An enrichment technique for bending analysis of in-plane heterogeneous thin plates with weak singularities

Abstract

Static solution of thin elastic plate problems with in-plane varying thickness or material properties having weak point singularities (e.g. crack tip or notches) is studied using a novel enrichment technique. Since the smooth basis functions are not capable of adapting to the adjacency of the singular edge point, enrichment bases called Equilibrated Singular Basis Functions (EqSBFs) are added to improve the solution quality. A combination of Chebyshev polynomials of the first kind and trigonometric functions are used as basis functions. The equilibrium equation is enforced by a weighted residual approach over a fictitious domain which contains the main problem domain. The total integration process is replaced by a composition of normalized pre-evaluated integrals, thus speeding up the procedure considerably. The novelty of the paper is that the proposed method can automatically identify and reproduce the enriching terms corresponding to the singularity order of the problem, which is an advantage with respect to the similar methods that need a priori knowledge of the analytical singularity order. Although the proposed technique is developed in the context of boundary methods, it may also be useful in other enriched methods such as XFEM.

Appendices

Appendix A The angular integrals of the coefficient matrix $\mathbf{A} ^S$

$$\begin{gathered}\bar{I}_{m,n,q}^1=\int _0^{2\pi } \mathrm{cos}^4\theta \mathrm{sin} m\bar{\theta } \mathrm{sin}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^2=\int _0^{2\pi } \mathrm{cos}^4\theta \mathrm{sin}m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^3=\int _0^{2\pi } \mathrm{cos}^3\theta \mathrm{sin}\theta \mathrm{sin}m\bar{\theta } \mathrm{sin}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^4=\int _0^{2\pi } \mathrm{cos}^3\theta \mathrm{sin} \theta \mathrm{sin}m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^5=\int _0^{2\pi } \mathrm{cos}^2\theta \mathrm{sin}^2\theta \mathrm{sin}m\bar{\theta } \mathrm{sin}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^6=\int _0^{2\pi } \mathrm{cos}^2\theta \mathrm{sin}^2\theta \mathrm{sin}m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^7=\int _0^{2\pi } \mathrm{cos}\theta \mathrm{sin}^3\theta \mathrm{sin}m\bar{\theta } \mathrm{sin}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^8=\int _0^{2\pi } \mathrm{cos}\theta \mathrm{sin}^3\theta \mathrm{sin}m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q\bar{\theta }d \bar{\theta }\\ \bar{I}_{m,n,q}^9=\int _0^{2\pi } \mathrm{sin}^4\theta \mathrm{sin}m\bar{\theta } \mathrm{sin}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{10}=\int _0^{2\pi } \mathrm{sin}^4\theta \mathrm{sin}m \bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{11}=\int _0^{2\pi } \mathrm{cos}^4\theta \mathrm{sin}m\bar{\theta }\mathrm{cos}n \bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{12}=\int _0^{2\pi } \mathrm{cos}^4\theta \mathrm{sin}m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{13}=\int _0^{2\pi } \mathrm{cos}^3\theta \mathrm{sin}\theta \mathrm{sin}m\bar{\theta } \mathrm{cos}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{14}=\int _0^{2\pi } \mathrm{cos}^3\theta \mathrm{sin} \theta \mathrm{sin}m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{15}=\int _0^{2\pi } \mathrm{cos}^2\theta \mathrm{sin}^2\theta \mathrm{sin}m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{sin}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{16}=\int _0^{2\pi }\mathrm{cos}^2\theta \mathrm{sin}^2\theta \mathrm{sin}m\bar{\theta }\mathrm{cos}n \bar{\theta }\mathrm{cos}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{17}=\int _0^{2\pi } \mathrm{cos}\theta \mathrm{sin}^3\theta \mathrm{sin}m\bar{\theta } \mathrm{cos}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{18}=\int _0^{2\pi } \mathrm{cos}\theta \mathrm{sin}^3 \theta \mathrm{sin}m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{19}=\int _0^{2\pi } \mathrm{sin}^4\theta \mathrm{sin}m\bar{\theta }\mathrm{cos}n\bar{\theta } \mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{20}=\int _0^{2\pi } \mathrm{sin}^4\theta \mathrm{sin} m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{21}=\int _0^{2\pi } \mathrm{cos}^4\theta \mathrm{cos}m\bar{\theta }\mathrm{sin}n\bar{\theta } \mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{22}=\int _0^{2\pi } \mathrm{cos}^4\theta \mathrm{cos} m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{23}=\int _0^{2\pi } \mathrm{cos}^3\theta \mathrm{sin}\theta \mathrm{cos}m\bar{\theta } \mathrm{sin}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{24}=\int _0^{2\pi } \mathrm{cos}^3\theta \mathrm{sin} \theta \mathrm{cos}m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{25}=\int _0^{2\pi } \mathrm{cos}^2\theta \mathrm{sin}^2\theta \mathrm{cos}m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{sin}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{26}=\int _0^{2\pi } \mathrm{cos}^2\theta \mathrm{sin}^2\theta \mathrm{cos}m\bar{\theta }\mathrm{sin}n \bar{\theta }\mathrm{cos}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{27}=\int _0^{2\pi } \mathrm{cos}\theta \mathrm{sin}^3\theta \mathrm{cos}m\bar{\theta } \mathrm{sin}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{28}=\int _0^{2\pi } \mathrm{cos}\theta \mathrm{sin}^3 \theta \mathrm{cos}m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{29}=\int _0^{2\pi } \mathrm{sin}^4\theta \mathrm{cos}m\bar{\theta }\mathrm{sin}n\bar{\theta } \mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{30}=\int _0^{2\pi } \mathrm{sin}^4\theta \mathrm{cos} m\bar{\theta }\mathrm{sin}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{31}=\int _0^{2\pi } \mathrm{cos}^4\theta \mathrm{cos}m\bar{\theta }\mathrm{cos}n \bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta } \\ \bar{I}_{m,n,q}^{32}=\int _0^{2\pi } \mathrm{cos}^4\theta \mathrm{cos}m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{33}=\int _0^{2\pi } \mathrm{cos}^3\theta \mathrm{sin}\theta \mathrm{cos}m\bar{\theta } \mathrm{cos}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{34}=\int _0^{2\pi } \mathrm{cos}^3\theta \mathrm{sin} \theta \mathrm{cos}m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{35}=\int _0^{2\pi } \mathrm{cos}^2\theta \mathrm{sin}^2\theta \mathrm{cos}m\bar{\theta }\mathrm{cos}n\bar{\theta } \mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{36}=\int _0^{2\pi } \mathrm{cos}^2 \theta \mathrm{sin}^2\theta \mathrm{cos}m\bar{\theta } \mathrm{cos}n\bar{\theta }\mathrm{cos}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{37}=\int _0^{2\pi } \mathrm{cos}\theta \mathrm{sin}^3\theta \mathrm{cos}m\bar{\theta } \mathrm{cos}n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{38}=\int _0^{2\pi } \mathrm{cos}\theta \mathrm{sin}^3 \theta \mathrm{cos}m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{39}=\int _0^{2\pi } \mathrm{sin}^4\theta \mathrm{cos}m\bar{\theta }\mathrm{cos} n\bar{\theta }\mathrm{sin}q\bar{\theta }d\bar{\theta }\\ \bar{I}_{m,n,q}^{40}=\int _0^{2\pi } \mathrm{sin}^4\theta \mathrm{cos}m\bar{\theta }\mathrm{cos}n\bar{\theta }\mathrm{cos}q \bar{\theta }d\bar{\theta }\\ \bar{\theta }=\gamma \theta \qquad m,n=0,...,O_t^S \qquad q=0,...,m_{\theta }\end{gathered}$$

Appendix B Matrix entries of $\mathbf{A} ^S$ in (45)

We defined $\hat{\mathbf{I }}_k =(\tilde{\mathbf{I }}_k)_{m+1,n+1,all},\qquad k=1,...,20$

$$\begin{gathered}\mathbf{A} _{2m+1,2n+1}^S\\ \quad =\sum \limits _{p=0}^{m_r}[(\beta ^3/\gamma )((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{3})^T (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{22}) (\mathbf{A} _1)_{all,all,p+1}\\ \qquad +\{(\beta ^2(\beta -1)/\gamma ) ((\hat{\mathbf{I }}_{1})^T\tilde{\mathbf{c }}_D^{11} +(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12} +4\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21}) +(\hat{\mathbf{I }}_{5})^T\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta ^2/\gamma )((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{12}+(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}\beta ^2 ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}) +(\hat{\mathbf{I }}_{9})^T\\ \qquad (\tilde{\mathbf{c }}_D^{22}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21}))\}(\mathbf{A} _2)_{all,all,p+1}\\ \qquad +\{\bar{\mathbf{D }}_{\theta 2}\beta \gamma ((\hat{\mathbf{I }}_{1})^T\tilde{\mathbf{c }}_D^{12} +(\hat{\mathbf{I }}_{3})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})-(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}\beta ((\hat{\mathbf{I }}_{7})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{12})+(\hat{\mathbf{I }}_{9})^T\\ \qquad (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22}))\}(\mathbf{A} _3)_{all,all,p+1}\\ \qquad +\{(\beta ^2(\beta -1)/\gamma )((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})\qquad +(\hat{\mathbf{I }}_{5})^T\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta ^2/\gamma )((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{21}+(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{12})\\ \qquad +2m\beta ^2((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}) +(\hat{\mathbf{I }}_{14})^T\\ \qquad (\tilde{\mathbf{c }}_D^{22}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{12}))\}(\mathbf{A} _4)_{all,all,p+1}\\ \qquad +\{(\beta (\beta -1)^2/\gamma )((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta (\beta -1)/\gamma )((\hat{\mathbf{I }}_{1})^T (\tilde{\mathbf{c }}_D^{12}+\tilde{\mathbf{c }}_D^{21}) +(\hat{\mathbf{I }}_{3})^T\\ \qquad (2\tilde{\mathbf{c }}_D^{11} -8\tilde{\mathbf{c }}_D^{33}+2\tilde{\mathbf{c }}_D^{22}) +(\hat{\mathbf{I }}_{5})^T(\tilde{\mathbf{c }}_D^{12} +\tilde{\mathbf{c }}_D^{21}))\\ \qquad +(\beta /\gamma )((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{22}+(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{11})\\ \qquad +2m\beta ((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +2m\beta (\beta -1)((\hat{\mathbf{I }}_{12})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{14})^T (\tilde{\mathbf{c }}_D^{22}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{12}))\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}\beta ((\hat{\mathbf{I }}_{7})^T (\tilde{\mathbf{c }}_D^{22}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}\beta (\beta -1) ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\\ \qquad +4m\bar{\mathbf{D }}_{\theta 1}\beta \gamma ((\hat{\mathbf{I }}_{16})^T\tilde{\mathbf{c }}_D^{33} +(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-\tilde{\mathbf{c }}_D^{12} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21} +\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{33})\}(\mathbf{A} _5)_{all,all,p+1}\\ \qquad +\{-\bar{\mathbf{D }}_{\theta 2}(\beta -1)\gamma ((\hat{\mathbf{I }}_{1})^T\tilde{\mathbf{c }}_D^{12} -(\hat{\mathbf{I }}_{3})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{22}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad -4m\bar{\mathbf{D }}_{\theta 1}\gamma ((\hat{\mathbf{I }}_{16})^T\tilde{\mathbf{c }}_D^{33} -(\hat{\mathbf{I }}_{18})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{33})\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}(\beta -1) ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}((\hat{\mathbf{I }}_{7})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\\ \qquad +2m\bar{\mathbf{D }}_{\theta 2}\gamma ^2 ((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\} (\mathbf{A} _6)_{all,all,p+1}\\ \qquad +\{-m^2\beta \gamma ((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{21}-(\hat{\mathbf{I }}_{3})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{5})^T (\tilde{\mathbf{c }}_D^{12}))\\ \qquad +2m\beta ((\hat{\mathbf{I }}_{12})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\} (\mathbf{A} _7)_{all,all,p+1}\\ \qquad +\{-m^2(\beta -1)\gamma ((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{21}-(\hat{\mathbf{I }}_{3})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{12})\\ \qquad -m^2\gamma ((\hat{\mathbf{I }}_{1})^T\tilde{\mathbf{c }}_D^{22}+(\hat{\mathbf{I }}_{3})^T(\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{5})^T(\tilde{\mathbf{c }}_D^{11}))\\ \qquad -4m\bar{\mathbf{D }}_{\theta 1}\gamma ((\hat{\mathbf{I }}_{16})^T\tilde{\mathbf{c }}_D^{33} -(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}-\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{33})\\ \qquad +2m(\beta -1)((\hat{\mathbf{I }}_{12})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\\ \qquad +2m((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{12}+2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\\ \qquad +2m^2\bar{\mathbf{D }}_{\theta 1}\gamma ^2 ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\} (\mathbf{A} _8)_{all,all,p+1}\\ \qquad +\{4m\bar{\mathbf{D }}_{\theta 1}(\gamma /\beta ) ((\hat{\mathbf{I }}_{16})^T(\tilde{\mathbf{c }}_D^{33} +(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-\tilde{\mathbf{c }}_D^{12} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{33}))\\ \qquad +m^2\bar{\mathbf{D }}_{\theta 2}(\gamma ^3/\beta ) ((\hat{\mathbf{I }}_{1})^T\tilde{\mathbf{c }}_D^{22} +(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{11})\\ \qquad +2m^2\bar{\mathbf{D }}_{\theta 1}(\gamma ^2/\beta ) ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\\ \qquad +2m\bar{\mathbf{D }}_{\theta 2}(\gamma ^2/\beta ) ((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\} (\mathbf{A} _9)_{all,all,p+1}]\\ \mathbf{A} _{2m+2,2n+1}^S\\ \quad =\sum \limits _{p=0}^{m_r}[(\beta ^3/\gamma ) ((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{11} +(\hat{\mathbf{I }}_{8})^T(\tilde{\mathbf{c }}_D^{12} +4\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{10})^T\tilde{\mathbf{c }}_D^{22}) (\mathbf{A} _1)_{all,all,p+1}\\ \qquad +\{(\beta ^2(\beta -1)/\gamma )((\hat{\mathbf{I }}_{6})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta ^2/\gamma )((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{12}+(\hat{\mathbf{I }}_{8})^T(\tilde{\mathbf{c }}_D^{11}-4\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{10})^T\tilde{\mathbf{c }}_D^{21})\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}\beta ^2 ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\} (\mathbf{A} _2)_{all,all,p+1}\\ \qquad +\{-\bar{\mathbf{D }}_{\theta 2}\beta \gamma ((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{12} +(\hat{\mathbf{I }}_{8})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})-(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}\beta ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\} (\mathbf{A} _3)_{all,all,p+1}\\ \qquad +\{(\beta ^2(\beta -1)/\gamma )((\hat{\mathbf{I }}_{6})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta ^2/\gamma )((\hat{\mathbf{I }}_{6})^T \tilde{\mathbf{c }}_D^{21}+(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{12})\\ \qquad +2m\beta ^2((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\} (\mathbf{A} _4)_{all,all,p+1}\\ \qquad +\{(\beta (\beta -1)^2/\gamma )((\hat{\mathbf{I }}_{6})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta (\beta -1)/\gamma )((\hat{\mathbf{I }}_{6})^T (\tilde{\mathbf{c }}_D^{12}+\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{8})^T(2\tilde{\mathbf{c }}_D^{11} -8\tilde{\mathbf{c }}_D^{33}+2\tilde{\mathbf{c }}_D^{22}) +(\hat{\mathbf{I }}_{10})^T (\tilde{\mathbf{c }}_D^{12} +\tilde{\mathbf{c }}_D^{21}))\\ \qquad +(\beta /\gamma )((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{22} +(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21}) +(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{11})\\ \qquad +2m\beta ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +2m\beta (\beta -1)((\hat{\mathbf{I }}_{17})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}\beta ((\hat{\mathbf{I }}_{2})^T (\tilde{\mathbf{c }}_D^{22}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}\beta (\beta -1) ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\\ \qquad -4m\bar{\mathbf{D }}_{\theta 1}\beta \gamma ((\hat{\mathbf{I }}_{11})^T\tilde{\mathbf{c }}_D^{33} +(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-\tilde{\mathbf{c }}_D^{12} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21} +\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{33})\}(\mathbf{A} _5)_{all,all,p+1}\\ \qquad +\{\bar{\mathbf{D }}_{\theta 2}(\beta -1)\gamma ((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{12} -(\hat{\mathbf{I }}_{8})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{22}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad +4m\bar{\mathbf{D }}_{\theta 1}\gamma ((\hat{\mathbf{I }}_{11})^T\tilde{\mathbf{c }}_D^{33} -(\hat{\mathbf{I }}_{13})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{33})\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}(\beta -1) ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}((\hat{\mathbf{I }}_{2})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\\ \qquad -2m\bar{\mathbf{D }}_{\theta 2}\gamma ^2 ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\} (\mathbf{A} _6)_{all,all,p+1}\\ \qquad +\{m^2\beta \gamma ((\hat{\mathbf{I }}_{6})^T \tilde{\mathbf{c }}_D^{21}-(\hat{\mathbf{I }}_{8})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{10})^T (\tilde{\mathbf{c }}_D^{12}))\\ \qquad +2m\beta ((\hat{\mathbf{I }}_{17})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\} (\mathbf{A} _7)_{all,all,p+1}\\ \qquad +\{m^2(\beta -1)\gamma ((\hat{\mathbf{I }}_{6})^T \tilde{\mathbf{c }}_D^{21}-(\hat{\mathbf{I }}_{8})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{12})\\ \qquad +m^2\gamma ((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{22} +(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{10})^T (\tilde{\mathbf{c }}_D^{11}))\\ \qquad +4m\bar{\mathbf{D }}_{\theta 1}\gamma ((\hat{\mathbf{I }}_{11})^T\tilde{\mathbf{c }}_D^{33} -(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}-\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{33})\\ \qquad +2m(\beta -1)((\hat{\mathbf{I }}_{17})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\\ \qquad +2m((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\\ \qquad +2m^2\bar{\mathbf{D }}_{\theta 1}\gamma ^2 ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\} (\mathbf{A} _8)_{all,all,p+1}\\ \qquad -\{4m\bar{\mathbf{D }}_{\theta 1}(\gamma /\beta ) ((\hat{\mathbf{I }}_{11})^T(\tilde{\mathbf{c }}_D^{33} +(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-\tilde{\mathbf{c }}_D^{12} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{33}))\\ \qquad +m^2\bar{\mathbf{D }}_{\theta 2}(\gamma ^3/\beta ) ((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{22} +(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{11})\\ \qquad +2m^2\bar{\mathbf{D }}_{\theta 1}(\gamma ^2/\beta ) ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\\ \qquad -2m\bar{\mathbf{D }}_{\theta 2}(\gamma ^2/\beta ) ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\} (\mathbf{A} _9)_{all,all,p+1}]\\ \mathbf{A} _{2m+1,2n+2}^S\nonumber \\ \quad =\sum \limits _{p=0}^{m_r}[(\beta ^3/\gamma ) ((\hat{\mathbf{I }}_{11})^T\tilde{\mathbf{c }}_D^{11} +(\hat{\mathbf{I }}_{13})^T(\tilde{\mathbf{c }}_D^{12} +4\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{15})^T\tilde{\mathbf{c }}_D^{22}) (\mathbf{A} _1)_{all,all,p+1}\\ \qquad +\{(\beta ^2(\beta -1)/\gamma )((\hat{\mathbf{I }}_{11})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta ^2/\gamma )((\hat{\mathbf{I }}_{11})^T \tilde{\mathbf{c }}_D^{12}+(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}\beta ^2 ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\} (\mathbf{A} _2)_{all,all,p+1}\\ \qquad +\{-\bar{\mathbf{D }}_{\theta 2}\beta \gamma ((\hat{\mathbf{I }}_{11})^T\tilde{\mathbf{c }}_D^{12} +(\hat{\mathbf{I }}_{13})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})-(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}\beta ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\} (\mathbf{A} _3)_{all,all,p+1}\\ \qquad +\{(\beta ^2(\beta -1)/\gamma )((\hat{\mathbf{I }}_{11})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta ^2/\gamma )((\hat{\mathbf{I }}_{11})^T \tilde{\mathbf{c }}_D^{21}+(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{12})\\ \qquad -2m\beta ^2((\hat{\mathbf{I }}_{2})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\} (\mathbf{A} _4)_{all,all,p+1}\\ \qquad +\{(\beta (\beta -1)^2/\gamma )((\hat{\mathbf{I }}_{11})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta (\beta -1)/\gamma )((\hat{\mathbf{I }}_{11})^T (\tilde{\mathbf{c }}_D^{12}+\tilde{\mathbf{c }}_D^{21}) +(\hat{\mathbf{I }}_{13})^T\\ \qquad (2\tilde{\mathbf{c }}_D^{11}-8\tilde{\mathbf{c }}_D^{33} +2\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{15})^T (\tilde{\mathbf{c }}_D^{12}+\tilde{\mathbf{c }}_D^{21}))\\ \qquad +(\beta /\gamma )((\hat{\mathbf{I }}_{11})^T \tilde{\mathbf{c }}_D^{22}+(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{11})\\ \qquad -2m\beta ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad -2m\beta (\beta -1)((\hat{\mathbf{I }}_{2})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}\beta ((\hat{\mathbf{I }}_{17})^T (\tilde{\mathbf{c }}_D^{22}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}\beta (\beta -1) ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\\ \qquad -4m\bar{\mathbf{D }}_{\theta 1}\beta \gamma ((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{33} +(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-\tilde{\mathbf{c }}_D^{12} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21} +\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{33})\}(\mathbf{A} _5)_{all,all,p+1}\\ \qquad +\{\bar{\mathbf{D }}_{\theta 2}(\beta -1)\gamma ((\hat{\mathbf{I }}_{11})^T\tilde{\mathbf{c }}_D^{12} -(\hat{\mathbf{I }}_{13})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{22}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad +4m\bar{\mathbf{D }}_{\theta 1}\gamma ((\hat{\mathbf{I }}_{6})^T \tilde{\mathbf{c }}_D^{33}-(\hat{\mathbf{I }}_{8})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{33})\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}(\beta -1) ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\\ \qquad +2\bar{\mathbf{D }}_{\theta 1}((\hat{\mathbf{I }}_{17})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\\ \qquad +2m\bar{\mathbf{D }}_{\theta 2}\gamma ^2 ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\} (\mathbf{A} _6)_{all,all,p+1}\\ \qquad +\{m^2\beta \gamma ((\hat{\mathbf{I }}_{11})^T \tilde{\mathbf{c }}_D^{21}-(\hat{\mathbf{I }}_{13})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{15})^T (\tilde{\mathbf{c }}_D^{12}))\\ \qquad -2m\beta ((\hat{\mathbf{I }}_{2})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\} (\mathbf{A} _7)_{all,all,p+1}\\ \qquad +\{m^2(\beta -1)\gamma ((\hat{\mathbf{I }}_{11})^T \tilde{\mathbf{c }}_D^{21}-(\hat{\mathbf{I }}_{13})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{12})\\ \qquad +m^2\gamma ((\hat{\mathbf{I }}_{11})^T\tilde{\mathbf{c }}_D^{22} +(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{15})^T (\tilde{\mathbf{c }}_D^{11}))\\ \qquad +4m\bar{\mathbf{D }}_{\theta 1}\gamma ((\hat{\mathbf{I }}_{6})^T\tilde{\mathbf{c }}_D^{33} -(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}-\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{33})\\ \qquad -2m(\beta -1)((\hat{\mathbf{I }}_{2})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\\ \qquad -2m((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\\ \qquad -2m^2\bar{\mathbf{D }}_{\theta 1}\gamma ^2 ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\} (\mathbf{A} _8)_{all,all,p+1}\\ \qquad +\{-4m\bar{\mathbf{D }}_{\theta 1}(\gamma /\beta ) ((\hat{\mathbf{I }}_{6})^T(\tilde{\mathbf{c }}_D^{33} +(\hat{\mathbf{I }}_{8})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-\tilde{\mathbf{c }}_D^{12} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{10})^T \tilde{\mathbf{c }}_D^{33}))\\ \qquad +m^2\bar{\mathbf{D }}_{\theta 2}(\gamma ^3/\beta ) ((\hat{\mathbf{I }}_{11})^T\tilde{\mathbf{c }}_D^{22} +(\hat{\mathbf{I }}_{13})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{15})^T \tilde{\mathbf{c }}_D^{11})\\ \qquad -2m^2\bar{\mathbf{D }}_{\theta 1}(\gamma ^2/\beta ) ((\hat{\mathbf{I }}_{17})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{19})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\\ \qquad +2m\bar{\mathbf{D }}_{\theta 2}(\gamma ^2/\beta ) ((\hat{\mathbf{I }}_{2})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{4})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\} (\mathbf{A} _9)_{all,all,p+1}]\\ \mathbf{A} _{2m+2,2n+2}^S\\ \quad =\sum \limits _{p=0}^{m_r}[(\beta ^3/\gamma ) ((\hat{\mathbf{I }}_{16})^T\tilde{\mathbf{c }}_D^{11} +(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12} +4\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21}) +(\hat{\mathbf{I }}_{20})^T\tilde{\mathbf{c }}_D^{22}) (\mathbf{A} _1)_{all,all,p+1}\\ \qquad +\{(\beta ^2(\beta -1)/\gamma )((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta ^2/\gamma )((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{12}+(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}\beta ^2 ((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\} (\mathbf{A} _2)_{all,all,p+1}\\ \qquad +\{-\bar{\mathbf{D }}_{\theta 2}\beta \gamma ((\hat{\mathbf{I }}_{16})^T\tilde{\mathbf{c }}_D^{12} +(\hat{\mathbf{I }}_{18})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})-(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}\beta ((\hat{\mathbf{I }}_{12})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\} (\mathbf{A} _3)_{all,all,p+1}\\ \qquad +\{(\beta ^2(\beta -1)/\gamma )((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta ^2/\gamma )((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{21}+(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{12})\\ \qquad -2m\beta ^2((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\} (\mathbf{A} _4)_{all,all,p+1}\\ \qquad +\{(\beta (\beta -1)^2/\gamma )((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{11}+(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{22})\\ \qquad +(\beta (\beta -1)/\gamma )((\hat{\mathbf{I }}_{16})^T (\tilde{\mathbf{c }}_D^{12}+\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{18})^T(2\tilde{\mathbf{c }}_D^{11} -8\tilde{\mathbf{c }}_D^{33}+2\tilde{\mathbf{c }}_D^{22}) +(\hat{\mathbf{I }}_{20})^T(\tilde{\mathbf{c }}_D^{12} +\tilde{\mathbf{c }}_D^{21}))\\ \qquad +(\beta /\gamma )((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{22}+(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{11})\\ \qquad -2m\beta ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad -2m\beta (\beta -1)((\hat{\mathbf{I }}_{7})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}\beta ((\hat{\mathbf{I }}_{12})^T (\tilde{\mathbf{c }}_D^{22}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}\beta (\beta -1) ((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\\ \qquad +4m\bar{\mathbf{D }}_{\theta 1}\beta \gamma ((\hat{\mathbf{I }}_{1})^T\tilde{\mathbf{c }}_D^{33} +(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-\tilde{\mathbf{c }}_D^{12} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21} +\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{33})\}(\mathbf{A} _5)_{all,all,p+1}\\ \qquad +\{-\bar{\mathbf{D }}_{\theta 2}(\beta -1)\gamma ((\hat{\mathbf{I }}_{16})^T\tilde{\mathbf{c }}_D^{12} -(\hat{\mathbf{I }}_{18})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{22}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{11})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{21})\\ \qquad -4m\bar{\mathbf{D }}_{\theta 1}\gamma ((\hat{\mathbf{I }}_{1})^T\tilde{\mathbf{c }}_D^{33} -(\hat{\mathbf{I }}_{3})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}+\tilde{\mathbf{c }}_D^{21} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{33})\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}(\beta -1) ((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\\ \qquad -2\bar{\mathbf{D }}_{\theta 1}((\hat{\mathbf{I }}_{12})^T (\tilde{\mathbf{c }}_D^{21}+2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\\ \qquad +2m\bar{\mathbf{D }}_{\theta 2}\gamma ^2 ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\} (\mathbf{A} _6)_{all,all,p+1}\\ \qquad +\{m^2\beta \gamma ((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{21}-(\hat{\mathbf{I }}_{18})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{20})^T (\tilde{\mathbf{c }}_D^{12}))\\ \qquad -2m\beta ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\} (\mathbf{A} _7)_{all,all,p+1}\\ \qquad +\{m^2(\beta -1)\gamma ((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{21}-(\hat{\mathbf{I }}_{18})^T\\ \qquad (-\tilde{\mathbf{c }}_D^{11}+4\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{12})\\ \qquad +m^2\gamma ((\hat{\mathbf{I }}_{16})^T \tilde{\mathbf{c }}_D^{22}+(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{20})^T (\tilde{\mathbf{c }}_D^{11}))\\ \qquad -4m\bar{\mathbf{D }}_{\theta 1}\gamma ((\hat{\mathbf{I }}_{1})^T \tilde{\mathbf{c }}_D^{33}-(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}-\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{33})\\ \qquad -2m(\beta -1)((\hat{\mathbf{I }}_{7})^T (\tilde{\mathbf{c }}_D^{11}-2\tilde{\mathbf{c }}_D^{33} -\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22}))\\ \qquad -2m((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21}))\\ \qquad +2m^2\bar{\mathbf{D }}_{\theta 1}\gamma ^2 ((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{22})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{11} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12}))\} (\mathbf{A} _8)_{all,all,p+1}\\ \qquad +\{4m\bar{\mathbf{D }}_{\theta 1}(\gamma /\beta ) ((\hat{\mathbf{I }}_{1})^T(\tilde{\mathbf{c }}_D^{33} +(\hat{\mathbf{I }}_{3})^T\\ \qquad (\tilde{\mathbf{c }}_D^{11}-\tilde{\mathbf{c }}_D^{12} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21} +\tilde{\mathbf{c }}_D^{22})+(\hat{\mathbf{I }}_{5})^T \tilde{\mathbf{c }}_D^{33}))\\ \qquad +m^2\bar{\mathbf{D }}_{\theta 2}(\gamma ^3/\beta ) ((\hat{\mathbf{I }}_{16})^T\tilde{\mathbf{c }}_D^{22} +(\hat{\mathbf{I }}_{18})^T\\ \qquad (\tilde{\mathbf{c }}_D^{12}+4\tilde{\mathbf{c }}_D^{33} +\tilde{\mathbf{c }}_D^{21})+(\hat{\mathbf{I }}_{20})^T \tilde{\mathbf{c }}_D^{11})\\ \qquad +2m^2\bar{\mathbf{D }}_{\theta 1}(\gamma ^2/\beta ) ((\hat{\mathbf{I }}_{12})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{21})\\ \qquad +(\hat{\mathbf{I }}_{14})^T(\tilde{\mathbf{c }}_D^{12} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\\ \qquad +2m\bar{\mathbf{D }}_{\theta 2}(\gamma ^2/\beta ) ((\hat{\mathbf{I }}_{7})^T(\tilde{\mathbf{c }}_D^{22} -2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{12})\\ \qquad +(\hat{\mathbf{I }}_{9})^T(\tilde{\mathbf{c }}_D^{21} +2\tilde{\mathbf{c }}_D^{33}-\tilde{\mathbf{c }}_D^{11}))\} (\mathbf{A} _9)_{all,all,p+1} \end{gathered}$$