# Matrix Method of Defect Analysis for Structures with Areas of Considerable Stiffness Differences
## 2 Materials and Methods
### Stiffness Matrix Formulation for Four-Node Element  
The conception of multi-area elements, based on the example of one-dimensional finite elements, was presented in [36], where verification tests were carried out using the example of a one-dimensional finite element with linear and broken shape functions. The tests performed confirmed the correctness of the assumptions and showed that it is possible to increase the accuracy of the solution up to 100%, i.e., to achieve an exact solution (called the reference solution). Further development of the method allowed for extension of the concept to two-dimensional elements. The area of the four-node finite element was divided into nine component sub-areas with dimensions defined in natural coordinates \(\xi\) and \(\eta\) (see Figure 5a). This allows the insertion of inclusions of any location within the area of the element. The well-known technique of finite element integration and summation in sub-areas is used. However, this technique with linear shape functions is not effective in the case of significant stiffness differences between the component sub-areas. The problem is solved by using the method of adjusting the shape functions distribution to the changes in the component sub-areas stiffness. The original function for modification of the linear distribution of the shape functions has been developed. The stiffness matrix of the multi-area finite element (Figure 5) can be described by the expression:  
\[ \begin{split}\mathbf{K_{e}}=\mathbf{K}_{1}+\mathbf{K}_{2}+\ldots+ \mathbf{K}_{n}=\\ =t_{1}\cdot\iint\limits_{\Omega_{1}}\mathbf{K_{e}^{1}}(\xi,\eta) \ d\xi\ d\eta+t_{2}\iint\limits_{\Omega_{2}}\mathbf{K_{e}^{2}}(\xi,\eta)\ d\xi\ d\eta+\ldots+\ t_{n}\cdot\iint\limits_{\Omega_{n}}\mathbf{K_{e}^{n}}(\xi, \eta)\ d\xi\ d\eta=\sum\limits_{k=1}^{n}\mathbf{K_{e}^{k}}\end{split}  \tag{6}\]  
where \(t_{i}\) (for \(i\) = 1, 2, \(\ldots\), \(n\))--thickness of individual sub-area, and \(\Omega_{i}\) (for \(i\) = 1, 2, \(\ldots\), \(n\))--area of the individual integration sub-area. In the standard approach, it is assumed that the shape functions do not change and remain linear (Figure 5a). Such an interpretation is justified for small differences in the stiffness of individual sub-areas of the finite element, because the effect of changing the displacement field is not so significant. For greater differences in stiffness, such an approach leads to increasingly large calculation errors and modification of the displacement field inside the finite element is obvious. Modification of the displacement field is in the form of broken shape functions (Figure 5a). For a two-dimensional element, the shape functions are the surface resulting from the components about the \(\xi\) and \(\eta\) axes, described by the coefficients \(\text{m}_{1}{}^{\prime}\), \(\text{m}_{2}{}^{\prime}\), \(\ldots\), \(\text{m}_{1}{}^{\prime}\) and \(\text{m}_{1}{}^{\prime\prime}\), \(\text{m}_{2}{}^{\prime\prime}\), \(\ldots\), \(\text{m}_{1}{}^{\prime\prime}\), respectively (Figure 5a). As a result of such a modification, the shape functions in the form of piecewise affine lines are obtained. The stiffness matrix of the single sub-area \(\mathbf{K_{e}^{k}}\) is formulated using eight parameters: \(\xi_{1}\), \(\xi_{2}\), \(\eta_{1}\), \(\eta_{2}\), \(\text{m}_{1}{}^{\prime}\), \(\text{m}_{2}{}^{\prime}\), \(\text{m}_{1}{}^{\prime\prime}\), \(\text{m}_{2}{}^{\prime\prime}\). Similarly to the case of a one-dimensional element [36], the stiffness matrix of a multi-area two-dimensional element is proposed to be determined in an explicit form, which will increase the computational efficiency in relation to numerical integration. Based on expression (6), it should be noted that the matrix of a multi-area element is a sum of the component matrices. The optimal solution is, therefore, to determine the sub-area matrix in a repeatable form.