## 2 Linear Dynamic Analysis with Condensed Localized Mortar Method
### Linear Dynamic Finite Element Analysis Considering Contact Condition  
This subsection presents the basic concept of the 2D linear dynamic finite element analysis used to solve the impact and frictionless contact problems by employing the condensed mortar method. Figure 2 presents a schematic diagram of the 2D contact problem. Even though only two contacting parts are considered in Figure 2, it can be easily extended to multiple parts. The initial boundary value problem of the general structural mechanics comprises a set of combined second-order partial differential equations and a given set of initial and boundary conditions [21].  
Figure 1: Flowchart for linear dynamic contact analysis.  
Figure 2: Schematic description for contact problem.  
The initial boundary value problem for the linear dynamic structural mechanics is given as:  
\[\begin{array}{l}\text{Div}\textbf{P}+\textbf{b}=\rho\tilde{\textbf{u}}\text{ in }\Omega\times\text{ }[0,\text{ T}]\\ \textbf{u}=\textbf{u}\text{ on }\Gamma_{u}\times\text{ }[0,\text{ T}]\\ \textbf{P}\cdot\textbf{N}=\textbf{t}\text{ on }\Gamma_{t}\times\text{ }[0,\text{ T}]\end{array} \tag{1}\]  
where \(\Omega\) is the domain occupied by all the material points, \(\Gamma_{u}\) and \(\Gamma_{t}\) denote the Dirichlet and Neumann boundary, where displacements \(\textbf{u}\) and tractions \(\textbf{t}\) are prescribed. **P** represents the first Piola-Kirchhoff stress tensor, **b** represents an external body force vector, \(\rho\) is the density, **u** is the displacement vector, and the overdot indicates the time derivative.  
In order to address the impact condition with the frictionless contact between adjoined parts, the Karush-Kuhn-Tucker conditions must be satisfied in the normal direction  
\[g_{n}\geq 0,\text{ }\tau_{n}\leq 0,\text{ }\tau_{n}g_{n}=0 \tag{2}\]  
where \(\tau_{n}\) represents the normal component of the contact traction and \(g_{n}\) indicates a gap function in the normal direction, which is defined by:  
\[g_{n}=-\textbf{n}\textbf{x}^{(1)}\bigg{[}\textbf{x}^{(1)}-\tilde{\textbf{x}}^ {(2)}\bigg{]} \tag{3}\]  
where \(\tilde{\textbf{x}}^{(2)}\) is defined by projecting the point \(\textbf{x}^{(1)}\) onto the surface \(\Gamma_{c}\) in the domain \(\Omega_{2}\) along the normal direction **n**. The projected point \(\tilde{\textbf{x}}^{(2)}\) can be obtained by using a well-known contact search algorithm (e.g., [22]).  
When the finite element method is employed to perform structural analysis based on Equations (1)-(3), the momentum equation under an external load can be given by:  
\[\textbf{M}\tilde{\textbf{u}}+\textbf{f}_{\text{int}}(\textbf{u})+\textbf{f}_ {c}(\textbf{u},\boldsymbol{\lambda})-\textbf{f}_{ext}=0 \tag{4}\]  
where **M** is the global mass matrix and \(\textbf{f}_{\text{int}}\), \(\textbf{f}_{\text{ext}}\) and \(\textbf{f}_{c}\) represent the global internal, external, and contact force vectors, respectively. Here, the mass matrix **M** is assumed to be constant.  
In order to obtain the discrete single unknown \(\textbf{u}_{n+1}\) in single step '\(n+1\)', Newmark approximation [23] is used to describe other quantities as a function of \(\textbf{u}_{n+1}\).  
\[\begin{array}{l}\dot{\textbf{u}}_{n+1}=\frac{\gamma}{\beta\Lambda\Gamma}( \textbf{u}_{n+1}-\textbf{u}_{n})-\frac{\gamma-\beta}{\beta\Lambda}\textbf{u}_ {n}-\frac{\gamma-2\beta}{2\beta}\Lambda t\tilde{\textbf{u}}_{n}\\ \tilde{\textbf{u}}_{n+1}=\frac{\gamma}{\beta\Lambda t^{2}}(\textbf{u}_{n+1}- \textbf{u}_{n})-\frac{1}{\beta\Lambda t}\dot{\textbf{u}}_{n}-\frac{1^{2}-2 \beta}{2\beta}\tilde{\textbf{u}}_{n}\end{array} \tag{5}\]  
where \(\beta\) and \(\gamma\) represent the algorithmic parameters for the Newmark method, typically \(2\beta=\gamma=1/2\).