1. Constrained gradient-based optimization
1.1 Optimization problem formulation
DAFoam solves constrained nonlinear optimization problems using gradient-based optimization algorithms, with the gradients efficiently computed by the discrete adjoint method. In such problems, an objective function $f(\vec{x})$ is minimized by changing the design variables ($\vec{x}$), subject to certain constraints $\vec{g}(\vec{x})$ and $\vec{h}(\vec{x})$.
\[\begin{aligned} \text{minimize: } & f(\vec{x}), \\ \text{with respect to: } & \vec{x} \\ \text{subject to: } & \vec{h}(\vec{x}) = 0, \\ & \vec{g}(\vec{x}) \le 0. \end{aligned}\]Here, $f$ is a scalar objective function to be minimized, and $\vec{x}$ is the vector of design variables, i.e., the variables we can modify in the design. $\vec{g}(\vec{x})$ and $\vec{h}(\vec{x})$ are the inequality and equality constraint functions, respectively. Here, we want to enforce $\vec{h}(\vec{x}) = 0$ and $\vec{g}(\vec{x}) \le 0$, thereby restricting the feasible design space. Note that $f$, $\vec{g}$, and $\vec{h}$ must be either explicit or implicit functions of the design variables $\vec{x}$. $f(\vec{x})$, $\vec{h}(\vec{x})$, and $\vec{g}(\vec{x})$ can be either linear or nonlinear functions.
A typical example is airfoil aerodynamic shape optimization, where the objective function is the drag coefficient $C_d$, the design variables $\vec{x}$ are control points that change the airfoil shape, and the constraints include a lift (equality) constraint ($C_l=0.5$) and a moment (inequality) constraint $C_m \le b$. Here both the objective and constraint functions are implicit functions of the design variables.
1.2 Iterative optimization process
In general, the above optimization problem can not be solved analytically, so we need to use an iterative approach to find the optimal solution $\vec{x}$. The iterative optimization process starts with an initial guess of the design variables $\vec{x}_0$, called initial design or baseline design. Then, an optimization algorithm compute a search direction and step size to update the design variable using
\[\vec{x}^{n+1} = \vec{x}^n + \alpha^n \vec{d}^n\]Here $n$ is the optimization iteration number, $\alpha$ is a scalar step size, and $\vec{d}$ is the search direction vector. $\vec{d}$ and $\vec{x}$ have the same size.
An optimizer typically first $\vec{d}$ is typically computed based on the gradients $\text{d}f/\text{d}\vec{x}$.
