Help debugging finite element solution in nonlinear elasticity
I'm writing some code to solve problems in nonlinear elasticity using finite element methods. I have been following <PERSON>'s book but I am having trouble with some nagging details.
My question is related to which configuration variables to use for calculating different quantities. I'm working on static problems in the updated Lagrangian formulation, so for each "time" $t$ I have a loading coefficient $\lambda_t$, and a displacement $\sideset{^t}{}{U}$. In chapter 6, <PERSON> derives a linearization of the principle of virtual work, resulting in the equation (6.103): \begin{equation} \left(\sideset{^t_{t\,}}{\text{L}}{K}+\sideset{^t{t\,}}{\text{NL}}{K}\right)U=\sideset{^{t+\Delta t}}{}R-\sideset{^t_t}{}F \end{equation} where $\sideset{^t{t\,}}{\text{L}}{K}$ is the "linear" part of the incremental stiffness matrix in the configuration with displacement $\sideset{^t}{}{U}$, $\sideset{^t{t\,}}{_\text{NL}}{K}$ is the "nonlinear" part, $U$ is the displacement increment, $\sideset{^{t+\Delta t}}{}R$ is the externally applied load at "time" $t+\Delta t$, and $\sideset{^t_t}{}F$ is the nodal point force resulting from the stress, at "time" $t$.
My question is about actually implementing this as a system of nonlinear equations to be solved via <PERSON>'s method. 935
As I understand it, the idea is to consider a function $H(\sideset{^{t}}{}U)=\sideset{^{t+\Delta t}}{}R-\sideset{^t_t}{}F$, where $\sideset{^t_t}{}F$ is a function of $\sideset{^{t}}{}U$ via the stress.
We want to find a displacement $\sideset{^{t+\Delta t}}{}U$ s.t. the unbalanced force vector is zero: \begin{equation} 0=H(\sideset{^{t+\Delta t}}{}U)\approx H(\sideset{^t}{}U)+\nabla H(\sideset{^t}{}U)\left(\sideset{^{t+\Delta t}}{}U-\sideset{^t}{}U\right)=\sideset{^{t+\Delta t}}{}R-\sideset{^t_t}{}F - \left(\sideset{^t_{t\,}}{\text{L}}{K}+\sideset{^t{t\,}}{\text{NL}}{K}\right)U \end{equation} so that $\left(\sideset{^t{t\,}}{\text{L}}{K}+\sideset{^t{t\,}}{_\text{NL}}{K}\right)$ is minus the Jacobian of the unbalanced force vector $\sideset{^{t+\Delta t}}{}R-\sideset{^t_t}{}F$. Is this interpretation correct? Is the sum of the incremental stiffness matrices derived by <PERSON> exactly equal to minus the Jacobian of the unbalanced force vector, when all are evaluated in the same configuration? In implementing <PERSON>'s method, an iteration is introduced, so that we can write $U^{(k)}$ for the incremental displacement at the $k$th iteration. How does this affect the terms in the system of equations? At iteration $k$, are the integrals still calculated over the volume $\sideset{^t}{}V$, or over $\sideset{^{t+\Delta t}}{}V^{(k-1)}$? What about the stress, and the stiffness matrices? 804