## 3 Modelling Strategies
### Numerical Solvers: Explicit and Implicit Methods  
Time integration methods are numerical techniques developed to approximate the equations of motion using a step-by-step direct integration procedure. Equations (1) and (2) are the linear and nonlinear equations of motion of a discrete structural, respectively [65]:  
\[\mathbf{M}\ddot{\mathbf{U}}+\mathbf{C}\dot{\mathbf{U}}+\mathbf{K}\mathbf{U}=\mathbf{R}(t) \tag{1}\]  
\[\mathbf{M}\ddot{\mathbf{U}}+\mathbf{C}\dot{\mathbf{U}}+\mathbf{N}(\mathbf{U})=\mathbf{R}(t) \tag{2}\]  
where \(\mathbf{M}\) is the structural mass matrix, \(\mathbf{C}\) is the viscous damping matrix, \(\mathbf{K}\) is the linear stiffness matrix, \(\mathbf{N}\) is the vector of nonlinear internal forces related to the current vector displacements \(\mathbf{U}\) at time \(t\), and \(\mathbf{R}(t)\) is the vector of time-dependent external forces. The subscript \((\cdot)\) indicates time derivatives of the displacement vector \(\mathbf{U}\). The problem consists of finding the time history solution for \(\mathbf{U}=\mathbf{U}(t)\), with the defined initial conditions \(\mathbf{U}(0)=\mathbf{U}_{o}\) and \(\hat{\mathbf{U}}(0)=\hat{\mathbf{U}}_{o}\), which for seismic actions are equal to zero. Direct numerical integration methods discretise the time \(t\) into time intervals \(\Delta t\) and assume that Equations (1) and (2) are solved for each time interval considering variations in displacements, velocities, and accelerations at each time step. Therefore, the total response is a sequence of independent problems, where the conditions change from one step to another [66]. Depending on the assumptions made about these variations, different integration schemes are derived, which can be further classified into explicit and implicit methods [67]. In the explicit methods, the solution at time \(t+\Delta t\) is calculated considering the equilibrium conditions at time \(t\), using the known values of \(\mathbf{U}\), \(\hat{\mathbf{U}}\), and \(\hat{\mathbf{U}}\) found in the previous step. The vector displacement (at the end of the time interval) is not a function of the loads or stiffness (at the end of the time step). As a result, no iteration is needed, and the method can directly proceed to the next time interval [66,67]. The approach allows for the incorporation of very complex constitutive laws and does not require the inversion of the stiffness matrix, resulting in a low computational cost per time step. Nevertheless, explicit methods are conditionally stable, meaning that the time increments should be small enough to ensure numerical stability. The time interval depends on the highest frequency of the discrete system without damping [67]. The introduction of damping affects the time step, reducing its length (Section 5). The main explicit solvers include the second-order central difference methods, which are suitable for large-scale models in stress-type analysis, and the fourth-order Runge-Kutta method [67]. In implicit solvers, the solution for displacements at the current increment \(\Delta t\) requires the velocities and accelerations of the current time itself, so trial values are assumed and refined by an iterative process [66,67]. This implies that the stiffness matrix should be inverted at each time step for the calculation of displacements. Implicit methods are normally unconditionally stable, where the maximum time interval is determined by the desired accuracy [67]. Conversely, the time step may be much larger than in explicit methods, and, thus, fewer intervals are required. The implicit time integration schemes include the Newmark methods, [PERSON]-\(\theta\) method, and Houbolt methods [68]. Among these, Newmark methods have better accuracy in structural problems. A generalized form of Newmark methods is the Hilber-Hughes-Taylor (HHT) method. It introduces the parameter \(\alpha\) in the original Newmark formulation to improve the stability with larger steps.