## 3 Modeling Framework
### Ground Structure Method  
The ground-structure technique, which was first introduced in the 1960s by [PERSON], is usually used for the truss topology design problem. In this approach, a large number of potential nodes and an even larger number of potential bars are distributed over a design domain. With a continuum of geometrically restricted conditions, this sizing reformulation ultimately generates the truss-layout by removing unnecessary members [15].  
A stable truss system that is sufficient to prevent rigid body motion should satisfy the force equilibrium equations to obtain the minimum volume. The formulation is  
\[\begin{array}{l}\min_{\mathbf{a}}\quad V=\mathbf{l}^{T}\mathbf{a}\\ \text{s.t.}\quad\mathbf{B}^{T}\mathbf{n}=\mathbf{f}\\ -\sigma_{\mathcal{C}}a_{i}\leq n_{i}\leq\sigma_{T}a_{i}=1,2\ldots N_{b}\end{array}, \tag{1}\]  
where \(V\) is the the truss' volume; and \(a_{i}\), \(l_{i}\), and \(\sigma_{i}\) are the cross-sectional area, length, and stress, respectively, of the _ith_ truss member (or all \(N_{b}\) members in the truss). \(\mathbf{n}\) is internal bar forces. The parameters \(N_{n}\) and \(N_{sup}\) are the numbers of nodes and components with supports, respectively, and \(N_{dof}\) are the free nodal components with \(N_{dof}=\mathbf{n}\)\(N_{n}-N_{sup}\) for n = 2 in a two-dimensional ground structure. \(\mathbf{B}^{T}\) is the nodal equilibrium matrix of size \(N_{dof}\times N_{b}\), \(f\) is the force's node, and its  
Figure 4: A flow chart of the hybrid topology optimizing procedure for the curved stiffener layout design. L-systems, Lindenmeyer systems.  
number is \(N_{dd}\); **n** is a vector with an internal (axial) force for all members in the ground structure. Stress limits in tension are \(\sigma_{T}>0\) and compression is \(\sigma_{C}<0\).  
To convert the constraint into its equalities, the slack variables \(s^{+}\) and \(s^{-}\) are used, and the optimization problem in (1) becomes a linear programming problem, as follows:  
\[\begin{array}{ll}\underset{\textbf{s}^{+}\textbf{s}^{-}}{\text{min}}&V^{*}= \frac{V}{\sigma_{T}}=\textbf{I}^{T}(\textbf{s}^{+}+\kappa\textbf{s}^{-})\\ \text{s.t.}&\textbf{B}^{T}(\textbf{s}^{+}\textbf{-}\textbf{-}\textbf{-})= \textbf{f}\\ &s_{i}^{+}\cdot s_{i}^{-}\geq 0\end{array} \tag{2}\]  
where \(a_{i}=s_{i}^{+}/\sigma_{T}+s_{i}^{-}/\sigma_{C}\) and \(n_{i}=s_{i}^{+}-s_{i}^{-}\), \(\kappa=\sigma_{T}/\sigma_{C}\). Only one of \(s_{i}^{+}\) and \(s_{i}^{-}\) is non-zero. The member experiences tension if \(s_{i}^{+}>0\) and compression if \(s_{i}^{+}>0\). Using matrix notation, (2) can be rewritten as Equation (3) and can be solved more efficiently using the interior-point algorithm [16; 17]:  
\[\begin{array}{ll}\underset{\textbf{s}^{+}\textbf{s}^{-}}{\text{min}}&V^{*}= \frac{V}{\sigma_{T}}=\left\{\begin{array}{cc}\textbf{I}^{T}&\text{x}\textbf{ I}^{T}\\ 1\times 2N_{b}&2N_{b}\times 1\\ \end{array}\right\}\\ \text{s.t.}&\left[\begin{array}{cc}\textbf{B}^{T}&-\textbf{B}^{T}\\ N_{dd}\times 2N_{b}&2N_{b}\times 1\\ \end{array}\right]\left\{\begin{array}{cc}\textbf{s}^{+}\\ \textbf{s}^{-}\\ 2N_{b}\times 1\\ \end{array}\right\}=\underset{\textbf{s}_{dd}\times 1}{\textbf{s}^{+},s_{i}^{-} \geq 0}\\ \end{array} \tag{3}\]  
There are three major reasons to choose the ground-structure technique as the first topology method to reflect the layout pattern of the given surface domain: (i) the use of a plate-like design domain for the wing surface; (ii) the length of this columnar structure is much greater than its dimensions (diameter or side lengths), and its prime characteristics are along the carbon fiber, which acts like a truss structure; (iii) evenly distributed grid vertexes, which can discretize the whole structure's information and function as knots along the detail stiffeners' path.