# A Level-Set-Based Density Method for Buckling Optimization of Structure with Curved Grid Stiffeners
## 2 Representation of Grid-Stiffener Model  
### Definition of Stiffener Height  
The total height \(h_{e}\) at the center of the \(e\)-th element of the grid-stiffened structure is obtained by adding the base height \(h_{b}\) and the stiffener height \(h_{s}\). \(h_{e}\) is calculated from the function \(h(\mathbf{x})\) at the element center \(\mathbf{x}_{e}=(x_{e},y_{e})\), and \(h(\mathbf{x})\) is given by  
\[h(\mathbf{x})=h_{b}+\tilde{\rho}(\mathbf{x})h_{s} \tag{1}\]  
where \(\tilde{\rho}(\mathbf{x})\) is regarded as the real density of stiffeners, describing their layout. When \(\tilde{\rho}(\mathbf{x})\) is 1, it is considered as stiffeners being present, while values close to 0 indicate the absence of stiffeners. The real density \(\tilde{\rho}(\mathbf{x})\) is derived through a threshold projection of the function \(\rho(\mathbf{x})\). The threshold projection employed in this paper is an approximate Heaviside function [32]  
\[\tilde{\rho}(\mathbf{x})=\frac{\tanh(\tau\eta)+\tanh[\tau(\rho(\mathbf{x})-\eta)]}{\tanh( \tau\eta)+\tanh[\tau(1-\eta)]} \tag{2}\]  
where the value of parameter \(\tau\) affects the approximation of Equation (2) to the ideal Heaviside function; \(\eta\) controls the location of the step point of the function. As shown in Figure 1, as \(\tau\) increases, this function approaches the ideal Heaviside function more closely. In this paper, \(\tau\) is set to 7. In addition, as depicted in Figure 1, when \(\eta=0.5\), the Step Point 1 is positioned at \(x=0.5\). When \(\eta=0.8\), Step Point 2 of the approximate Heaviside function is located at \(x=0.8\). Moreover, this parameter is related to the width of the stiffeners, as discussed later, and can be treated as a design variable in the optimization. The function \(\rho(\mathbf{x})\) is obtained by taking the maximum value from two functions, \(\Gamma(\mathbf{x})\) and \(\Theta(\mathbf{x})\), i.e.,  
\[\rho(\mathbf{x})=\max\{\Gamma(\mathbf{x}),\Theta(\mathbf{x})\} \tag{3}\]  
where the function \(\max\{\cdot,\cdot\}\) is described as [15]  
\[\max\{s,t\}=\frac{1}{2}[s+t+\text{abs}_{\varepsilon}(s-t)] \tag{4}\]  
where \(\text{abs}_{\varepsilon}(a)=\sqrt{a^{2}+\varepsilon}\); and \(\varepsilon\) is set to 0.01. The functions \(\Gamma(\mathbf{x})\) and \(\Theta(\mathbf{x})\) are derived from two functions \(\gamma(\mathbf{x})\) and \(\theta(\mathbf{x})\) through cosine transformations [33], i.e.,  
\[\Gamma(\mathbf{x}) =\frac{1}{2}+\frac{1}{2}\cos(\frac{2\pi}{\mu_{1}}\gamma(\mathbf{x})) \tag{5}\] \[\Theta(\mathbf{x}) =\frac{1}{2}+\frac{1}{2}\cos(\frac{2\pi}{\mu_{2}}\theta(\mathbf{x})) \tag{6}\]  
where \(\gamma(\mathbf{x})\) and \(\theta(\mathbf{x})\) represent fundamental level set functions; \(\mu_{1}\) and \(\mu_{2}\) correspond to the wavelengths of two cosine functions. It can be easily observed that the ranges of the functions \(\Gamma(\mathbf{x})\) and \(\Theta(\mathbf{x})\) are both between 0 and 1. When \(\tau\) in Equation (2) is sufficiently  
Figure 1: The effect of parameters \(\tau\) and \(\eta\) on the approximate Heaviside function. large and the norms of gradient vector of the fundamental level set functions \(\gamma(\mathbf{x})\) and \(\theta(\mathbf{x})\) equal 1, the width \(w_{s}\) of the stiffeners can be expressed as [15]  
\[w_{s}=\frac{\mu}{\pi}\arccos(2\eta-1) \tag{7}\]  
where \(\mu\) can be \(\mu_{1}\) or \(\mu_{2}\). From Equation (7), it can be observed that the width \(w_{s}\) varies in proportion to the wavelength \(\mu\) of the cosine functions, and there is a relationship between the parameters \(w_{s}\) and \(\eta\). Therefore, \(\eta\) can be used as a design variable to change the width of the stiffeners. Based on the aforementioned theories, an example process for generating grid stiffeners is provided. Figure 2 shows the images of the fundamental level set functions \(\gamma(\mathbf{x})\) and \(\theta(\mathbf{x})\). And Figure 3 shows their corresponding contours. It can be observed that the contours of the level set functions are at \(0^{\circ}\) and \(90^{\circ}\).