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A 12-node conforming straight-sided quadrilateral element with high-order completeness (QH12-C1)

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Abstract

A new 12-node conforming quadrilateral element with high-order completeness, denoted as QH12-C1 was proposed in this paper. The main steps are outlined as follows: first, build the expression of the interpolation displacement function satisfying the requirements for the high order completeness in the global coordinate system; Second, transform the displacement function expression by global coordinates into isoparametric coordinate and find the relationship of two series coefficients of two kind displacement function expression; Third, modify the displacement function expression to satisfy the requirements of the nodal freedoms and inter-element boundary continuity. The key to the new element construction is the acquisition of the linear relationship expressions among twenty-four coefficients of element displacement interpolation polynomial in the global coordinate system and isoparametric coordinate system. As a result, the relationship between cube completeness and inter-element continuity is explicitly given, and the proof of completeness and continuity was conducted to guarantee the validity of the derivation results theoretically. Because the explicit expressions of the element shape function are given, the calculation time consumption of the element shape function is greatly saved. Further, to verify the correctness of the theoretical work, nine numerical examples were executed. The computation results from these examples demonstrate that the QH12-C1 possessed of excellent performances, including high simulation accuracy, fast convergence speed, insensitivity to mesh distortion, and excellent monotone convergence.

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Authors and Affiliations

  1. Civil Engineering School, Central South University, Changsha, 410075, China

    Guoxiang Zhang & Junyu Xiang

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  1. Guoxiang Zhang
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  2. Junyu Xiang
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Correspondence to Guoxiang Zhang.

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Appendix A

Appendix A

$$\left\{ \begin{aligned} H_{ 1 , 1} &= \frac{{3_{1} }}{{32m_{0} }}[192A^{7} (m_{x} - m_{y} ) + 6m_{x}^{2} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} - 16A^{6} (9m_{x}^{2} + 10m_{x} m_{y} + 9m_{y}^{2} ) \hfill \\ &\quad + 16A^{5} (9m_{x}^{3} + 8m_{x}^{2} m_{y} - 8m_{x} m_{y}^{2} - 9m_{y}^{3} ) + 4A^{3} m_{x} m_{y} (m_{x}^{3} - 13m_{x}^{2} m_{y} + 13m_{x} m_{y}^{2} - m_{y}^{3} ) \hfill \\ &\quad - 16A^{4} (6m_{x}^{4} + 5m_{x}^{3} m_{y} + 6m_{x}^{2} m_{y}^{2} + 5m_{x} m_{y}^{3} + 6m_{y}^{4} ) + 2Am_{x} m_{y} ( - 3m_{x}^{5} + 3m_{x}^{4} m_{y} \hfill \\ &\quad + 10m_{x}^{3} m_{y}^{3} - 10m_{x}^{2} m_{y}^{3} - 3m_{x} m_{y}^{4} + 3m_{y}^{5} ) + A^{2} (9m_{x}^{6} + 30m_{x}^{5} m_{y} - 45m_{x}^{4} m_{y}^{2} - 100m_{x}^{3} m_{y}^{3} \hfill \\&\quad - 45m_{x}^{2} m_{y}^{4} + 30m_{x} m_{y}^{5} + 9m_{y}^{6} )] \hfill \\ H_{1,2} &= \frac{3}{{32m_{0} }}[192A^{7} (m_{x} + m_{y} ) + 6m_{x}^{2} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ &\quad- 16A^{6} (9m_{x}^{2} - 10m_{x} m_{y} + 9m_{y}^{2} ) + 16A^{5} (9m_{x}^{3} - 8m_{x}^{2} m_{y} - 8m_{x} m_{y}^{2} + 9m_{y}^{3} ) \hfill \\ &\quad- 4A^{3} m_{x} m_{y} (m_{x}^{3} + 13m_{x}^{2} m_{y} + 13m_{x} m_{y}^{2} + m_{y}^{3} ) - 16A^{4} (6m_{x}^{4} - 5m_{x}^{3} m_{y} \hfill \\ &\quad+ 6m_{x}^{2} m_{y}^{2} - 5m_{x} m_{y}^{3} + 6m_{y}^{4} ) + 2Am_{x} m_{y} (3m_{x}^{5} + 3m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{3} \hfill \\&\quad - 10m_{x}^{2} m_{y}^{3} + 3m_{x} m_{y}^{4} + 3m_{y}^{5} ) + A^{2} (9m_{x}^{6} - 30m_{x}^{5} m_{y} - 45m_{x}^{4} m_{y}^{2} + 100m_{x}^{3} m_{y}^{3} \hfill \\ &\quad- 45m_{x}^{2} m_{y}^{4} - 30m_{x} m_{y}^{5} + 9m_{y}^{6} )] \hfill \\ H_{1,3} &= - \frac{3}{{32m_{0} }}[192A^{7} (m_{x} - m_{y} ) - 6m_{x}^{2} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ &\quad+ 16A^{6} (9m_{x}^{2} + 10m_{x} m_{y} + 9m_{y}^{2} ) + 16A^{5} (9m_{x}^{3} + 8m_{x}^{2} m_{y} - 8m_{x} m_{y}^{2} + 9m_{y}^{3} ) \hfill \\ &\quad+ 4A^{3} m_{x} m_{y} (m_{x}^{3} - 13m_{x}^{2} m_{y} + 13m_{x} m_{y}^{2} - m_{y}^{3} ) + 16A^{4} (6m_{x}^{4} + 5m_{x}^{3} m_{y} + 6m_{x}^{2} m_{y}^{2} + 5m_{x} m_{y}^{3} + 6m_{y}^{4} ) \hfill \\&\quad+ 2Am_{x} m_{y} ( - 3m_{x}^{5} + 3m_{x}^{4} m_{y} + 10m_{x}^{3} m_{y}^{3} - 10m_{x}^{2} m_{y}^{3} - 3m_{x} m_{y}^{4} + 3m_{y}^{5} ) + A^{2} ( - 9m_{x}^{6} - 30m_{x}^{5} m_{y} \hfill \\&\quad + 45m_{x}^{4} m_{y}^{2} + 100m_{x}^{3} m_{y}^{3} + 45m_{x}^{2} m_{y}^{4} - 30m_{x} m_{y}^{5} - 9m_{y}^{6} )] \hfill \\ \end{aligned} \right.$$
(65)
$$\left\{ \begin{aligned} H_{ 1 , 4} &= - \frac{3}{{32m_{0} }}[192A^{7} (m_{x} + m_{y} ) - 6m_{x}^{2} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ &\quad+ 16A^{6} (9m_{x}^{2} - 10m_{x} m_{y} + 9m_{y}^{2} ) + 16A^{5} (9m_{x}^{3} - 8m_{x}^{2} m_{y} - 8m_{x} m_{y}^{2} + 9m_{y}^{3} ) - 4A^{3} m_{x} m_{y} (m_{x}^{3} + 13m_{x}^{2} m_{y} \hfill \\ &\quad+ 13m_{x} m_{y}^{2} + m_{y}^{3} ) + 16A^{4} (6m_{x}^{4} - 5m_{x}^{3} m_{y} + 6m_{x}^{2} m_{y}^{2} - 5m_{x} m_{y}^{3} + 6m_{y}^{4} ) + 2Am_{x} m_{y} (3m_{x}^{5} + 3m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{3} \hfill \\&\quad - 10m_{x}^{2} m_{y}^{3} + 3m_{x} m_{y}^{4} + 3m_{y}^{5} ) + A^{2} ( - 9m_{x}^{6} + 30m_{x}^{5} m_{y} + 45m_{x}^{4} m_{y}^{2} - 100m_{x}^{3} m_{y}^{3} + 45m_{x}^{2} m_{y}^{4} + 30m_{x} m_{y}^{5} - 9m_{y}^{6} )] \hfill \\ H_{1,5} &= - \frac{{9m_{x} }}{{64m_{0} }}[128A^{7} - 96A^{6} (m_{x} + 3m_{y} ) + 4m_{x}^{3} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} ) + 16A^{5} (6m_{x}^{2} + 27m_{x} m_{y} + m_{y}^{2} ) \hfill \\ &\quad- 8A^{3} m_{y} (9m_{x}^{3} - 5m_{x}^{2} m_{y} - 36m_{x} m_{y}^{2} + 6m_{y}^{3} ) - 16A^{4} (4m_{x}^{3} + 9m_{x}^{2} m_{y} + 2m_{x} m_{y}^{2} + 9m_{y}^{3} ) \hfill \\ &\quad+ Am_{y} ( - 9m_{x}^{5} + 3m_{x}^{4} m_{y} + 36m_{x}^{3} m_{y}^{2} - 28m_{x}^{2} m_{y}^{3} - 27m_{x} m_{y}^{4} + 9m_{y}^{5} ) \hfill \\&\quad+ 6A^{2} (m_{x}^{5} + 9m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{2} - 30m_{x}^{2} m_{y}^{3} + 5m_{x} m_{y}^{4} + 9m_{y}^{5} )] \hfill \\ H_{1,6} &= - \frac{{9m_{x} u_{6} }}{{64m_{0} }}[128A^{7} - 96A^{6} (m_{x} - 3m_{y} ) \hfill \\ &\quad+ 4m_{x}^{3} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} ) + 16A^{5} (6m_{x}^{2} - 27m_{x} m_{y} + m_{y}^{2} ) + 8A^{3} m_{y} (9m_{x}^{3} + 5m_{x}^{2} m_{y} - 36m_{x} m_{y}^{2} - 6m_{y}^{3} ) \hfill \\ &\quad- 16A^{4} (4m_{x}^{3} - 9m_{x}^{2} m_{y} + 2m_{x} m_{y}^{2} - 9m_{y}^{3} ) + Am_{y} (9m_{x}^{5} + 3m_{x}^{4} m_{y} - 36m_{x}^{3} m_{y}^{2} - 28m_{x}^{2} m_{y}^{3} + 27m_{x} m_{y}^{4} + 9m_{y}^{5} ) \hfill \\ &\quad+6A^{2} (m_{x}^{5} - 9m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{2} + 30m_{x}^{2} m_{y}^{3} + 5m_{x} m_{y}^{4} - 9m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(66)
$$\left\{ \begin{aligned} H_{ 1 , 7} &= - \frac{{9m_{y} u_{7} }}{{64m_{0} }}[128A^{7} + 96A^{6} (3m_{x} - m_{y} ) - 4m_{x}^{4} m_{y}^{3} + 4m_{x}^{2} m_{y}^{5} + 16A^{5} (m_{x}^{2} - 27m_{x} m_{y} + 6m_{y}^{2} ) \hfill \\&\quad- 8A^{3} m_{x} (6m_{x}^{3} + 36m_{x}^{2} m_{y} - 5m_{x} m_{y}^{2} - 9m_{y}^{3} ) + 16A^{4} (9m_{x}^{3} - 2m_{x}^{2} m_{y} + 9m_{x} m_{y}^{2} - 4m_{y}^{3} ) \hfill \\&\quad + Am_{x} (9m_{x}^{5} + 27m_{x}^{4} m_{y} - 28m_{x}^{3} m_{y}^{2} - 36m_{x}^{2} m_{y}^{3} + 3m_{x} m_{y}^{4} + 9m_{y}^{5} ) \hfill \\&\quad- 6A^{2} (9m_{x}^{5} - 5m_{x}^{4} m_{y} - 30m_{x}^{3} m_{y}^{2} + 10m_{x}^{2} m_{y}^{3} + 9m_{x} m_{y}^{4} - m_{y}^{5} )] \hfill \\ H_{1,8} &= - \frac{{9m_{y} }}{{64m_{0} }}[128A^{7} - 96A^{6} (3m_{x} + m_{y} ) - 4m_{x}^{4} m_{y}^{3} + 4m_{x}^{2} m_{y}^{5} + 16A^{5} (m_{x}^{2} + 27m_{x} m_{y} + 6m_{y}^{2} ) \hfill \\ &\quad-8A^{3} m_{x} (6m_{x}^{3} - 36m_{x}^{2} m_{y} - 5m_{x} m_{y}^{2} + 9m_{y}^{3} ) + 16A^{4} (9m_{x}^{3} + 2m_{x}^{2} m_{y} + 9m_{x} m_{y}^{2} + 4m_{y}^{3} ) \hfill \\ &\quad+ Am_{x} (9m_{x}^{5} - 27m_{x}^{4} m_{y} - 28m_{x}^{3} m_{y}^{2} + 36m_{x}^{2} m_{y}^{3} + 3m_{x} m_{y}^{4} - 9m_{y}^{5} ) \hfill \\ &\quad+6A^{2} (9m_{x}^{5} + 5m_{x}^{4} m_{y} - 30m_{x}^{3} m_{y}^{2} - 10m_{x}^{2} m_{y}^{3} + 9m_{x} m_{y}^{4} + m_{y}^{5} )] \hfill \\ H_{1,9} &= \frac{{9m_{x} }}{{64m_{0} }}[128A^{7} + 96A^{6} (m_{x} + 3m_{y} ) - 4m_{x}^{5} m_{y}^{2} + 4m_{x}^{3} m_{y}^{4} + 16A^{5} (6m_{x}^{2} + 27m_{x} m_{y} + m_{y}^{2} ) \hfill \\ &\quad-8A^{3} m_{y} (9m_{x}^{3} - 5m_{x}^{2} m_{y} - 36m_{x} m_{y}^{2} + 6m_{y}^{3} ) + 16A^{4} (4m_{x}^{3} + 9m_{x}^{2} m_{y} + 2m_{x} m_{y}^{2} + 9m_{y}^{3} ) \hfill \\&\quad + Am_{x} ( - 9m_{x}^{5} + 3m_{x}^{4} m_{y} + 36m_{x}^{3} m_{y}^{2} - 28m_{x}^{2} m_{y}^{3} - 27m_{x} m_{y}^{4} + 9m_{y}^{5} ) - \hfill \\ &\quad6A^{2} (m_{x}^{5} + 9m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{2} - 30m_{x}^{2} m_{y}^{3} + 5m_{x} m_{y}^{4} + 9m_{y}^{5} )] \hfill \\ &\quad 0m_{x}^{3} m_{y}^{2} - 10m_{x}^{2} m_{y}^{3} + 9m_{x} m_{y}^{4} + m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(67)
$$\left\{ \begin{aligned} H_{1,10} &= \frac{{9m_{x} }}{{64m_{0} }}[128A^{7} + 96A^{6} (m_{x} - 3m_{y} ) - 4m_{x}^{5} m_{y}^{2} + 4m_{x}^{3} m_{y}^{4} + 16A^{5} (6m_{x}^{2} - 27m_{x} m_{y} + m_{y}^{2} ) + \hfill \\ &\quad+8A^{3} m_{y} (9m_{x}^{3} + 5m_{x}^{2} m_{y} - 36m_{x} m_{y}^{2} - 6m_{y}^{3} ) + 16A^{4} (4m_{x}^{3} - 9m_{x}^{2} m_{y} + 2m_{x} m_{y}^{2} - 9m_{y}^{3} ) \hfill \\ &\quad+ Am_{x} (9m_{x}^{5} + 3m_{x}^{4} m_{y} - 36m_{x}^{3} m_{y}^{2} - 28m_{x}^{2} m_{y}^{3} + 27m_{x} m_{y}^{4} + 9m_{y}^{5} ) - \hfill \\ &\quad- 6A^{2} (m_{x}^{5} - 9m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{2} + 30m_{x}^{2} m_{y}^{3} + 5m_{x} m_{y}^{4} - 9m_{y}^{5} )] \hfill \\ H_{1,11} &= \frac{{9m_{x} }}{{64m_{0} }}[128A^{7} - 96A^{6} (3m_{x} - m_{y} ) + 4m_{x}^{4} m_{y}^{3} - 4m_{x}^{2} m_{y}^{5} + 16A^{5} (m_{x}^{2} - 27m_{x} m_{y} + 6m_{y}^{2} ) + \hfill \\ &\quad+ 8A^{3} m_{x} (6m_{x}^{3} - 5m_{y}^{2} m_{x} + 36m_{y} m_{x}^{2} - 9m_{y}^{3} ) - 16A^{4} (9m_{x}^{3} - 2m_{x}^{2} m_{y} + 9m_{x} m_{y}^{2} - 4m_{y}^{3} ) \hfill \\ &\quad+ Am_{x} (9m_{x}^{5} + 27m_{x}^{4} m_{y} - 28m_{x}^{3} m_{y}^{2} - 36m_{x}^{2} m_{y}^{3} + 3m_{x} m_{y}^{4} + 9m_{y}^{5} ) + \hfill \\ &\quad+ 6A^{2} (9m_{x}^{5} - 5m_{x}^{4} m_{y} - 30m_{x}^{3} m_{y}^{2} + 10m_{x}^{2} m_{y}^{3} + 9m_{x} m_{y}^{4} - m_{y}^{5} )] \hfill \\ &\quad H_{1,12} + \frac{{9m_{x} }}{{64m_{0} }}[128A^{7} + 96A^{6} (3m_{x} + m_{y} ) + 4m_{x}^{4} m_{y}^{3} - 4m_{x}^{2} m_{y}^{5} + 16A^{5} (m_{x}^{2} - 27m_{x} m_{y} + 6m_{y}^{2} ) - \hfill \\ &\quad- 8A^{3} m_{x} (6m_{x}^{3} - 5m_{y}^{2} m_{x} - 36m_{y} m_{x}^{2} + 9m_{y}^{3} ) - 16A^{4} (9m_{x}^{3} + 2m_{x}^{2} m_{y} + 9m_{x} m_{y}^{2} + 4m_{y}^{3} ) \hfill \\&\quad + Am_{x} (9m_{x}^{5} { - }27m_{x}^{4} m_{y} - 28m_{x}^{3} m_{y}^{2} + 36m_{x}^{2} m_{y}^{3} + 3m_{x} m_{y}^{4} { - }9m_{y}^{5} ) + \hfill \\ &\quad+ 6A^{2} (9m_{x}^{5} + 5m_{x}^{4} m_{y} - 30m_{x}^{3} m_{y}^{2} { - }10m_{x}^{2} m_{y}^{3} + 9m_{x} m_{y}^{4} + m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(68)
$$\left\{ \begin{aligned} H_{2,1} &= \frac{3}{{32m_{0} }}[48A^{6} (3m_{x}^{2} - 2m_{x} m_{y} + m_{y}^{2} ) + 16A^{5} m_{y} (6m_{x}^{2} + 5m_{x} m_{y} + 3m_{y}^{3} ) \hfill \\ &\quad+ 4A^{3} m_{y}^{3} (9m_{x}^{2} + 10m_{x} m_{y} + 9m_{y}^{2} ) + m_{x} m_{y}^{2} (m_{x} - m_{y} )^{2} (7m_{x}^{3} + 13m_{x}^{2} m_{y} + 3m_{x} m_{y}^{2} - 3m_{y}^{3} ) + \hfill \\&\quad+ Am_{x} m_{y}^{2} ( - 5m_{x}^{4} - 6m_{x}^{3} m_{y} + 20m_{x}^{2} m_{y}^{2} + 6m_{x} m_{y}^{3} - 15m_{y}^{4} ) \hfill \\&\quad - 8A^{4} (9m_{x}^{4} + 3m_{x}^{3} m_{y} + 2m_{x}^{2} m_{y}^{2} - 8m_{x} m_{y}^{3} - 6m_{y}^{4} ) \hfill \\&\quad + A^{2} (9m_{x}^{6} - 63m_{x}^{4} m_{y}^{2} + 6m_{x}^{3} m_{y}^{3} - 59m_{x}^{2} m_{y}^{4} + 2m_{x} m_{y}^{5} + 9m_{y}^{6} )] \hfill \\H_{2,2} &= - \frac{3}{{32m_{0} }}[48A^{6} (3m_{x}^{2} + 2m_{x} m_{y} + m_{y}^{2} ) - 16A^{5} m_{y} (6m_{x}^{2} - 5m_{x} m_{y} + 3m_{y}^{3} ) \hfill \\&\quad - 4A^{3} m_{y}^{3} (9m_{x}^{2} - 10m_{x} m_{y} + 9m_{y}^{2} ) + m_{x} m_{y}^{2} (m_{x} + m_{y} )^{2} (7m_{x}^{3} - 13m_{x}^{2} m_{y} + 3m_{x} m_{y}^{2} + 3m_{y}^{3} ) + \hfill \\&\quad Am_{x} m_{y}^{2} ( - 5m_{x}^{4} + 6m_{x}^{3} m_{y} + 20m_{x}^{2} m_{y}^{2} - 6m_{x} m_{y}^{3} - 15m_{y}^{4} ) \hfill \\&\quad - 8A^{4} (9m_{x}^{4} - 3m_{x}^{3} m_{y} + 2m_{x}^{2} m_{y}^{2} + 8m_{x} m_{y}^{3} - 6m_{y}^{4} ) \hfill \\&\quad + A^{2} (9m_{x}^{6} - 63m_{x}^{4} m_{y}^{2} - 6m_{x}^{3} m_{y}^{3} - 59m_{x}^{2} m_{y}^{4} - 2m_{x} m_{y}^{5} + 9m_{y}^{6} )] \hfill \\H_{2,3}& = - \frac{3}{{32m_{0} }}[48A^{6} (3m_{x}^{2} - 2m_{x} m_{y} + m_{y}^{2} ) - 16A^{5} m_{y} (6m_{x}^{2} + 5m_{x} m_{y} + 3m_{y}^{3} ) \hfill \\&\quad - 4A^{3} m_{y}^{3} (9m_{x}^{2} + 10m_{x} m_{y} + 9m_{y}^{2} ) + m_{x} m_{y}^{2} (m_{x} - m_{y} )^{2} (7m_{x}^{3} + 13m_{x}^{2} m_{y} + 3m_{x} m_{y}^{2} - 3m_{y}^{3} ) \hfill \\&\quad + Am_{x} m_{y}^{2} (5m_{x}^{4} + 6m_{x}^{3} m_{y} - 20m_{x}^{2} m_{y}^{2} - 6m_{x} m_{y}^{3} + 15m_{y}^{4} ) \hfill \\&\quad - 8A^{4} (9m_{x}^{4} + 3m_{x}^{3} m_{y} + 2m_{x}^{2} m_{y}^{2} - 8m_{x} m_{y}^{3} - 6m_{y}^{4} ) \hfill \\&\quad + A^{2} (9m_{x}^{6} - 63m_{x}^{4} m_{y}^{2} + 6m_{x}^{3} m_{y}^{3} - 59m_{x}^{2} m_{y}^{4} + 2m_{x} m_{y}^{5} + 9m_{y}^{6} )] \hfill \\ \end{aligned} \right.$$
(69)
$$\left\{ \begin{aligned} H_{2,4} &= \frac{3}{{32m_{0} }}[48A^{6} (3m_{x}^{2} + 2m_{x} m_{y} + m_{y}^{2} ) + 16A^{5} m_{y} (6m_{x}^{2} - 5m_{x} m_{y} + 3m_{y}^{3} ) \hfill \\&\quad + 4A^{3} m_{y}^{3} (9m_{x}^{2} - 10m_{x} m_{y} + 9m_{y}^{2} ) + m_{x} m_{y}^{2} (m_{x} + m_{y} )^{2} (7m_{x}^{3} - 13m_{x}^{2} m_{y} + 3m_{x} m_{y}^{2} + 3m_{y}^{3} ) \hfill \\&\quad + Am_{x} m_{y}^{2} (5m_{x}^{4} - 6m_{x}^{3} m_{y} - 20m_{x}^{2} m_{y}^{2} + 6m_{x} m_{y}^{3} + 15m_{y}^{4} ) \hfill \\&\quad - 8A^{4} (9m_{x}^{4} - 3m_{x}^{3} m_{y} + 2m_{x}^{2} m_{y}^{2} + 8m_{x} m_{y}^{3} - 6m_{y}^{4} ) \hfill \\&\quad + A^{2} (9m_{x}^{6} - 63m_{x}^{4} m_{y}^{2} - 6m_{x}^{3} m_{y}^{3} - 59m_{x}^{2} m_{y}^{4} - 2m_{x} m_{y}^{5} + 9m_{y}^{6} )] \hfill \\H_{2,5} &= - \frac{{9m_{x} }}{{128m_{0} }}[64A^{6} (9m_{x} - 2m_{y} ) + 48A^{4} m_{y}^{3} (m_{x} + 3m_{y} ) + 32A^{5} m_{y} (4m_{x} + 9m_{y} ) \hfill \\&\quad - 16A^{4} (18m_{x}^{3} + 2m_{x}^{2} m_{y} + 9m_{x} m_{y}^{2} + m_{y}^{3} ) - 2Am_{y}^{2} (9m_{x}^{4} - 2m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} + 18m_{x} m_{y}^{3} + 27m_{y}^{4} ) \hfill \\&\quad + m_{y}^{2} (27m_{x}^{5} - m_{x}^{4} m_{y} - 54m_{x}^{3} m_{y}^{2} + 10m_{x}^{2} m_{y}^{3} + 27m_{x} m_{y}^{4} - 9m_{y}^{5} ) \hfill \\&\quad + 4A^{2} (9m_{x}^{5} - 54m_{x}^{3} m_{y}^{2} - 4m_{x}^{2} m_{y}^{3} - 99m_{x} m_{y}^{4} + 12m_{y}^{5} )] \hfill \\H_{2,6} &= \frac{{9m_{x} }}{{128m_{0} }}[64A^{6} (9m_{x} + 2m_{y} ) - 48A^{4} m_{y}^{3} (m_{x} - 3m_{y} ) - 32A^{5} m_{y} (4m_{x} - 9m_{y} ) \hfill \\&\quad - 16A^{4} (18m_{x}^{3} - 2m_{x}^{2} m_{y} + 9m_{x} m_{y}^{2} - m_{y}^{3} ) - 2Am_{y}^{2} (9m_{x}^{4} + 2m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 18m_{x} m_{y}^{3} + 27m_{y}^{4} ) + \hfill \\&\quad m_{y}^{2} (27m_{x}^{5} + m_{x}^{4} m_{y} - 54m_{x}^{3} m_{y}^{2} - 10m_{x}^{2} m_{y}^{3} + 27m_{x} m_{y}^{4} + 9m_{y}^{5} ) \hfill \\&\quad + 4A^{2} (9m_{x}^{5} - 54m_{x}^{3} m_{y}^{2} + 4m_{x}^{2} m_{y}^{3} - 99m_{x} m_{y}^{4} - 12m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(70)
$$\left\{ \begin{aligned} H_{2,7} &= \frac{{9m_{y}^{2} }}{{128m_{0} }}[64A^{6} + 32A^{5} (9m_{x} + 2m_{y} ) + 48A^{3} (3m_{x} - m_{y} )m_{y}^{2} \hfill \\&\quad - 16A^{4} (m_{x}^{2} + 27m_{x} m_{y} - 4m_{y}^{2} ) - 4A^{2} (3m_{x}^{4} + 18m_{x}^{3} m_{y} - 8m_{x}^{2} m_{y}^{2} - 18m_{x} m_{y}^{3} - 3m_{y}^{4} ) \hfill \\&\quad - 2Am_{x} (9m_{x}^{4} - 6m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} + 22m_{x} m_{y}^{3} + 27m_{y}^{4} ) \hfill \\&\quad + m_{x} (3m_{x}^{5} + 9m_{x}^{4} m_{y} - 14m_{x}^{3} m_{y}^{2} - 18m_{x}^{2} m_{y}^{3} + 11m_{x} m_{y}^{4} + 9m_{y}^{5} )] \hfill \\H_{2,8} &= - \frac{{9m_{y}^{2} }}{{128m_{0} }}[ - 64A^{6} + 32A^{5} (9m_{x} + 2m_{y} ) + 48A^{3} (3m_{x} + m_{y} )m_{y}^{2} \hfill \\&\quad + 16A^{4} (m_{x}^{2} - 27m_{x} m_{y} - 4m_{y}^{2} ) + 4A^{2} (3m_{x}^{4} - 18m_{x}^{3} m_{y} - 8m_{x}^{2} m_{y}^{2} + 18m_{x} m_{y}^{3} - 3m_{y}^{3} ) \hfill \\&\quad - 2Am_{x} (9m_{x}^{4} + 6m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 22m_{x} m_{y}^{3} + 27m_{y}^{4} ) \hfill \\&\quad + m_{x} ( - 3m_{x}^{5} + 9m_{x}^{4} m_{y} + 14m_{x}^{3} m_{y}^{2} - 18m_{x}^{2} m_{y}^{3} - 11m_{x} m_{y}^{4} + 9m_{y}^{5} )] \hfill \\ H_{2,9} &= \frac{{9m_{x} }}{{128m_{0} }}[64A^{6} (9m_{x} - 2m_{y} ) - 48A^{3} m_{y}^{3} (m_{x} + 3m_{y} ) - 32A^{5} m_{y} (4m_{x} + 9m_{y} ) \hfill \\&\quad - 16A^{4} (18m_{x}^{3} + 2m_{x}^{2} m_{y} + 9m_{x} m_{y}^{2} + m_{y}^{3} ) + 2Am_{y}^{2} (9m_{x}^{4} - 2m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 18m_{x} m_{y}^{3} + 27m_{y}^{4} ) \hfill \\&\quad + m_{y}^{2} (27m_{x}^{5} - m_{x}^{4} m_{y} - 54m_{x}^{3} m_{y}^{2} + 10m_{x}^{2} m_{y}^{3} + 27m_{x} m_{y}^{4} - 9m_{y}^{5} ) \hfill \\&\quad + 4A^{2} (9m_{x}^{5} - 54m_{x}^{3} m_{y}^{2} - 4m_{x}^{2} m_{y}^{3} - 99m_{x} m_{y}^{4} + 12m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(71)
$$\left\{ \begin{aligned} H_{2,10} &= - \frac{{9m_{x} }}{{128m_{0} }}[64A^{6} (9m_{x} + 2m_{y} ) + 48A^{3} m_{y}^{3} (m_{x} - 3m_{y} ) + 32A^{5} m_{y} (4m_{x} - 9m_{y} ) \hfill \\&\quad - 16A^{4} (18m_{x}^{4} - 2m_{x}^{2} m_{y} + 9m_{x} m_{y}^{2} - m_{y}^{3} ) + 2Am_{y}^{2} (9m_{x}^{4} + 2m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 18m_{x} m_{y}^{3} + 27m_{y}^{4} ) \hfill \\&\quad + m_{y}^{2} (27m_{x}^{5} + m_{x}^{4} m_{y} - 54m_{x}^{3} m_{y}^{2} - 10m_{x}^{2} m_{y}^{3} + 27m_{x} m_{y}^{4} + 9m_{y}^{5} ) \hfill \\&\quad + 4A^{2} (9m_{x}^{5} - 54m_{x}^{3} m_{y}^{2} + 4m_{x}^{2} m_{y}^{3} - 99m_{x} m_{y}^{4} - 12m_{y}^{5} )] \hfill \\ H_{2,11} &= - \frac{{9m_{y}^{2} }}{{128m_{o} }}[64A^{6} - 32A^{5} (9m_{x} - 2m_{y} ) - 48A^{3} (3m_{x} - m_{y} )m_{y}^{2} - 16A^{4} (m_{x}^{2} + 27m_{y} m_{x} - 4m_{y}^{2} ) \hfill \\&\quad - 4A^{2} (3m_{x}^{3} + 18m_{x}^{3} m_{y} - 8m_{x}^{2} m_{y}^{2} - 18m_{x} m_{y}^{3} - 3m_{y}^{4} ) \hfill \\&\quad + 2Am_{x} (9m_{x}^{4} - 6m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} + 22m_{x} m_{y}^{3} + 27m_{y}^{4} ) \hfill \\&\quad + m_{x} (3m_{x}^{5} + 9m_{x}^{4} m_{y} - 14m_{x}^{3} m_{y}^{2} - 18m_{x}^{2} m_{y}^{3} + 11m_{x} m_{y}^{4} + 9m_{y}^{5} )] \hfill \\&\quad H_{2,12} = \frac{{9m_{y}^{2} }}{{128m_{o} }}[ - 64A^{6} - 32A^{5} (9m_{x} + 2m_{y} ) - 48A^{3} (3m_{x} + m_{y} )m_{y}^{2} + 16A^{4} (m_{x}^{2} - 27m_{y} m_{x} - 4m_{y}^{2} ) \hfill \\&\quad + 4A^{2} (3m_{x}^{4} - 18m_{x}^{3} m_{y} - 8m_{x}^{2} m_{y}^{2} + 18m_{x} m_{y}^{3} - 3m_{y}^{4} ) + 2Am_{x} (9m_{x}^{4} + 6m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 22m_{x} m_{y}^{3} + 27m_{y}^{4} ) \hfill \\&\quad + m_{x} ( - 3m_{x}^{5} + 9m_{x}^{4} m_{y} + 14m_{x}^{3} m_{y}^{2} - 18m_{x}^{2} m_{y}^{3} - 11m_{x} m_{y}^{4} + 9m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(72)
$$\left\{ \begin{aligned} H_{3,1} &= \frac{3}{{32m_{0} }}[48A^{6} (m_{x}^{2} - 2m_{x} m_{y} + 3m_{y}^{2} ) - 16A^{5} m_{x} (3m_{x}^{2} + 5m_{x} m_{y} + 6m_{y}^{2} ) \hfill \\&\quad - 4A^{3} m_{x}^{3} (9m_{x}^{2} + 10m_{x} m_{y} + 9m_{y}^{2} ) - m_{x}^{2} (m_{x} - m_{y} )^{2} m_{y} (3m_{x}^{3} - 3m_{x}^{2} m_{y} - 13m_{x} m_{y}^{2} - 7m_{y}^{3} ) \hfill \\&\quad + 8A^{4} (6m_{x}^{4} + 8m_{x}^{3} m_{y} - 2m_{x}^{2} m_{y}^{2} - 3m_{x} m_{y}^{3} - 9m_{y}^{4} ) \hfill \\&\quad + A^{2} (9m_{x}^{6} + 2m_{x}^{5} m_{y} - 59m_{x}^{4} m_{y}^{2} + 6m_{x}^{3} m_{y}^{3} - 63m_{x}^{2} m_{y}^{4} + 9m_{y}^{6} )] \hfill \\ H_{3,2} &= \frac{3}{{32m_{0} }}[48A^{6} (m_{x}^{2} + 2m_{x} m_{y} + 3m_{y}^{2} ) - 16A^{5} m_{x} (3m_{x}^{2} - 5m_{x} m_{y} + 6m_{y}^{2} ) \hfill \\&\quad - 4A^{3} m_{x}^{3} (9m_{x}^{2} - 10m_{x} m_{y} + 9m_{y}^{2} ) + m_{x}^{2} (m_{x} + m_{y} )^{2} m_{y} (3m_{x}^{3} + 3m_{x}^{2} m_{y} - 13m_{x} m_{y}^{2} + 7m_{y}^{3} ) \hfill \\&\quad + 8A^{4} (6m_{x}^{4} - 8m_{x}^{3} m_{y} - 2m_{x}^{2} m_{y}^{2} + 3m_{x} m_{y}^{3} - 9m_{y}^{4} ) \hfill \\&\quad + A^{2} (9m_{x}^{6} - 2m_{x}^{5} m_{y} - 59m_{x}^{4} m_{y}^{2} - 6m_{x}^{3} m_{y}^{3} - 63m_{x}^{2} m_{y}^{4} + 9m_{y}^{6} )] \hfill \\ H_{3,3}& = - \frac{3}{{32m_{0} }}[48A^{6} (m_{x}^{2} - 2m_{x} m_{y} + 3m_{y}^{2} ) + 16A^{5} m_{x} (3m_{x}^{2} + 5m_{x} m_{y} + 6m_{y}^{2} ) \hfill \\&\quad + 4A^{3} m_{x}^{3} (9m_{x}^{2} + 10m_{x} m_{y} + 9m_{y}^{2} ) - m_{x}^{2} (m_{x} - m_{y} )^{2} m_{y} (3m_{x}^{3} - 3m_{x}^{2} m_{y} - 13m_{x} m_{y}^{2} - 7m_{y}^{3} ) \hfill \\&\quad + 8A^{4} (6m_{x}^{4} + 8m_{x}^{3} m_{y} - 2m_{x}^{2} m_{y}^{2} - 3m_{x} m_{y}^{3} - 9m_{y}^{4} ) \hfill \\&\quad + A^{2} (9m_{x}^{6} + 2m_{x}^{5} m_{y} - 59m_{x}^{4} m_{y}^{2} + 6m_{x}^{3} m_{y}^{3} - 63m_{x}^{2} m_{y}^{4} + 9m_{y}^{6} )] \hfill \\ \end{aligned} \right.$$
(73)
$$\left\{ \begin{aligned} H_{3,4} = - \frac{3}{{32m_{0} }}[48A^{6} (m_{x}^{2} + 2m_{x} m_{y} + 3m_{y}^{2} ) + 16A^{5} m_{x} (3m_{x}^{2} - 5m_{x} m_{y} + 6m_{y}^{2} ) \hfill \\ + 4A^{3} m_{x}^{3} (9m_{x}^{2} - 10m_{x} m_{y} + 9m_{y}^{2} ) + m_{x}^{2} (m_{x} + m_{y} )^{2} m_{y} (3m_{x}^{3} + 3m_{x}^{2} m_{y} - 13m_{x} m_{y}^{2} + 7m_{y}^{3} ) \hfill \\ + 8A^{4} (6m_{x}^{4} - 8m_{x}^{3} m_{y} - 2m_{x}^{2} m_{y}^{2} + 3m_{x} m_{y}^{3} - 9m_{y}^{4} ) + A^{2} (9m_{x}^{6} - 2m_{x}^{5} m_{y} - 59m_{x}^{4} m_{y}^{2} - 6m_{x}^{3} m_{y}^{3} - 63m_{x}^{2} m_{y}^{4} + 9m_{y}^{6} )] \hfill \\ H_{3,5} = - \frac{{9m_{x}^{2} }}{{128m_{0} }}[64A^{4} - 48A^{3} m_{x}^{2} (m_{x} + 3m_{y} ) - 32A^{5} (2m_{x} + 9m_{y} ) + 16A^{4} (4m_{x}^{2} + 27m_{x} m_{y} - m_{y}^{2} ) \hfill \\ + 4A^{2} (3m_{x}^{4} - 18m_{x}^{3} m_{y} + 8m_{x}^{2} m_{y}^{2} + 18m_{x} m_{y}^{3} - 3m_{y}^{4} ) + 2Am_{y} (27m_{x}^{4} - 22m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} + 6m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ + m_{y} ( - 9m_{x}^{5} + 11m_{x}^{4} m_{y} + 18m_{x}^{3} m_{y}^{2} - 14m_{x}^{2} m_{y}^{3} - 9m_{x} m_{y}^{4} + 3m_{y}^{5} )] \hfill \\ H_{3,6} = - \frac{{9m_{x}^{2} }}{{128m_{0} }}[64A^{4} - 48A^{3} m_{x}^{2} (m_{x} - 3m_{y} ) - 32A^{5} (2m_{x} - 9m_{y} ) + 16A^{4} (4m_{x}^{2} - 27m_{x} m_{y} - m_{y}^{2} ) \hfill \\ + 4A^{2} (3m_{x}^{4} + 18m_{x}^{3} m_{y} + 8m_{x}^{2} m_{y}^{2} - 18m_{x} m_{y}^{3} - 3m_{y}^{4} ) - 2Am_{y} (27m_{x}^{4} + 22m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 6m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ + m_{y} (9m_{x}^{5} + 11m_{x}^{4} m_{y} - 18m_{x}^{3} m_{y}^{2} - 14m_{x}^{2} m_{y}^{3} + 9m_{x} m_{y}^{4} + 3m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(74)
$$\left\{ \begin{aligned} H_{3,7} = - \frac{{9m_{x}^{2} }}{{128m_{0} }}[32A^{5} m_{x} (9m_{x} - 4m_{y} ) + 48A^{3} m_{x}^{3} (3m_{x} - m_{y} ) + 64A^{6} (2m_{x} + 9m_{y} ) \hfill \\ + 16A^{4} (m_{x}^{3} - 9m_{x}^{2} m_{y} + 2m_{x} m_{y}^{2} - 18m_{y}^{3} ) - 2Am_{x}^{2} (27m_{x}^{4} - 18m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} + 2m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ - 4A^{2} (12m_{x}^{5} + 99m_{x}^{4} m_{y} - 4m_{x}^{3} m_{y}^{2} + 54m_{x}^{2} m_{y}^{3} - 9m_{y}^{5} ) \hfill \\ + m_{x}^{2} (9m_{x}^{5} + 27m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{3} - 54m_{x}^{2} m_{y}^{3} + m_{x} m_{y}^{4} + 27m_{y}^{5} )] \hfill \\ H_{3,8} = - \frac{{9m_{x}^{2} }}{{128m_{0} }}[ - 32A^{5} m_{x} (9m_{x} + 4m_{y} ) - 48A^{3} m_{x}^{3} (3m_{x} + m_{y} ) + 64A^{6} (2m_{x} - 9m_{y} ) \hfill \\ + 16A^{4} (m_{x}^{3} + 9m_{x}^{2} m_{y} + 2m_{x} m_{y}^{2} + 18m_{y}^{3} ) + 2Am_{x}^{2} (27m_{x}^{4} + 18m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 2m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ - 4A^{2} (12m_{x}^{5} - 99m_{x}^{4} m_{y} - 4m_{x}^{3} m_{y}^{2} - 54m_{x}^{2} m_{y}^{3} + 9m_{y}^{5} ) \hfill \\ + m_{x}^{2} (9m_{x}^{5} - 27m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{3} + 54m_{x}^{2} m_{y}^{3} + m_{x} m_{y}^{4} - 27m_{y}^{5} )] \hfill \\ H_{3,9} = \frac{{9m_{x}^{2} }}{{128m_{0} }}[64A^{6} + 48A^{3} m_{x}^{2} (m_{x} + 3m_{y} ) + 32A^{5} (2m_{x} + 9m_{y} ) + 16A^{4} (4m_{x}^{2} + 27m_{x} m_{y} - m_{y}^{2} ) \hfill \\ + 4A^{2} (3m_{x}^{4} - 18m_{x}^{3} m_{y} + 8m_{x}^{2} m_{y}^{2} + 18m_{x} m_{y}^{3} - 3m_{y}^{4} ) \hfill \\ - 2Am_{y} (27m_{x}^{4} - 22m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} + 6m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ + m_{y} ( - 9m_{x}^{5} + 11m_{x}^{4} m_{y} + 18m_{x}^{3} m_{y}^{2} - 14m_{y}^{2} m_{y}^{3} - 9m_{x} m_{y}^{4} + 3m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(75)
$$\left\{ \begin{aligned} H_{3,10} = \frac{{9m_{x}^{2} }}{{128m_{0} }}[64A^{6} + 48A^{3} m_{x}^{2} (m_{x} - 3m_{y} ) + 32A^{5} (2m_{x} - 9m_{y} ) + 16A^{4} (4m_{x}^{2} - 27m_{x} m_{y} - m_{y}^{2} ) \hfill \\ + 4A^{2} (3m_{x}^{4} + 18m_{x}^{3} m_{y} + 8m_{x}^{2} m_{y}^{2} - 18m_{x} m_{y}^{3} - 3m_{y}^{4} ) \hfill \\ + 2Am_{y} (27m_{x}^{4} + 22m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 6m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ + m_{y} (9m_{x}^{5} + 11m_{x}^{4} m_{y} - 18m_{x}^{3} m_{y}^{2} - 14m_{y}^{2} m_{y}^{3} + 9m_{x} m_{y}^{4} + 3m_{y}^{5} )] \hfill \\ H_{3,11} = \frac{{9m_{y} }}{{128m_{0} }}[ - 32A^{5} m_{x} (9m_{x} - 4m_{y} ) - 48A^{3} m_{x}^{3} (3m_{x} - m_{y} ) + 64A^{6} (2m_{x} + 9m_{y} ) \hfill \\ + 16A^{4} (m_{x}^{3} - 9m_{x}^{2} m_{y} + 2m_{x} m_{y}^{2} - 18m_{y}^{3} ) + 2Am_{x}^{2} (27m_{x}^{4} - 18m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} + 2m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ - 4A^{4} (12m_{x}^{5} + 99m_{x}^{4} m_{y} - 4m_{x}^{3} m_{y}^{2} + 54m_{x}^{2} m_{y}^{3} - 9m_{y}^{5} ) \hfill \\ + m_{x}^{2} (9m_{x}^{5} + 27m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{2} - 54m_{x}^{2} m_{y}^{3} + m_{x} m_{y}^{4} + 27m_{y}^{5} )] \hfill \\ H_{3,12} = \frac{{9m_{y} }}{{128m_{0} }}[32A^{5} m_{x} (9m_{x} + 4m_{y} ) + 48A^{3} m_{x}^{3} (3m_{x} + m_{y} ) + 64A^{6} (2m_{x} - 9m_{y} ) \hfill \\ + 16A^{4} (m_{x}^{3} + 9m_{x}^{2} m_{y} + 2m_{x} m_{y}^{2} + 18m_{y}^{3} ) - 2Am_{x}^{2} (27m_{x}^{4} + 18m_{x}^{3} m_{y} - 36m_{x}^{2} m_{y}^{2} - 2m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ - 4A^{4} (12m_{x}^{5} - 99m_{x}^{4} m_{y} - 4m_{x}^{3} m_{y}^{2} - 54m_{x}^{2} m_{y}^{3} + 9m_{y}^{5} ) \hfill \\ + m_{x}^{2} (9m_{x}^{5} - 27m_{x}^{4} m_{y} - 10m_{x}^{3} m_{y}^{2} + 54m_{x}^{2} m_{y}^{3} + m_{x} m_{y}^{4} - 27m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(76)
$$\left\{ \begin{aligned} H_{4,1} = \frac{1}{{64Am_{0} }}[10m_{x}^{2} (m_{x} - m_{y} )^{3} m_{y}^{2} (m_{x} + m_{y} )^{2} - 20Am_{x}^{2} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ + 16A^{5} m_{x} m_{y} (9m_{x}^{2} + 10m_{x} m_{y} + 9m_{y}^{2} ) + 144A^{6} (m_{x}^{3} - m_{x}^{2} m_{y} + m_{x} m_{y}^{2} - m_{y}^{3} ) + \hfill \\ 4A^{3} m_{x} m_{y} (9m_{x}^{4} + 10m_{x}^{3} m_{y} + 18m_{x}^{2} m_{y}^{2} + 10m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ - 8A^{4} (9m_{x}^{5} + 9m_{x}^{4} m_{y} + 10m_{x}^{3} m_{y}^{2} - 10m_{x}^{2} m_{y}^{3} - 9m_{x} m_{y}^{4} - 9m_{y}^{5} ) + \hfill \\ A^{2} (9m_{x}^{7} - 9m_{x}^{6} m_{y} - 65m_{x}^{5} m_{y}^{2} + 65m_{x}^{4} m_{y}^{3} - 65m_{x}^{3} m_{y}^{4} + 65m_{x}^{2} m_{y}^{5} + 9m_{x} m_{y}^{6} - 9m_{y}^{7} )] \hfill \\ H_{4,2} = \frac{2}{{64Am_{0} }}[ - 10m_{x}^{2} (m_{x} - m_{y} )^{3} m_{y}^{2} (m_{x} + m_{y} )^{2} + 20Am_{x}^{2} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ + 16A^{5} m_{x} m_{y} (9m_{x}^{2} - 10m_{x} m_{y} + 9m_{y}^{2} ) - 144A^{6} (m_{x}^{3} + m_{x}^{2} m_{y} + m_{x} m_{y}^{2} + m_{y}^{3} ) \hfill \\ + 4A^{3} m_{x} m_{y} (9m_{x}^{4} - 10m_{x}^{3} m_{y} + 18m_{x}^{2} m_{y}^{2} - 10m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ + 8A^{4} (9m_{x}^{5} - 9m_{x}^{4} m_{y} + 10m_{x}^{3} m_{y}^{2} + 10m_{x}^{2} m_{y}^{3} - 9m_{x} m_{y}^{4} + 9m_{y}^{5} ) + \hfill \\ A^{2} ( - 9m_{x}^{7} - 9m_{x}^{6} m_{y} - 65m_{x}^{5} m_{y}^{2} + 65m_{x}^{4} m_{y}^{3} + 65m_{x}^{3} m_{y}^{4} + 65m_{x}^{2} m_{y}^{5} - 9m_{x} m_{y}^{6} - 9m_{y}^{7} )] \hfill \\ H_{4,3} = \frac{1}{{64Am_{0} }}[ - 10m_{x}^{2} (m_{x} - m_{y} )^{3} m_{y}^{2} (m_{x} + m_{y} )^{2} - 20Am_{x}^{2} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ + 16A^{5} m_{x} m_{y} (9m_{x}^{2} + 10m_{x} m_{y} + 9m_{y}^{2} ) - 144A^{6} (m_{x}^{3} - m_{x}^{2} m_{y} + m_{x} m_{y}^{2} - m_{y}^{3} ) \hfill \\ + 4A^{3} m_{x} m_{y} (9m_{x}^{4} + 10m_{x}^{3} m_{y} + 18m_{x}^{2} m_{y}^{2} + 10m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ + 8A^{4} (9m_{x}^{5} + 9m_{x}^{4} m_{y} + 10m_{x}^{3} m_{y}^{2} - 10m_{x}^{2} m_{y}^{3} - 9m_{x} m_{y}^{4} - 9m_{y}^{5} ) + \hfill \\ A^{2} ( - 9m_{x}^{7} + 9m_{x}^{6} m_{y} + 65m_{x}^{5} m_{y}^{2} - 65m_{x}^{4} m_{y}^{3} + 65m_{x}^{3} m_{y}^{4} - 65m_{x}^{2} m_{y}^{5} - 9m_{x} m_{y}^{6} + 9m_{y}^{7} )] \hfill \\ \end{aligned} \right.$$
(77)
$$\left\{ \begin{aligned} H_{4,4} = \frac{1}{{64Am_{0} }}[10m_{x}^{2} (m_{x} - m_{y} )^{3} m_{y}^{2} (m_{x} + m_{y} )^{2} + 20Am_{x}^{2} m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ + 16A^{5} m_{x} m_{y} (9m_{x}^{2} - 10m_{x} m_{y} + 9m_{y}^{2} ) + 144A^{6} (m_{x}^{3} + m_{x}^{2} m_{y} + m_{x} m_{y}^{2} + m_{y}^{3} ) \hfill \\ + 4A^{3} m_{x} m_{y} (9m_{x}^{4} - 10m_{x}^{3} m_{y} + 18m_{x}^{2} m_{y}^{2} - 10m_{x} m_{y}^{3} + 9m_{y}^{4} ) \hfill \\ - 8A^{4} (9m_{x}^{5} - 9m_{x}^{4} m_{y} + 10m_{x}^{3} m_{y}^{2} + 10m_{x}^{2} m_{y}^{3} - 9m_{x} m_{y}^{4} + 9m_{y}^{5} ) + \hfill \\ A^{2} (9m_{x}^{7} + 9m_{x}^{6} m_{y} - 65m_{x}^{5} m_{y}^{2} - 65m_{x}^{4} m_{y}^{3} - 65m_{x}^{3} m_{y}^{4} - 65m_{x}^{2} m_{y}^{5} + 9m_{x} m_{y}^{6} + 9m_{y}^{7} )] \hfill \\ H_{4,5} = - \frac{{9m_{x}^{2} }}{{64Am_{0} }}[16A^{6} (3m_{x} - m_{y} ) + 16A^{5} m_{y} (m_{x} + 3m_{y} ) + (3m_{x} - m_{y} )m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ - 8A^{4} m_{x} (3m_{x}^{2} + m_{x} m_{y} + 6m_{y}^{2} ) + 4A^{3} m_{y} (m_{x}^{3} + 3m_{x}^{2} m_{y} + m_{x} m_{y}^{2} + 3m_{y}^{3} ) \hfill \\ - 2Am_{y}^{2} (3m_{x}^{4} - 2m_{x}^{3} m_{y} - 6m_{x}^{2} m_{y}^{2} + 2m_{x} m_{y}^{3} + 3m_{y}^{4} ) \hfill \\ + A^{2} (3m_{x}^{5} - m_{x}^{4} m_{y} - 12m_{x}^{3} m_{y}^{2} - 4m_{x}^{2} m_{y}^{3} - 39m_{x} m_{y}^{4} + 5m_{y}^{5} )] \hfill \\ H_{4,6} = \frac{{9m_{x}^{2} }}{{64Am_{0} }}[16A^{6} (3m_{x} + m_{y} ) - 16A^{5} m_{y} (m_{x} - 3m_{y} ) + (3m_{x} + m_{y} )m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ - 8A^{4} m_{x} (3m_{x}^{2} - m_{x} m_{y} + 6m_{y}^{2} ) - 4A^{3} m_{y} (m_{x}^{3} - 3m_{x}^{2} m_{y} + m_{x} m_{y}^{2} - 3m_{y}^{3} ) \hfill \\ - 2Am_{y}^{2} (3m_{x}^{4} + 2m_{x}^{3} m_{y} - 6m_{x}^{2} m_{y}^{2} - 2m_{x} m_{y}^{3} + 3m_{y}^{4} ) \hfill \\ + A^{2} (3m_{x}^{5} + m_{x}^{4} m_{y} - 12m_{x}^{3} m_{y}^{2} + 4m_{x}^{2} m_{y}^{3} - 39m_{x} m_{y}^{4} - 5m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(78)
$$\left\{ \begin{aligned} H_{4,7} = \frac{{9m_{y}^{2} }}{{64Am_{0} }}[16A^{6} (m_{x} + 3m_{y} ) + 16A^{5} m_{x} (3m_{x} - m_{y} ) + (m_{x} + 3m_{y} )m_{x}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ - 8A^{4} m_{y} (3m_{y}^{2} - m_{x} m_{y} + 6m_{x}^{2} ) + 4A^{3} m_{x} (3m_{x}^{3} - m_{x}^{2} m_{y} + 3m_{x} m_{y}^{2} - m_{y}^{3} ) \hfill \\ - 2Am_{x}^{2} (3m_{x}^{4} - 2m_{x}^{3} m_{y} - 6m_{x}^{2} m_{y}^{2} + 2m_{x} m_{y}^{3} + 3m_{y}^{4} ) \hfill \\ + A^{2} ( - 5m_{x}^{5} - 39m_{x}^{4} m_{y} + 4m_{x}^{3} m_{y}^{2} - 12m_{x}^{2} m_{y}^{3} + m_{x} m_{y}^{4} + 3m_{y}^{5} )] \hfill \\ H_{4,8} = \frac{{9m_{y}^{2} }}{{64Am_{0} }}[16A^{6} (m_{x} - 3m_{y} ) - 16A^{5} m_{x} (3m_{x} + m_{y} ) + (m_{x} - 3m_{y} )m_{x}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ + 8A^{4} m_{y} (3m_{y}^{2} + m_{x} m_{y} + 6m_{x}^{2} ) - 4A^{3} m_{x} (3m_{x}^{3} + m_{x}^{2} m_{y} + 3m_{x} m_{y}^{2} + m_{y}^{3} ) \hfill \\ - 2Am_{x}^{2} (3m_{x}^{4} + 2m_{x}^{3} m_{y} - 6m_{x}^{2} m_{y}^{2} - 2m_{x} m_{y}^{3} + 3m_{y}^{4} ) \hfill \\ + A^{2} ( - 5m_{x}^{5} + 39m_{x}^{4} m_{y} + 4m_{x}^{3} m_{y}^{2} + 12m_{x}^{2} m_{y}^{3} + m_{x} m_{y}^{4} - 3m_{y}^{5} )] \hfill \\ H_{4,9} = \frac{{9m_{x}^{2} }}{{64Am_{0} }}[16A^{6} (3m_{x} - m_{y} ) - 16A^{5} m_{y} (m_{x} + 3m_{y} ) + (3m_{x} - m_{y} )m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ - 8A^{4} m_{x} (3m_{x}^{2} + m_{x} m_{y} + 6m_{y}^{2} ) - 4A^{3} m_{y} (m_{x}^{3} + 3m_{x}^{2} m_{y} + m_{x} m_{y}^{2} + 3m_{y}^{3} ) + \hfill \\ 2Am_{y}^{2} (3m_{x}^{4} - 2m_{x}^{3} m_{y} - 6m_{x}^{2} m_{y}^{2} + 2m_{x} m_{y}^{3} + 3m_{y}^{4} ) \hfill \\ + A^{2} (3m_{x}^{5} - m_{x}^{4} m_{y} - 12m_{x}^{3} m_{y}^{2} - 4m_{x}^{2} m_{y}^{3} - 39m_{x} m_{y}^{4} + 5m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(79)
$$\left\{ \begin{aligned} H_{4,10} = - \frac{{9m_{x}^{2} }}{{64Am_{0} }}[16A^{6} (3m_{x} + m_{y} ) + 16A^{5} m_{y} (m_{x} - 3m_{y} ) + (3m_{x} + m_{y} )m_{y}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ - 8A^{4} m_{x} (3m_{x}^{2} - m_{x} m_{y} + 6m_{y}^{2} ) + 4A^{3} m_{y} (m_{x}^{3} - 3m_{x}^{2} m_{y} + m_{x} m_{y}^{2} - 3m_{y}^{3} ) \hfill \\ + 2Am_{y}^{2} (3m_{x}^{4} + 2m_{x}^{3} m_{y} - 6m_{x}^{2} m_{y}^{2} - 2m_{x} m_{y}^{3} + 3m_{y}^{4} ) \hfill \\ + A^{2} (3m_{x}^{5} + m_{x}^{4} m_{y} - 12m_{x}^{3} m_{y}^{2} + 4m_{x}^{2} m_{y}^{3} - 39m_{x} m_{y}^{4} - 5m_{y}^{5} )] \hfill \\ H_{4,11} = - \frac{{9m_{y}^{2} }}{{64Am_{0} }}[16A^{6} (m_{x} + 3m_{y} ) + 16A^{5} m_{x} ( - 3m_{x} + m_{y} ) + (m_{x} + 3m_{y} )m_{x}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ - 8A^{4} m_{y} (3m_{y}^{2} - m_{x} m_{y} + 6m_{x}^{2} ) - 4A^{3} m_{x} (3m_{x}^{3} - m_{x}^{2} m_{y} + 3m_{x} m_{y}^{2} - m_{y}^{3} ) \hfill \\ + 2Am_{x}^{2} (3m_{x}^{4} - 2m_{x}^{3} m_{y} - 6m_{x}^{2} m_{y}^{2} + 2m_{x} m_{y}^{3} + 3m_{y}^{4} ) \hfill \\ + A^{2} ( - 5m_{x}^{5} - 39m_{x}^{4} m_{y} + 4m_{x}^{3} m_{y}^{2} - 12m_{x}^{2} m_{y}^{3} + m_{x} m_{y}^{4} + 3m_{y}^{5} )] \hfill \\ H_{4,12} = - \frac{{9m_{y}^{2} }}{{64Am_{0} }}[16A^{6} (m_{x} - 3m_{y} ) + 16A^{5} m_{x} (3m_{x} + m_{y} ) + (m_{x} - 3m_{y} )m_{x}^{2} (m_{x}^{2} - m_{y}^{2} )^{2} \hfill \\ + 8A^{4} m_{y} (3m_{y}^{2} + m_{x} m_{y} + 6m_{x}^{2} ) + 4A^{3} m_{x} (3m_{x}^{3} + m_{x}^{2} m_{y} + 3m_{x} m_{y}^{2} + m_{y}^{3} ) + \hfill \\ 2Am_{x}^{2} (3m_{x}^{4} + 2m_{x}^{3} m_{y} - 6m_{x}^{2} m_{y}^{2} - 2m_{x} m_{y}^{3} + 3m_{y}^{4} ) \hfill \\ + A^{2} ( - 5m_{x}^{5} + 39m_{x}^{4} m_{y} + 4m_{x}^{3} m_{y}^{2} + 12m_{x}^{2} m_{y}^{3} + m_{x} m_{y}^{4} - 3m_{y}^{5} )] \hfill \\ \end{aligned} \right.$$
(80)

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Zhang, G., Xiang, J. A 12-node conforming straight-sided quadrilateral element with high-order completeness (QH12-C1). Engineering with Computers 38, 453–479 (2022). https://doi.org/10.1007/s00366-020-01075-y

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