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3.8.2 BFGS Method
The BFGS method updates the stiffness matrix (or rather its inverse) to provide an approximation to the exact matrix. A displacement increment is defined as and an increment in the residual is defined as The updated matrix should satisfy the quasi-Newton equation: In order to calculate this update, as displacement increment is first calculated: This displacement vector defines a “direction” for the actual displacement increment. A line search (see next section) can now be applied to determine the optimal displacement increment: where is determined from the line search. With the updated position calculated, can be evaluated. Also, using equations (3.8.2-1) and (3.8.2-2), and can be evaluted. The stiffness update can now be expressed as where the matrix is an matrix of the simple form: The vectors and are given by The vector is equal to and has already been calculated.
To avoid numerically dangerous updates, the condition number of the updating matrix is calculated: The update is not performed when this number exceeds a preset tolerance.
Considering the actual computations involved, it should be noted that using the matrix updates defined above, the calculation of the search direction in (3.8.2-4) can be rewritten as, Hence, the search direction can be computed without explicitly calculating the updated matrices or performing any additional costly matrix factorizations as required in the full Newton-Raphson method.