *p _{1}* and

*p*are somewhat arbitrary, but we should

_{2}*not*use (0.00, 1.00) to avoid layers with zero thickness.

The resulting variable definitions in HEEDS MDO look like this:

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: *r _{1}*,

*r*,

_{2}*r*and

_{3}*r*. For performance and manufacturability reasons, the design variables must satisfy the relationships:

_{4}*r _{1 }≤ r_{2 }≤ r_{3 }≤ r_{4} ≤ R *(2)

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.

- Define the maximum radius
*R*as a*constant parameter*and assign its specified value as the baseline. - Define four
*continuous*variables*p*,_{1}*p*,_{2}*p*, and_{3}*p*with a range (0.0, 1.0) for each one. Think of these as_{4}*percentages*. These will be our four independent variables in the problem. - Define four
*dependent*thickness variables*r*,_{1}*r*,_{2}*r*,_{3}*r*as follows:_{4 }

* r _{1}* =

*p**

_{1}*R*So

*p*is a percentage of the total radius

_{1}*r _{2}* =

*r*+

_{1}*p** (

_{2}*R*–

*r*) So

_{1}*p*is a percentage of the remaining radius after

_{2}*r*is determined

_{1}*r _{3}* =

*r** (

_{2 }+ p_{3}*R*–

*r*) So

_{2}*p*is a percentage of the remaining radius after

_{3}*r*is determined

_{2}*r _{4}* =

*r*+

_{3}*p** (

_{4}*R*–

*r*) So

_{3}*p*is a percentage of the remaining radius after

_{4}*r*is determined

_{3}Because the constraint in Equation (2) allows for the radii of two adjacent control points to be equal, the range used for *p _{1}*,

*p*,

_{2}*p*, and

_{3}*p*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.

_{4}The resulting variable definitions in HEEDS MDO look like this:

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**.