THEORY
The Sensitivity Equation Method (SEM) is a method for approximately solving infinite dimensional optimal design problems. The SEM couples a trust-region quasi-Newton optimization algorithm with gradient information by approximately solving the sensitivity equation for design sensitivities. The sensitivity equation is a linear partial differential equation (PDE) which describes the influence of a design parameter on the global system.
An optimal design problem is often achieved by cascading simulation software into optimization algorithms. In that cascading approach is important to carefully pass information between the simulation and the optimizer. A simulation code is used to produce a finite dimensional model which is used to supply approximate function evaluations to the optimization algorithm. All the numerous variations on this theme may be formulated as approximate~then~optimize approaches. The sensitivity equation method views the simulation scheme as a device to produce approximations of both the cost function and the sensitivities. In that way the approximate derivatives are passed to the optimizer along with the approximate function evaluations.
A gradient based optimization algorithm investigates descent directions to iterate the design parameters to their opimal values. The descent direction is determined using a local approximation of the cost functional. Newton's method requires both the gradient and the Hessian of the cost functional to formulate a first and a second approximation.Taking steps which minimize this approximation produce a sequence of parameters which converge quadratically as they approach the minimizer of the cost functional. Since the computation of Hessian, or an approximation of it, can be very expensive we consider of quasi~Newton methods which relay on approximation of the Hessian using only function and gradient information at the iterates.
The optimal design methods can be generated
using design sensitivities, which are derivativies of the state
(flow) variables (i.e density, pressure, velocity for an aerodynamic design
problem) with respect to the design parameters (thickness, camber, length
of the airfoil). Sensitivities are used to evaluate an expression for the
gradient of the cost functional (i.e maximum lift coefficient CL, or minimum
drag coefficient CD), obtained using implicit differentiation to the approximate
cost functional f(x;u). The term 
is
the sensitivity of the approximate state variables x to the design parameter
u. So if the sensitivities are known, the gradient of the approximate
cost functional is computed. Thus the approximation of the gradient together
with the approximate function evaluations is used in a gradient based optimization
algorithm and iterated to convergence.
Since the gradient of the cost functional
needs the approximations for the sensitivities of the state variables it
is necessary to use a method to calculate them. The design sensitivities
can be computed using a semi-analytical method, or the finite
difference sensitivity method or the sensitivity equation method,
which has advantages over the previous methods in that there is no need
to remesh or to compute the mesh sensitivities. The so~called sensitivity
equation can be derived by applying implicit differentiation to the
governing state equations (Euler equations for inviscid flow or Navier-Stokes
equations for viscous flow) with the corresponding boundary conditions.
The resulting sensitivity equation often has similar structure to the state
equation. Furthermore, since the design variables rarely appear in the
state equation, the sensitivity equation is linear in .
This can lead to relatively inexpensive approximations of the sensitivies.
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