## 2 Theory of Energy Conserving Integration
### Improved Iteration for Hybrid Simulation  
Substituting Equations (3)-(7) into Equation (8) yields a nonlinear algebraic equation expressed as  
\[\mathbf{M}(2\frac{\mathbf{u}_{n+1}-\mathbf{u}_{n}}{\Delta t^{2}}-2\mathbf{v} _{n})+\mathbf{C}\frac{\mathbf{u}_{n+1}-\mathbf{u}_{n}}{\Delta t}+\mathbf{q}_{ \rm in,m}(\frac{\mathbf{u}_{n+1}+\mathbf{u}_{n}}{2})=\frac{\mathbf{q}_{\rm ex, \it n+1}+\mathbf{q}_{\rm ex,\it n}}{2} \tag{17}\]  
The above equation can be considered as an equivalent static equation written as  
\[\mathbf{R}(\mathbf{u}_{n+1})=\mathbf{F}_{n+1} \tag{18}\]  
in which \(\mathbf{R}(\mathbf{u}_{n+1})\) and \(\mathbf{F}_{n+1}\) are the equivalent internal and external forces respectively. With Newton-type iteration, the displacement increment from \(t_{\rm i}\) to \(t_{\rm i+1}\) can be obtained as  
Figure 1: Beam element in different coordinate systems, (**a**) global system; (**b**) basic (simply supported) system. \[\Delta\mathbf{u}=\sum\mathbf{K}^{-1}[\mathbf{F}_{n+1}-\mathbf{R}(\mathbf{u}_{n+1,i})] \tag{19}\]  
where \(\mathbf{K}\) is the equivalent initial or tangent stiffness matrix; subscript \(i\) represents the number of iterative step. Defining the exact secant stiffness \(\mathbf{K}_{\text{sec}}\) as the stiffness with which the exact solution is obtained with just one iteration, we have  
\[\Delta\mathbf{u}_{\text{T}}=\mathbf{K}_{\text{sec}}^{-1}[\mathbf{F}_{n+1}- \mathbf{R}(\mathbf{u}_{n})] \tag{20}\]  
In each iterative step, the overshoot can be identified by a difference vector written as  
\[\Phi_{i}=\left|\Delta\mathbf{u}_{n+1,i}\right|-\left|\Delta\mathbf{u}_{\text{ T}}\right| \tag{21}\]  
in which \(\mid\mid\) denotes absolute value of a vector. If some components of \(\mathbf{K}\) are smaller than those of \(\mathbf{K}_{\text{sec}}\), the relevant component of \(\mathbf{\Phi}\) could be negative, which indicates overshoot taking place. For hybrid tests, a possible significant problem arising from iteration is the overshooting with which the displacement exceeds the true value and then goes back, approaching the true value. The overshoot would bring artificial hysteresis for inelastic materials, and hence should be avoided or at least confined at an acceptable level. In a hybrid simulation with Newton-type iteration algorithm, the stiffness matrix used for the iteration consists of two parts: one from the numerical substructure and the other from the experimental substructure. The former is assembled and updated from the FE program, while the latter is usually fixed and the initial stiffness is quite often an option. In some cases, this iterative stiffness could be \"smaller\" than exact secant stiffness, and hence produce the reciprocating iterative phenomenon [22]. This iteration will lead to artificial numerical dissipation and inaccurate structure response. The authors of [22] presented a solution of introducing a reducing parameter in every iteration, which can reduce the possibility of spurious loading/unloading cycles. However, in the same condition of convergence criterion, it takes much more computation time to reach a satisfactory equilibrium, and the uniform displacement increment cannot be ensured. In general, the application of the iterative stiffness matrix always makes the overshooting phenomenon take place in the first or second iteration and disappear in the following iterations of each time step. Thus, the current paper presents a solution of increment reduction factor \(\theta\) only for the first one or two iterations. The reduction factor \(\theta\) of displacement increment is used in first or first two iterations for the balance of overshoot remedy and computation efficiency.