# Design and Mechanical Properties of Negative Poisson's Ratio Structure-Based Topology Optimization
## 2 Design of a Negative Poisson's Ratio Structure Based on the Variable Density Method
### Objective Function  
Assuming the relationship between stress and strain is:  
\[\sigma_{ij}=C_{ijkl}\epsilon_{kl} \tag{1}\]  
where the expression of \(C_{ijkl}\) for the homogenized elastic tensor is:  
\[C_{ijkl}^{H}=\frac{1}{|Y|}\int\limits_{Y}C_{ijpq}(\varepsilon_{pq}^{0(kl)}- \varepsilon_{pq}^{*(kl)})dY \tag{2}\]  
where \(Y\) is the base cell domain, \(C_{ijpq}\) is the elasticity tensor in index notation, \(\varepsilon_{pq}^{0(kl)}\) is the unit test strain fields, and \(\varepsilon_{pq}^{*(kl)}\) is the periodic fluctuation strain fields. According to [PERSON]'s [44] proposed theory of element interconnectivity, further calculations for topology optimization are facilitated. The Formula (2) is rewritten as follows:  
\[Q_{ijkl}^{H}=\frac{1}{|Y|}\sum_{e=1}^{N}{{{\left({{\mathbf{u}}_{e}^{A(ij)}} \right)}^{T}}{k_{e}}{\mathbf{u}}_{e}^{A(kl)}} \tag{3}\]  
where N is the number of elements obtained by dividing the entire design domain into N parts, \({\mathbf{u}}_{e}^{A(ij)}\) is the element displacement vector for load case _ij_, and \({k_{e}}\) is the stiffness. The objective of topology optimization in this paper is to design a suitable structure in a two-dimensional plane and then finally convert \(Q_{ijkl}^{H}\) into \(Q_{ij}^{H}\) in a two-dimensional problem. To do this, we rewrite Equation (4) in extended form, as shown below:  
\[ \begin{bmatrix}C_{11}^{H}&C_{12}^{H}&C_{13}^{H}\\ C_{21}^{H}&C_{22}^{H}&C_{23}^{H}\\ C_{31}^{H}&C_{32}^{H}&C_{33}^{H}\end{bmatrix} = \begin{bmatrix}Q_{11}&Q_{12}&Q_ {13}\\ Q_{21}&Q_{22}&Q_{23}\\ Q_{31}&Q_{32}&Q_{33}\end{bmatrix}  \tag{4}\]  
where \(C_{11}^{H}\) and \(C_{22}^{H}\) are the theoretical equivalent moduli of elasticity in directions 1 and 2, respectively, \(Q_{11}\) and \(Q_{22}\) are the principal strain energies in directions 1 and 2, respectively, and \(E_{12}^{H}\) and \(C_{12}\) are theoretical shear modulus and the shear strain energy, respectively. \[Q_{ij}=\frac{1}{|Y|}\sum_{e=1}^{N}{q_{e}^{(ij)}} \tag{5}\]  
\[{q_{e}^{(ij)}}={{\left({{\mathbf{u}}_{e}^{A(i)}}\right)}^{T}}{k_{e}}{\mathbf{u }}_{e}^{A(i)} \tag{6}\]  
where \({q_{e}^{(ij)}}\) is the element e mutual energy, \({\mathbf{u}}_{e}^{A(i)}\) is the element displacement vector for load case _ij_, and \({k_{e}}\) is the element stiffness matrix. The method uses [PERSON]'s modulus based on the pseudo-density of the element. \[{E_{e}}({x_{e}})={E_{min}}+{x_{e}^{p}}({E_{0}}-{E_{min}}),\;{x_{e}}\in[0,1] \tag{7}\]  
In the equation, \({E_{e}}\) is the [PERSON]'s modulus of the final material, \({E_{min}}\) is the [PERSON]'s modulus of the substitute material, and \({E_{0}}\) is the [PERSON]'s modulus of the starting material. Directly using the [PERSON]'s ratio formula -\({C_{12}}/{C_{11}}\) as the objective function and the existing optimization criteria for iterative optimization can be highly challenging. However, based on the formulation proposed by [PERSON] [38], topology optimization introduces different relaxation factors into two-dimensional problems. This approach enables the optimized topology structure to maintain a negative Poisson's ratio while significantly improving the unidirectional stiffness of the resulting microstructure unit cells. The mathematical expression of the objective function in the final optimization problem is:  
\[ \begin{array}{l}\text{min:}c={C_{12}}-{\left[{{(0.8-\beta)}^{l}\cdot{C_{11} }+0.8^{l}\cdot{C_{22}}}\right]}\\ \text{Subject to : }\text{KU}^{A(kl)}=F^{(kl)},k,l=1,\ldots,d\\ \sum\limits_{e=1}^{N}{{{\left.{v_{e}}{x_{e}}\right.}\leq V}}\\ 0\leq{x_{e}}\leq 1,e=1,\ldots,N\end{array}