# Optimization of Sensor Positions and Orientations for Multiple Load Case Scenarios
## 2 Materials and Methods
### Optimization Problem Formulation
#### 2.1.6 Measured Strain Along the Sensor Evaluation  
Unlike temperature or other scalar measurements, strain at a point in 3D space in a material can be represented by the strain tensor. Although the primary purpose of a strain sensor is to detect the strain along the sensor, every real-life sensor has some sensitivity to perpendicular strain values. For this reason, a strain transformation method is used to incorporate the impacts of strains perpendicular to the sensor for comprehensive strain detection. The full strain transformation process for a 3D case is shown below. A 3D strain tensor \(\varepsilon\) is presented in Equation (13). \[\varepsilon= \begin{bmatrix}\varepsilon_{11}&\varepsilon_{12}&\varepsilon_{13 }\\.&\varepsilon_{22}&\varepsilon_{23}\\ (\text{sym.})&.&\varepsilon_{33}\end{bmatrix}  \tag{13}\]  
where \(\varepsilon_{11}\), \(\varepsilon_{22}\), and \(\varepsilon_{33}\) are strains along the sensor in \(x_{1},x_{2}\), and \(x_{3}\) directions, respectively. \(\varepsilon_{12}\), \(\varepsilon_{13}\), and \(\varepsilon_{23}\) are shear strains in the \(x_{1}x_{2}\), \(x_{1}x_{3}\), and \(x_{2}x_{3}\) planes, respectively. The transformed strain tensor \(\varepsilon_{\text{trans}}\) is obtained using rotation matrix \(\mathbf{R}\) and original strain tensor \(\varepsilon\). In three-dimensional strain transformation, the strain tensor \(\varepsilon\) in a global (original) coordinate system can be transformed to a new coordinate system using a rotation matrix \(\mathbf{R}\). The transformed strain tensor \(\varepsilon_{\text{trans}}\) can be expressed as  
\[\varepsilon_{\text{trans}}=\mathbf{R}^{\text{T}}\varepsilon\mathbf{R} \tag{14}\]  
Figure 5: Sensor response approximation in the line model. where **R** is a \(3\times 3\) orthonormal rotation matrix constructed from orthonormal basis vectors \(\mathbf{d},\mathbf{m},\) and \(\mathbf{p}.\)\(\mathbf{d}\) is the unit vector in detection of the sensor (see Equation (14)), and \(\mathbf{m},\mathbf{p}\) are unit vectors perpendicular to \(\mathbf{d}.\)  
\[\mathbf{R}=[\mathbf{d}\ \mathbf{m}\ \mathbf{p}]= \begin{bmatrix}d_{1}&m_{1}&p_{1} \\ d_{2}&m_{2}&p_{2}\\ d_{3}&m_{3}&p_{3}\end{bmatrix} = \begin{bmatrix}\cos(\Phi)*\cos(\theta)&-\cos( \theta)*\sin(\Phi)&\sin(\theta)\\ \cos(\Phi)*\sin(\theta)&-\sin(\theta)*\cos(\Phi)&-\cos(\theta)\\ \sin(\Phi)&-\cos(\Phi)&0\end{bmatrix}  \tag{15}\]  
These vectors are used in Equation (13) to extract strain components in their respective directions. The rotation matrix \(\mathbf{R}\) is used to compute all three components, \(\varepsilon_{d},\varepsilon_{m},\varepsilon_{p},\) by enabling the following transformation:  
\[\varepsilon_{\text{trans}}= \begin{bmatrix}\mathbf{d}^{\text{T}}\\ \mathbf{m}^{\text{T}}\\ \mathbf{p}^{\text{T}}\end{bmatrix} \varepsilon \begin{bmatrix}\mathbf{d}& \mathbf{m}&\mathbf{p}\end{bmatrix}  \tag{16}\]  
Expanding this matrix multiplication will give us  
\[\varepsilon_{\text{trans}}= \begin{bmatrix}\mathbf{d}^{\text{T}}\varepsilon \ \mathbf{d}&\mathbf{d}^{\text{T}}\varepsilon\ \mathbf{m}&\mathbf{d}^{\text{T}}\varepsilon\ \mathbf{p}\\ \mathbf{m}^{\text{T}}\varepsilon\ \mathbf{d}&\mathbf{m}^{\text{T}}\varepsilon\ \mathbf{m}&\mathbf{m}^{\text{T}}\varepsilon\ \mathbf{p}\\ \mathbf{p}^{\text{T}}\varepsilon\ \mathbf{d}&\mathbf{p}^{\text{T}}\varepsilon\ \mathbf{m}&\mathbf{p}^{\text{T}}\varepsilon\ \mathbf{p}\end{bmatrix}  \tag{17}\]  
Using the notations for these transformed components  
\[\varepsilon_{\text{trans}}= \begin{bmatrix}\varepsilon_{d}&\varepsilon_{dm}& \varepsilon_{dp}\\.&\varepsilon_{m}&\varepsilon_{mp}\\ (\text{sym}.)&.&\varepsilon_{p}\end{bmatrix}  \tag{18}\]  
In summary, in 3D space, the strain component in the sensor's detection direction is calculated using the following expression:  
\[\varepsilon_{d}=\mathbf{d}^{\text{T}}\varepsilon\mathbf{d} \tag{19}\]  
Expanding this expression, the strain along the sensor can be computed using a quadratic expression that integrates strain tensor and directional vector components.