Mathematical Formulation of the General Problem
2.2 Description of an Optimization Problem
If we can assume that the Objective Function, Υ, is continuous and differentiable, a solution to the problem,ξ∗, must satisfy theKarush-Kuhn-Tuckerconditions:
a(ξ∗) =0,b(ξ∗)≤0; (2.37a)
∇Υ(ξ∗) +λT∇a(ξ∗) +μT∇b(ξ∗) =0T, whereλ =0,μ≥0,μTb=0, (2.37b)
where ∇a= [∇a1,∇a2, ...,∇am]T is the Jacobian of a, and ∇ is the gradient with respect to the parametersξ. The first condition simply reiterates that the solution must meet all conditions defined by the constraints. Some may recognize the second condition to be simply that found in the theory of Lagrange multipliers which states that the gradients of the function and the constraining surfaces must be linear combinations of each other.
While the constraints must always be met, the inequality constraints can be further subdivided into two types: (1)activeand (2)inactiveconstraints. Active inequality constraints are those which actually play a part in determining the location of the minimizing solution; the solution lies on these constraining surfaces (b(ξ∗) = 0) for, in the absence of these surfaces, the solution would be located at a point located within the volume excluded by the active constraint. Inactive equality constraints are those which could be removed from the list of constraints as they have no impact on the solution, since the minimizing solution is located neither on the constraining surface (b(ξ∗) = 0), nor within the volume excluded by the constraint (b(ξ∗)>0). In the simplest sense, the number of degrees of freedom (design parameters) is equal to the total number of parameters, n, minus the number of equality constraints, m, minus the number of active inequality constraints, s. In theory, each equality constraint and active inequality constraint could be used to eliminate one of the parameters in terms of the others until there were no constraints, and a new function to minimize,Υ, that is only dependent on them−n−pparameters left; in practice this may be impossible to do.
The vector λ is a vector of Lagrange multipliers, and is restricted to all non-zero values; by nature, equality constraints are always active — a Lagrange multiplier with a value of zero would effectively remove the constraint and render it inactive. The vectorμis another vector of Lagrange
multipliers applied to the inequality constraints. Permitting elements of μ to be equal to zero is simply a combination of the active and inactive constraints; any zero value corresponds to an inactive constraint, which reflects that fact that it has no effect on the solution.
These conditions arenecessary, however they areinsufficientto guarantee that Υ(ξ∗) is a local constrained minimum. In order for the pointξ∗to be a minimizing solution, the Hessian of the La- grangian must be positive-definite in the local vicinity of the solution pointξ∗, where the Lagrangian is defined to be
L(ξ,λ,μ)≡Υ(ξ) +λTa(ξ) +μTb(ξ), (2.38)
and the Hessian is the gradient of the gradient with respect to the parameters ξ. This is akin to stating that any small step in any allowable direction away from the solution point results in an increase in the value of the objective function.
In this specific problem, we have created a model for which we can define a set of parameters that determine the trajectories of ions as they exit the engine and collide with neutrals. The subsequent evolution of each resulting CEX ion may result in a collision with some portion of the spacecraft and sputtering of material from the surface. It is this sputtering of sensitive areas of the spacecraft that we wish to minimize, and so defines what the Objective Function is. In Section 2.1 we derived the equations for which Chapters 3 and 4 are devoted to developing a method to solve. The end result is a model that computes the sputtering rate, per unit area, due to CEX ion collisions with a surface, based on a number of parameters chosen. It is this sputtering flux that we define to be the objective function.
The parameters ξ chosen to determine the value of the sputtered flux Υ will determine how sophisticated the model will need to be. A small, but by no means exhaustive list of possible quantities that could be chosen to be parameters are: physical geometry of the grid set, grid hole pattern, grid thickness, and beamlet structure. Due to the extensive previous work studying the ion optics for the NSTAR grid set, for this project we will use the NSTAR grid hole pattern, grid thickness, and beamlet structure as given conditions, and only parameters defining the physical shape of the grid set will be used as optimizing (design) parameters,ξ.
Further, the constraints imposed on the proposed optimization will be those physical constraints on the shape of the grid set such that there is a high confidence that the grids will survive the launch into space. While such quantities could include both the maximum tolerable stresses and displacements, we will consider only the maximum displacements in this study.
After developing the sputtering rate model in the following three chapters, the method by which we determine the displacements of any grid under a specified load is detailed in Chapter 6. The optimization procedure and results are presented in Chapter 7.