Prev Section 3.8: Newton-Raphson Method Up Section 3.8: Newton-Raphson Method Subsection 3.8.2: BFGS Method Next
3.8.1 Full Newton Method
The Newton-Raphson equation (3.1.1-1) can be written in terms of the discretized equilibrium equations that were derived in the previous section as follows: Since the virtual velocities are arbitrary, a discretized Newton-Raphson scheme can be formulated as follows: This is the basis of the Newton-Raphson method. For each iteration , both the stiffness matrix and the residual vector are re-evaluated and a displacement increment u is calculated by pre-multiplying both sides of the above equation by . This procedure is repeated until some convergence criteria are satisfied.
The formation of the stiffness matrix and, especially, calculation of its inverse, are computationally expensive. Quasi-Newton methods do not require the reevaluation of the stiffness matrix for every iteration. Instead, a quick update is calculated. One particular method that has been quite successful in the field of computational solid mechanics is the BFGS method, which is described in the next section.