2. 位移协调元的最小势能原理
根据式(1.1)，弹性体最小势能原理的泛函可以表示为
(2.1)\[\begin{equation}
\Pi = \iiint\limits_V {\left[ {A\left( {{\varepsilon _{ij}}} \right){\text{ - }}{f_i}{u_i}} \right]{\text{d}}V} - \iint\limits_{{S_p}} {{{\bar p}_i}{u_i}{\text{d}}S}
\end{equation}\]
其中应变和位移满足几何方程，位移边界条件为变分约束条件，物理方程为非变分约束条件。 首先对求解域进行离散，把求解域\(V\)分割为\(N\)个有限单元，其中\(m\)号有限单元子域为\({{V^{\left( m \right)}}}\)，外表面为\({{S^{\left( m \right)}}}\)。设在相邻有限元之间的交界面上位移函数\(u_{i}\) 是连续的，这种有限元称为位移协调元。位移协调元要求单元位移函数满足下列条件：
（1）在每个单元中是连续的和单值的；
（2）在单元的交界面上是协调的，即\(u_{i}^{(m)}=u_{i}^{\left(m^{\prime}\right)} \quad\left(\text { 在 } S^{\left(m m^{\prime}\right)} \text { 上 }\right)\)；
（3）所有含有 \(S_{u}\) 的单元，都要满足位移边界条件\(u_{i}=\bar{u}_{i} \quad\left(\text { 在 } S_{u} \text { 上 }\right)\)
如果位移函数的选择满足以上三个条件，则基于位移协调元的最小势能原理的泛函可以改写成：
(2.2)\[\begin{equation}
{\Pi ^*} = \sum\limits_{m = 1}^N {\left\{ {\iiint\limits_{{V^{\left( m \right)}}} {\left[ {{A^{\left( m \right)}}\left( {{\varepsilon _{ij}}} \right) - {f_i}u_i^{\left( m \right)}} \right]{\text{d}}V} - \iint\limits_{S_p^{\left( m \right)}} {{{\bar p}_i}u_i^{\left( m \right)}{\text{d}}S}} \right\}}
\label{eq:Pi*_1}
\end{equation}\]
对上式求一阶变分, 得
(2.3)\[\begin{equation}
\delta {\Pi ^*} = \sum\limits_{m = 1}^N {\left\{ {\iiint\limits_{{V^{\left( m \right)}}} {\left[ {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta \varepsilon _{ij}^{\left( m \right)} - {f_i}\delta u_i^{\left( m \right)}} \right]{\text{d}}V} - \iint\limits_{S_p^{\left( m \right)}} {{{\bar p}_i}\delta u_i^{\left( m \right)}{\text{d}}S}} \right\}}
\label{eq:delta_Pi*_1}
\end{equation}\]
对于第\(m\)个单元有
(2.4)\[\begin{equation}
\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta \varepsilon _{ij}^{\left( m \right)} = \frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta u_{i,j}^{\left( m \right)}
\end{equation}\]
(2.5)\[\begin{equation}
{\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta u_i^{\left( m \right)}} \right)_{,j}} = \frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta u_{i,j}^{\left( m \right)} + {\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}} \right)_{,j}}\delta u_i^{\left( m \right)}
\end{equation}\]
第\(m\)个单元上应用高斯散度定理得
(2.6)\[\begin{equation}
\iiint\limits_V {{{\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta u_i^{\left( m \right)}} \right)}_{,j}}{\text{d}}V} = \iint\limits_{{S^{\left( m \right)}}} {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}n_j^{\left( m \right)}\delta u_i^{\left( m \right)}{\text{d}}S}
\end{equation}\]
因此式(2.3)右边第一项
(2.7)\[\begin{split}\begin{equation}
\begin{array}{*{20}{l}}
{\iiint\limits_{{V^{\left( m \right)}}} {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta \varepsilon _{ij}^{\left( m \right)}{\text{d}}V}}&{ = \iiint\limits_{{V^{\left( m \right)}}} {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta u_{i,j}^{\left( m \right)}{\text{d}}V}} \\
{\text{ }}&{ = \iiint\limits_{{V^{\left( m \right)}}} {\left[ {{{\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}\delta u_i^{\left( m \right)}} \right)}_{,j}} - {{\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}} \right)}_{,j}}\delta u_i^{\left( m \right)}} \right]{\text{d}}V}} \\
{\text{ }}&{ = \iint\limits_{{S^{\left( m \right)}}} {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}n_j^{\left( m \right)}\delta u_i^{\left( m \right)}{\text{d}}S} - \iiint\limits_{{V^{\left( m \right)}}} {{{\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}} \right)}_{,j}}\delta u_i^{\left( m \right)}{\text{d}}V}} \\
{\text{ }}&{ = \iint\limits_{S_p^{\left( m \right)}} {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}n_j^{\left( m \right)}\delta u_i^{\left( m \right)}{\text{d}}S} + \iint\limits_{{S^{\left( {m{m^\prime }} \right)}}} {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}n_j^{\left( m \right)}\delta u_i^{\left( m \right)}{\text{d}}S} - \iiint\limits_{{V^{\left( m \right)}}} {{{\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}} \right)}_{,j}}\delta u_i^{\left( m \right)}{\text{d}}V}}
\end{array}
\end{equation}\end{split}\]
其中，\({{S^{\left( m \right)}}}\)由三部分组成
(2.8)\[\begin{equation}
{S^{(m)}} = S_p^{(m)} + S_u^{(m)} + {S^{\left( {m{m^\prime }} \right)}}
\end{equation}\]
\({S^{\left( {m{m^\prime }} \right)}}\)是相邻有限单元 \(m\) 与 \(m^{\prime}\) 之间的交界面。根据位移协调的条件，在单元交界面\({S^{\left( {m{m^\prime }} \right)}}\) 上有
(2.9)\[\begin{equation}
u_i^{\left( m \right)} = u_i^{\left( {{m^\prime }} \right)} = u_i^{\left( {m{m^\prime }} \right)} \quad\left(\text { 在 } S^{\left(m m^{\prime}\right)} \text { 上 }\right)
\end{equation}\]
或
(2.10)\[\begin{equation}
\delta u_i^{\left( m \right)} = \delta u_i^{\left( {{m^\prime }} \right)} = \delta u_i^{\left( {m{m^\prime }} \right)} \quad\left(\text { 在 } S^{\left(m m^{\prime}\right)} \text { 上 }\right)
\end{equation}\]
因此，式(2.3)可写成
(2.11)\[\begin{split}\begin{equation}
\begin{array}{*{20}{l}}
{\delta {\Pi ^*}}&{ = \sum\limits_{m = 1}^N {\left\{ { - \iiint\limits_{{V^{\left( m \right)}}} {\left[ {{{\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}} \right)}_{,j}} + {f_i}} \right]\delta u_i^{\left( m \right)}{\text{d}}V} + \iint\limits_{S_p^{\left( m \right)}} {\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}n_j^{\left( m \right)} - {{\bar p}_i}} \right)\delta u_i^{\left( m \right)}{\text{d}}S}} \right\}} } \\
{\text{ }}&{ + \sum\limits_{\left( {m{m^\prime }} \right)} {\iint\limits_{{S^{\left( {m{m^\prime }} \right)}}} {\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}n_j^{\left( m \right)} + \frac{{\partial {A^{\left( {{m^\prime }} \right)}}}}{{\partial \varepsilon _{ij}^{\left( {{m^\prime }} \right)}}}n_j^{\left( {{m^\prime }} \right)}} \right)\delta u_i^{\left( {m{m^\prime }} \right)}{\text{d}}S}} }
\end{array}
\end{equation}\end{split}\]
由于 \(\delta u_{i}^{(m)}\) 在 \({{V^{\left( m \right)}}}\) 中和在 \(S_p^{\left( m \right)}\) 上，\(\delta u_{i}^{\left(m m^{\prime}\right)}\) 在相邻有限元之间的交界面上，都是独立的变量，所以 \(\delta \Pi^{*}=0\) 给出了下列关系
(2.12)\[\begin{equation}
{\left( {\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}} \right)_{,j}} + {f_i} = 0 \quad \left( \text { 在 } V^{\left( m \right)} \text { 内 } \right)
\end{equation}\]
(2.13)\[\begin{equation}
\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}n_j^{\left( m \right)} - {{\bar p}_i} = 0 \quad \left( \text { 在 } S_p^{\left( m \right)} \text { 上 } \right)
\end{equation}\]
(2.14)\[\begin{equation}
\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}}n_j^{\left( m \right)} + \frac{{\partial {A^{\left( {{m^\prime }} \right)}}}}{{\partial \varepsilon _{ij}^{\left( {{m^\prime }} \right)}}}n_j^{\left( {{m^\prime }} \right)} = 0 \quad \left( \text { 在 } S^{\left(m m^{\prime}\right)} \text { 上 } \right)
\end{equation}\]
应用物理方程\(\frac{{\partial {A^{\left( m \right)}}}}{{\partial \varepsilon _{ij}^{\left( m \right)}}} = \sigma _{ij}^{\left( m \right)}\)，可得
(2.15)\[\begin{equation}
\sigma _{ij,j}^{\left( m \right)} + {f_i} = 0 \quad \left( \text { 在 } V^{\left( m \right)} \text { 内 } \right)
\label{eq:equilibrium_element_1}
\end{equation}\]
(2.16)\[\begin{equation}
\sigma _{ij}^{\left( m \right)} n_j - {{\bar p}_i} = 0 \quad \left( \text { 在 } S_p^{\left( m \right)} \text { 上 } \right)
\label{eq:sp_element_1}
\end{equation}\]
(2.17)\[\begin{equation}
\sigma _{ij}^{\left( m \right)}n_j^{\left( m \right)} + \sigma _{ij}^{\left( m^{\prime} \right)}n_j^{\left( {{m^\prime }} \right)} = 0 \quad \left( \text { 在 } S^{\left(m m^{\prime}\right)} \text { 上 } \right)
\label{eq:stress_continuity_1}
\end{equation}\]
这就是位移协调元的最小势能原理，以上各式表明，\(\Pi^{*}\) 取极值等效于弹性体各单元的平衡方程（式(2.15)）和单元边界上的力边界条件（式(2.16)），而且给出了相邻单元交界面上应力矢量的连续条件（式(2.17)）。值得指出的是，“在相邻单元的交界面上应力矢量是连续的”这一结论，它的前提是假定所选择的单元位移函数，不仅在单元交界面上是协调的，而且要使它满足有限元平衡方程（式(2.15)）和外力已知边界条件（式(2.16)），也就是有限元平衡方程和外力已知边界条件不致遭到破坏。