8.1 The Objective Function

This chapter describes the theoretical framework of FEBio's optimization module. This module can be used to optimize model parameters (like material parameters, or load values) in order to achieve an objective. The objective is generally the minimization of the objective function, to be introduced below.

The objective function, which is to be minimized, can be defined as follows,

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

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.