# Special Finite Elements with Adaptive Strain Field on the Example of One-Dimensional Elements
## 3 General Theoretical Assumptions of One-Dimensional Multi-Area Elements  
The conception of multi-area finite element has been presented and explained in Figure 6, on the example of one-dimensional element. Comparison between standard solution (Figure 6a) and multi-area elements (Figure 6b) shows that the deformation field in both cases is described by the shape functions [25; 26]. However, in multi-area elements, the shape function distribution is modified by the \(\chi\) parameter. This modification (\(\chi\)) causes that stiffness is redistributed inside the finite element. Therefore, the basic assumption of multi-area conception is appropriate adjustment of shape function distribution to changes of stiffness characteristics in sub-areas, as shown in Figure 6b. Shape functions modified in this way could be described for example using spline functions. Theoretical considerations concerning spline functions are included in chapters 14 and 15 of the work [27]. The use of spline functions in FEM calculations is presented in chapter 18 of work [28]. Figure 5: Diagram of the calculation error. Figure 6: Linear element with variable stiffness-geometric parameters: (**a**) Standard stiffness summation from sub-areas; (**b**) Multi-area element conception. However, the authors propose quite uncomplicated solution in form of modified stiffness summation from sub-areas, as it is described by Formula (1). The advantage of this solution is that there is no need for homogenization process. The stiffness matrix of a single sub-area is determined from the Formula (2). \[\mathbf{K}_{k}(\chi)=\int\limits_{\xi_{k}}\mathbf{B}_{k}(\chi)\cdot\mathbf{D}_{ k}\cdot\mathbf{B}_{k}(\chi)\cdot dv_{k} \tag{2}\]  
Global coordinates \(x\) and generalized dimensionless coordinates \(\xi\) are enclosed in the following intervals:  
\[x\in\langle-\frac{L}{2},+\frac{L}{2}\rangle,\quad\xi\in\langle-1,+1\rangle, \text{ where:}\quad\xi=\frac{2}{L}x. \tag{3}\]  
Shape functions expressed in natural coordinates \(\xi\) have the following forms for \(i\) and \(j\) node, respectively:  
\[ \begin{array}{c}N_{k}^{i}=\frac{\chi_{(1+1}\cdot\eta_{i+1}-\chi_{l}\cdot m_{ l}}{\xi_{l+1}-\xi_{l}}(\xi-\xi_{l})+\chi_{l}\cdot m_{l}\\ N_{k}^{i}=1-N_{k}^{i}\end{array}  \tag{4}\]  
The strain field:  
\[ \begin{array}{c}B_{k}=L\cdot N_{k}=\left|\frac{d}{d\xi}\right|\cdot\left|N_ {k}^{i}\quad N_{k}^{j}\quad\right|=\\ =\left|\begin{array}{cc}\frac{\chi_{(1+1}\cdot\eta_{i+1}-\chi_{l}\cdot m_{ l}}{\xi_{l+1}-\xi_{l}}&-\frac{\chi_{(1+1}\cdot\eta_{i+1}-\chi_{l}\cdot m_{ l}}{\xi_{l+1}-\xi_{l}}&\\ \end{array}\end{array} \right|=\left| \begin{array}{cc}b&-b\end{array} \right| \end{array} \tag{5}\]  
After substituting to the Equation (2), final formula has been obtained:  
\[\mathbf{K}_{k}(\chi)=\int\limits_{\xi_{k}}\mathbf{B}_{k}(\chi)\cdot\mathbf{D}_ {k}\cdot\mathbf{B}_{k}(\chi)\cdot dv_{k}=\int\limits_{\xi_{l}}^{\xi_{l+1}} \left| \begin{array}{cc}b\\ -b\end{array} \right|\cdot\left|E_{k}\right|\cdot\left| \begin{array}{cc}b&-b \end{array} \right|\cdot\left|\cdot\mathbf{J}\cdot d\xi \tag{6}\]  
where:  
\(\mathbf{B}_{k}\)--strain matrix of sub-area \(k\),  
\(\mathbf{D}_{k}\)--material matrix, \(\mathbf{D}_{k}^{k}=\left[E_{k}\right]\),  
\(\mathbf{N}_{k}\)--shape function matrix of sub-area \(k\),  
\(\mathbf{J}\)--Jakobian matrix equals \(2/L\),  
\(L\)--length of the whole element. After transformations, stiffness matrix of the component sub-area has the following explicit form:  
\[\begin{array}{c}/\mathbf{K}_{k}(\chi)=(EA)_{k}\cdot\left(\frac{(\chi_{(1+1 }\cdot\eta_{i+1}-\chi_{l}\cdot m_{l}))}{(\xi_{l+1}-\xi_{l})}\right)\cdot\int \limits_{\xi_{l}}^{\xi_{l+1}}\left[ \begin{array}{cc}1&-1\\ -1&1\end{array}