# An Investigation of Particle Swarm Optimization Topologies in Structural Damage Detection
## 2 Structural Damage Detection  
### Problem Formulation  
For an undamped system, the structure can be discretized into \(D\) elements while using the finite element methods (FEM), and the structural characteristic equation is expressed as:  
\[\left(\mathbf{K}\,-\,\lambda_{s}\,\mathbf{M}\right)\Phi_{s}\,=\,0,\,\,\,\left(s =1,2,\ldots,m\right) \tag{1}\]  
where \(\mathbf{M}\) is the global mass matrix, \(\mathbf{K}\) is the global stiffness matrix, \(\lambda_{s}\) indicates the \(s\)th eigenvalue of the structure, and \(\Phi_{s}\) denotes the \(s\)th mode shape. Subsequently, \(\mathbf{M}\) and \(\mathbf{K}\) are assembled by the corresponding element matrices:  
\[\mathbf{K}=\sum_{j=1}^{D}\mathbf{k}_{j} \tag{2a}\] \[\mathbf{M}=\sum_{j=1}^{D}\mathbf{m}_{j} \tag{2b}\]  
where \(\mathbf{k}_{j}\) is the \(j\)th element stiffness matrix, \(\mathbf{m}_{j}\) is the \(j\)th element mass matrix, and \(D\) is the number of elements of the structure.  
In this paper, an Euler-Bernoulli beam is modeled using the FEM. The beam is discretized into a number of \(D\) elements, with the displacement and slope as nodal degreesof freedom and cubic interpolation function. For a uniform beam of length \(L\), the mass matrix and the stiffness matrix of each beam element are given, as follows:  
\[\mathbf{m^{e}}=\frac{\rho Al}{420}\begin{bmatrix}156&22l&54&-13l\\ 22l&4l^{2}&13l&-3l^{2}\\ 54&13l&156&-22l\\ -13l&-3l^{2}&-22l&4l^{2}\end{bmatrix} \tag{3}\]  
\[\mathbf{k^{e}}=\frac{EI}{l^{3}}\begin{bmatrix}12&6l&-12&6l\\ 6l&4l^{2}&-6l&2l^{2}\\ -12&-6l&12&-6l\\ 6l&2l^{2}&-6l&4l^{2}\end{bmatrix} \tag{4}\]  
where \(l=L/D\) is the finite element length, \(\rho\) is the mass density, \(A\) is the cross-section area, \(E\) is Young's modulus, and \(l\) is the moment of inertia. Subsequently, convert the local coordinate system to the global coordinate system, the global mass matrix \(\mathbf{M}\), and global stiffness matrix \(\mathbf{K}\) can be obtained using Equations (2a) and (2b).  
A scenario should first be defined in order to simulate any damage in the structure. In fact, each damage scenario is a D-dimensional vector, and the \(j\)th component of the vector represents the damage to the structure. In this study, the damage is modeled using the [PERSON]'s modulus reduction, and the decrease in mass is negligible [22]. Assuming that the element stiffness of the structure is uniformly reduced after damage, then the reduction factor is used to indicate damage:  
\[\alpha_{j}\,=\,\frac{E\,-\,E_{j}}{E} \tag{5}\]  
where \(E\) and \(E_{j}\) are the [PERSON]'s modulus of the \(j\)th element before and after the damage. \(\alpha_{j}\in[0,1]\) and \(\alpha_{j}=0\) mean no damage, while \(\alpha_{j}=1\) indicates a complete loss of the element stiffness, and \(\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{D})\) is the damage vector of the structure.  
This definition is profitable, because it allows for estimating not only the severity of the damage, but also its location at the elemental level. When considering the damage vector \(\alpha\), the stiffness matrix \(\mathbf{K}\) after the damage is changed as:  
\[\mathbf{K}^{d}\,=\,\sum_{j=1}^{D}(1\,-\,\alpha_{j})\,\mathbf{k}_{j} \tag{6}\]  
where \(\mathbf{K}^{d}\) is the global stiffness matrix after damage.