## 2 Theory
### Filtering and Parameterizations
For instance, a general rectangular thin plate maintains its hardening nonlinearity, and the resultant curve passes through the target points while retaining the hardening nonlinearity to minimize the objective function. The resultant curve is depicted by a dotted line (...). If one uses a large number of target points, this issue can be resolved without any constraints. However, when using only a small number of target points without the slope constraint, the optimization design process always falls into the local minimum, and the F-D curve behaves as shown by the dotted line in Figure 1.  
Another slope can be defined at the target points by tracing the line that connects the origin and the target points and considering its angle with the force axis. These are denoted by \(\theta_{c}^{P1}\), \(\theta_{c}^{P2}\), and \(\theta_{c}^{P3}\). The slope \(\theta_{k}^{P}\) is associated with the secant stiffness matrix, and the slope \(\theta_{c}^{P}\) is associated with the inverse of the secant stiffness matrix. In a previous study [13], the tangential stiffness and secant stiffness matrices were used as objective functions, which were associated with slopes concerning \(\mathbf{K}_{T}^{P}\) and \(\theta_{k}^{P}\), respectively, to minimize compliance for nonlinear structures. Even if they eventually maximize the slopes in the F-D plane, the sensitivities to the compliances according to the two stiffness matrices described in detail in the reference are not directly related to slopes \(\theta_{k}^{P}\) and \(\theta_{c}^{P}\). Let us now discuss the sensitivity analysis related to the slope constraints. In a similar way to a previous study [34], the principle of virtual work is applied to derive the finite element equation in a matrix-vector form for the in-plane motion of the thin rectangular plate:  
\[\mathbf{K}\Big{(}\mathbf{u},\ \boldsymbol{\alpha},\overline{\boldsymbol{ \bar{\rho}}}\Big{)}\mathbf{u}\ =\ \Big{[}\mathbf{K}_{L}\Big{(}\boldsymbol{\alpha},\overline{ \boldsymbol{\bar{\rho}}}\Big{)}\ +\ \mathbf{K}_{NL}\Big{(}\mathbf{u},\ \boldsymbol{\alpha}, \overline{\boldsymbol{\bar{\rho}}}\Big{)}\Big{]}\mathbf{u}\ =\ \boldsymbol{\mathbf{f}}^{ext} \tag{14}\]  
where \(\mathbf{K}\) is the secant stiffness matrix, which is composed of the linear stiffness matrix \(\mathbf{K}_{L}\) and the nonlinear stiffness matrix \(\mathbf{K}_{NL}\). If a structure considered in the optimization problem is displaced by a concentrated force, (i.e., a single input case, in Equation (14), \(\boldsymbol{\mathbf{f}}_{k}^{ext}=f\) and \(\boldsymbol{\mathbf{f}}_{n}^{ext}\ =0\) ; \(\ n=1\ \sim\ N\,n\
eq i\)),  
Figure 1: Introduction of definable slopes in the force–displacement (F-D) curve and estimated F-D curves with and without a slope constraint in the optimization process. then the slope \(\theta_{c_{ij}}^{P}\) at the target point \(P\), which is defined for the external force at the \(i\)th degree of freedom and the displacement at \(j\)th degree of freedom, is represented by the inverse of the secant matrix. \[\theta_{c_{ij}}^{P}=\mathbf{K}_{ij}^{-1} \tag{15}\]  
The slope \(\theta_{c}^{P}\) is associated with the inverse of the matrix \(\mathbf{K}\), and its sensitivity can be derived by the direct differentiation method (DDM):  
\[ \begin{array}{rcl}\frac{d\mathbf{K}^{-1}}{d\overline{\rho}}&=&\frac{d\mathbf{ K}^{-1}}{d\overline{\rho}}\frac{\partial\mathbf{u}}{\partial\overline{\rho}}+ \frac{d\mathbf{K}^{-1}}{\partial\alpha}\frac{d\overline{\rho}}{\partial \overline{\rho}}&+&\frac{d\mathbf{K}^{-1}}{d\overline{\rho}}\\ &=&\mathbf{K}^{-1}\frac{d\overline{\rho}}{\partial\mathbf{u}}\mathbf{K}^{-1} \frac{d\overline{\rho}}{\partial\overline{\rho}}+\mathbf{K}^{-1}\frac{d \overline{\rho}}{\partial\alpha}\mathbf{K}^{-1}\frac{d\overline{\alpha}}{ \partial\overline{\rho}}&+&\mathbf{K}^{-1}\frac{d\overline{\rho}}{\partial \overline{\rho}}\mathbf{K}^{-1}\end{array}  \tag{16}\]  
Equation (16) represents the sensitivity of the inverse of the stiffness matrix \(\mathbf{K}\) with respect to the physical design density vector \(\overline{\rho}\).