$$\varphi\left(\mathbf{a}\right)=\sum\limits _{i=1}^{n}\left[y_{i}-f\left(x_{i};\mathbf{a}\right)\right]^{2}.$$

Here, the $\left(x_{i},y_{i}\right)$ are user-defined data pairs and $f\left(x;\mathbf{a}\right)$ is the function that extracts the corresponding data from the model. The optimization module tries to find the model parameters a that minimize the function $\varphi$ . It does this by repeatedly evaluating the function $y$ , which will usually call FEBio to solve a forward FE problem.