# A Point Crack Source Location Method without Velocity Information in Anisotropic Plates
## 2 Methodology
### Objective Function and Minimization  
A simple optimization problem for finding \((x,y)\) from Equation (4) is to minimize \(g(x,y,t,\alpha,s)\), which is defined as  
Figure 2: Illustrative figure of the simulated AE signal. The presence of noise in the AE signal changes the signal arrival time, which is determined by the threshold.  
\[g(x,y,t,\alpha,\mathbf{s})\] \[=\sum_{i\
eq j}\Bigl{[}(t-(t_{j}-s_{j}))^{2}\Bigl{(}(x-x_{i})^{2}+(y-y_ {i})^{2}\alpha^{2}\Bigr{)}-(t-(t_{i}-s_{i}))^{2}\Bigl{(}(x-x_{j})^{2}+(y-y_{j})^ {2}\alpha^{2}\Bigr{)}\Bigr{]}\]  
However, we have more variables (\(n+3\) variables: \(x\), \(y\), \(t\), \(\alpha\), \(s_{1}\), \(\cdots\), \(s_{n}\)) than the amount of measurement data (\(n\) data: \(t_{1}\), \(\cdots\), \(t_{n}\)). This lack of information can be overcome by introducing the regularization term \(|\mathbf{s}|^{2}\) into the optimization process as follows:  
\[(x,y,t,\alpha,\mathbf{s})=\operatorname*{argmin}_{(x,y,t,\alpha,\mathbf{s})}g (x,y,t,\alpha,\mathbf{s})+\lambda|\mathbf{s}|^{2} \tag{5}\]  
where \(\lambda\) is a given constant.  
There are various techniques for solving the unconstrained minimizing problem in Equation (5), such as the gradient descent method, conjugate gradient method, and [PERSON]'s method [28]. In this paper, we use the gradient descent method, which requires the gradient of \(f(x,y,t,\alpha,\mathbf{s})=g(x,y,t,\alpha,\mathbf{s})+\lambda|\mathbf{s}|^{2}\). Since the derivative result of the objective function \(f\) is complicated, we provide the gradient of \(f(x,y,t,\alpha,\mathbf{s})=g(x,y,t,\alpha,\mathbf{s})+\lambda|\mathbf{s}|^{2}\) here for the convenience of the reader: \(\
abla f=\Bigl{[}\frac{\partial f}{\partial x},\frac{\partial f}{\partial y },\frac{\partial f}{\partial t},\frac{\partial f}{\partial x},\frac{\partial f }{\partial s_{1}},\frac{\partial f}{\partial s_{2}},\cdots,\frac{\partial f}{ \partial s_{n}}\Bigr{]}^{T}\) where  
\[\begin{array}{rcl}\frac{\partial f}{\partial x}&=&4\sum_{i\
eq j}\Bigl{[}(t-( t_{j}-s_{j}))^{2}\bigl{(}(x-x_{i})^{2}+(y-y_{i})^{2}\alpha^{2}\bigr{)}-(t-(t_{i}-s_{i }))^{2}\bigl{(}(x-x_{j})^{2}+(y-y_{j})^{2}\alpha^{2}\bigr{)}\Bigr{]}\\ &&\qquad\qquad\times\Bigl{[}(t-(t_{j}-s_{j}))^{2}\bigl{(}x-x_{i})-(t-(t_{i}-s_ {i}))^{2}\bigl{(}x-x_{j})\Bigr{]}\end{array}\] \[\begin{array}{rcl}\frac{\partial f}{\partial y}&=&4\sum_{i\
eq j} \Bigl{[}(t-(t_{j}-s_{j}))^{2}\bigl{(}(x-x_{i})^{2}+(y-y_{i})^{2}\alpha^{2} \bigr{)}-(t-(t_{i}-s_{i}))^{2}\bigl{(}(x-x_{i})^{2}+(y-y_{j})^{2}\alpha^{2} \bigr{)}\Bigr{]}\\ &&\qquad\qquad\times\Bigl{[}(t-(t_{j}-s_{j}))^{2}\bigl{(}(x-x_{i})^{2}+(y-y_{ i})^{2}\alpha^{2}\bigr{)}-(t-(t_{i}-s_{i}))^{2}\bigl{(}(x-x_{j})^{2}+(y-y_{j})^{2} \alpha^{2}\bigr{)}\Bigr{]}\\ &&\qquad\qquad\times\Bigl{[}(t-(t_{j}-s_{j}))\bigl{(}(x-x_{i})^{2}+(y-y_{j})^{ 2}\alpha^{2}\bigr{)}-(t-(t_{i}-s_{i}))\bigl{(}(x-x_{j})^{2}+(y-y_{j})^{2} \alpha^{2}\bigr{)}\Bigr{]}\\ \frac{\partial f}{\partial\alpha}&=&4\sum_{i\
eq j}\Bigl{[}(t-(t_{j}-s_{j}))^ {2}((x-x_{i})^{2}+(y-y_{i})^{2}\alpha^{2})-(t-(t_{i}-s_{i}))^{2}\bigl{(}(x-x_{ j})^{2}+(y-y_{j})^{2}\alpha^{2}\bigr{)}\Bigr{]}\\ &&\qquad\qquad\times\Bigl{[}(t-(t_{j}-s_{j}))\bigl{(}(x-x_{i})^{2}+(y-y_{i})^{ 2}\alpha^{2}\bigr{)}-(t-(t_{i}-s_{i}))\bigl{(}(x-x_{j})^{2}+(y-y_{j})^{2} \alpha^{2}\bigr{)}\Bigr{]}\\ &&\qquad\qquad\times\Bigl{[}(t-(t_{j}-s_{j}))^{2}((x-x_{i})^{2}+(y-y_{i})^{ 2}\alpha^{2})-(t-(t_{i}-s_{i}))^{2}\bigl{(}(x-x_{j})^{2}+(y-y_{j})^{2}\alpha^{ 2}\bigr{)}\Bigr{]}\\ &&\qquad\qquad\times\Bigl{[}(t-(t_{j}-s_{j}))^{2}(y-y_{j})^{2}\alpha-(t-(t_{i}- s_{i}))^{2}(y-y_{j})^{2}\alpha\Bigr{]}\\ &&\qquad\qquad\frac{\partial f}{\partial\alpha_{\mathbf{k}}}&=&4\sum_{k\
eq i}(( x-x_{i})^{2}+(y-y_{i})^{2}\alpha^{2})(t-s_{k})\\ &&\qquad\qquad\times\bigl{[}\ ((x-x_{i})^{2}+(y-y_{i})^{2}\alpha^{2})(t-s_{k})^{2}-((x-x_{ k})^{2}+(y-y_{k})^{2}\alpha^{2})(t-s_{i})^{2}\ \bigr{]}\\ &&\qquad\qquad+2\lambda s_{k}\end{array}\]