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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: $$\begin{equation} \delta\mathbf{v}^{T}\cdot\mathbf{K}\cdot\mathbf{u}=-\delta\mathbf{v}^{T}\cdot\mathbf{R}.\label{eq355} \end{equation}$$ Since the virtual velocities $\delta\mathbf{v}$ are arbitrary, a discretized Newton-Raphson scheme can be formulated as follows: $$\begin{equation} \begin{array}{cc} \mathbf{K}\left(\mathbf{x}_{k}\right)\cdot\mathbf{u}=-\mathbf{R}\left(\mathbf{x}_{k}\right); & \mathbf{x}_{k+1}=\mathbf{x}_{k}+\mathbf{u}\end{array}.\label{eq356} \end{equation}$$ This is the basis of the Newton-Raphson method. For each iteration $k$ , 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 $\mathbf{K}^{-1}$ . This procedure is repeated until some convergence criteria are satisfied.