An energy–momentum conserving scheme for geometrically exact shells with drilling DOFs

Abstract

An energy–momentum conserving temporal integration scheme is presented for a recently proposed geometrically exact shell formulation with drilling degrees of freedom. The scheme is based on a novel idea of defining mixed discrete derivatives for holonomic constraint functions with displacements and rotations. By defining general discrete derivative expressions with unknown terms, the mixed discrete derivatives with second-order accuracy are constructed according to deformation modes to satisfy directionality and orthogonality properties simultaneously, thus preserving conservation laws of total energy and momenta. The analysis of shell structures is conducted using the weak form quadrature elements to ensure exact incorporation of constraints and conservation of total energy after discretization, as well as circumvent shear and membrane locking phenomena. Benchmark numerical examples are presented to demonstrate the validity of the present scheme.

Appendix: Expression of element tangent stiffness matrix

The element stiffness matrix ${\mathbf{K}}^{(e)}$ consists of three parts corresponding to ${\mathbf{G}}_{iner}^{(e)}$, ${\mathbf{G}}_{int}^{(e)}$ and ${\mathbf{G}}_{C}^{(e)}$ in the element residual force vector as

$$ {\mathbf{K}}^{(e)} = {\mathbf{K}}_{iner}^{(e)} + {\mathbf{K}}_{int}^{(e)} + {\mathbf{K}}_{c}^{(e)} . $$

(A.1)

The element inertial tangent stiffness matrix ${\mathbf{K}}_{iner}^{(e)}$ can be derived from Eq. (66) as

$$ {\mathbf{K}}_{iner}^{(e)} = \frac{1}{\Delta t}\mathop \sum \limits_{i = 1}^{M} \mathop \sum \limits_{j = 1}^{N} w_{i} w_{j} j_{0mij} \left| {\mathbf{J}} \right|_{ij} {\mathbf{A}}_{ij}^{T} \widetilde{{\mathbf{H}}}_{ij} {\mathbf{A}}_{ij} $$

(A.2)

with

$$ \widetilde{{\mathbf{H}}} = \left[ {\begin{array}{*{20}c} {\frac{2}{\Delta t}{\mathbf{M}}_{\rho 0} } & {0_{3 \times 3} } \\ {0_{3 \times 3} } & {\frac{1}{2}\left( {\widehat{{{\dot{\mathbf{t}}}}}_{n + 1} - \widehat{{{\dot{\mathbf{t}}}}}_{n} } \right){\mathbf{I}}_{\rho 0} \widehat{{\mathbf{t}}}_{n + 1} - \frac{2}{\Delta t}\widehat{{\mathbf{t}}}_{n + 1/2} {\mathbf{I}}_{\rho 0} \widehat{{\mathbf{t}}}_{n + 1} } \\ \end{array} } \right] $$

(A.3)

The element internal stiffness matrix corresponding to Eq. (69) is written as

$$ \begin{aligned} & {\mathbf{K}}_{int}^{(e)} = \sum\limits_{i = 1}^{M} \sum\limits_{j = 1}^{N} w_{i} w_{j} j_{0mij} \left| {\mathbf{J}} \right|_{ij} \left( {\mathbf{B}}_{ij(n + 1/2)}^{T} {\mathbf{D}}_{{ij\left( {t_{n} ,t_{n + 1} } \right)}} {\mathbf{B}}_{ij(n + 1)} \right. \\ &\quad \left. + {\mathbf{C}}_{ij(n + 1/2)}^{T} {\bar{\mathbf{J}}}_{ij}^{T} {\varvec{\Xi}}_{{ij(t_{n} ,t_{n + 1} )}} {\bar{\mathbf{J}}}_{ij} {\mathbf{C}}_{ij(n + 1)} + {\varvec{\Theta}}_{{ij\left( {t_{n} ,t_{n + 1} } \right)}} \right) \end{aligned} $$

(A.4)

where

$$ {\mathbf{D}} = D_{{\varvec{\upchi}}} {\mathbf{N}} $$

(A.5)

is the constitutive matrix while

$$ {\varvec{\Xi}} = \frac{1}{2}\left[ {\begin{array}{*{20}c} {n^{11} {\mathbf{I}}_{3 \times 3} } & {n^{12} {\mathbf{I}}_{3 \times 3} } & {q^{1} {\mathbf{I}}_{3 \times 3} } & {m^{11} {\mathbf{I}}_{3 \times 3} } & {m^{12} {\mathbf{I}}_{3 \times 3} } \\ {n^{12} {\mathbf{I}}_{3 \times 3} } & {n^{22} {\mathbf{I}}_{3 \times 3} } & {q^{2} {\mathbf{I}}_{3 \times 3} } & {m^{12} {\mathbf{I}}_{3 \times 3} } & {m^{22} {\mathbf{I}}_{3 \times 3} } \\ {q^{1} {\mathbf{I}}_{3 \times 3} } & {q^{2} {\mathbf{I}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } \\ {m^{11} {\mathbf{I}}_{3 \times 3} } & {m^{12} {\mathbf{I}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } \\ {m^{12} {\mathbf{I}}_{3 \times 3} } & {m^{22} {\mathbf{I}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } \\ \end{array} } \right] $$

(A.6)

$$ {\varvec{\Theta}}_{{ij\left( {t_{n} ,t_{n + 1} } \right)}} = {\text{diag}}\left[ {\begin{array}{*{20}c} \cdots & {\left[ {\begin{array}{*{20}c} {0_{3 \times 3} } & {0_{3 \times 3} } & {0_{3 \times 1} } \\ {0_{3 \times 3} } & {\overline{{\varvec{\Theta}}} } & {0_{3 \times 1} } \\ {0_{1 \times 3} } & {0_{1 \times 3} } & 0 \\ \end{array} } \right]_{{(ij)(kl)\left( {t_{n} ,t_{n + 1} } \right)}} } & \cdots \\ \end{array} } \right] $$

(A.7)

with

$$ \begin{aligned} & {\bar{\boldsymbol{\varTheta }}}_{{(ij)(kl)\left( {t_{n} ,t_{n + 1} } \right)}}\\ & = \frac{1}{2}\delta_{ik} \delta_{jl} \left( {q_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{1} {\hat{\mathbf{r}}},_{1(n + 1/2)} + q_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{2} {\hat{\mathbf{r}}},_{2(n + 1/2)} } \right)_{ij} {\hat{\mathbf{t}}}_{(kl)(n + 1)} \\ & \quad + \frac{1}{2}\delta_{jl} C_{ik}^{(M)} \left[ \left( {\bar{J}_{11} m_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{11} + \bar{J}_{12} m_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{12} } \right){\hat{\mathbf{r}}},_{1(n + 1/2)} \right. \\ & \quad \left. + \left( {\bar{J}_{11} m_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{12} + \bar{J}_{12} m_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{22} } \right){\hat{\mathbf{r}}},_{2(n + 1/2)} \right]_{ij} {\hat{\mathbf{t}}}_{(kl)(n + 1)} \\ & \quad + \frac{1}{2}\delta_{ik} C_{jl}^{(N)} \left[ \left( {\bar{J}_{21} m_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{11} + \bar{J}_{22} m_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{12} } \right){\hat{\mathbf{r}}},_{1(n + 1/2)} \right. \\ & \quad \left. + \left( {\bar{J}_{21} m_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{12} + \bar{J}_{22} m_{{\left( {t_{n} ,t_{n + 1} } \right)}}^{22} } \right){\hat{\mathbf{r}}},_{2(n + 1/2)} \right]_{ij} {\hat{\mathbf{t}}}_{(kl)(n + 1)} \\ \end{aligned} $$

(A.8)

According to Eq. (80), the element tangent stiffness matrix related to drilling constraint comprises of two parts corresponding to the original and additional terms in discrete derivatives, which has the form

$$ {\mathbf{K}}_{C}^{(e)} = \mathop \sum \limits_{i = 1}^{M} \mathop \sum \limits_{j = 1}^{N} w_{i} w_{j} j_{0mij} \left| {\mathbf{J}} \right|_{ij} \widetilde{{\mathbf{B}}}_{ij}^{T} \left( {{\mathbf{H}}_{ij}^{o} + {\mathbf{H}}_{ij}^{a} } \right)\widetilde{{\mathbf{B}}}_{ij} , $$

(A.9)

where

$$ {\mathbf{H}}^{o} = \left[ {\begin{array}{*{20}c} {\overline{{\mathbf{H}}}_{11}^{o} } & \cdots & {\overline{{\mathbf{H}}}_{14}^{o} } \\ \vdots & \ddots & \vdots \\ {\overline{{\mathbf{H}}}_{41}^{o} } & \cdots & {\overline{{\mathbf{H}}}_{44}^{o} } \\ \end{array} } \right] $$

(A.10)

and

$$ {\mathbf{H}}^{\alpha } = \left[ {\begin{array}{*{20}c} {\overline{{\mathbf{H}}}_{11}^{a} } & \cdots & {\overline{{\mathbf{H}}}_{14}^{a} } \\ \vdots & \ddots & \vdots \\ {\overline{{\mathbf{H}}}_{41}^{a} } & \cdots & {\overline{{\mathbf{H}}}_{44}^{a} } \\ \end{array} } \right] $$

(A.11)

With the expression

$$ \begin{aligned} \Delta & = {\bar{\boldsymbol{\tau }}}_{n + 1}^{T} {\varvec{\Lambda}}_{n + 1} {\bar{\boldsymbol{\tau }}}_{0} - {\bar{\boldsymbol{\tau }}}_{n}^{T} {\varvec{\Lambda}}_{n} {\bar{\boldsymbol{\tau }}}_{0} \\ &\quad - \left( {{\mathbf{r}}_{n + 1} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right)^{T} {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \\ & \quad - \left( {{\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} \varLambda_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \otimes {\mathbf{t}}_{0} + {\bar{\boldsymbol{\tau }}}_{n + 1/2} \otimes {\bar{\boldsymbol{\tau }}}_{0} } \right):\\ &\quad \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right) \\ \end{aligned} $$

(A.12)

and

$$ {\varvec{\Omega}}_{\alpha } = \frac{{3{\varvec{\uptau}}_{\alpha }^{T} {\varvec{\Lambda}}{\bar{\boldsymbol{\tau }}}_{0} \left( {{\varvec{\uptau}}_{\alpha } \otimes {\varvec{\uptau}}_{\alpha } } \right) - \left\| {{\varvec{\uptau}}_{\alpha } } \right\|^{2} \left( {{\varvec{\Lambda}}{\bar{\boldsymbol{\tau }}}_{0} \otimes {\varvec{\uptau}}_{\alpha } + {\varvec{\uptau}}_{\alpha } \otimes {\varvec{\Lambda}}{\bar{\boldsymbol{\tau }}}_{0} + {\varvec{\uptau}}_{\alpha }^{T} {\varvec{\Lambda}}{\bar{\boldsymbol{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} } \right)}}{{\left\| {{\varvec{\uptau}}_{\alpha } } \right\|^{5} }} $$

(A.13)

the submatrices in (A.10) and (A.11) are presented as follows:

$$ {\bar{\mathbf{H}}}_{\alpha \alpha }^{o} = \lambda_{n + 1} {\varvec{\Phi}}_{n + 1/2} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} $$

(A.14)

$$ \begin{aligned} {\bar{\mathbf{H}}}_{\alpha 3}^{o} & = \frac{1}{2}\lambda_{n + 1} {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\hat{\mathbf{t}}}_{n + 1} \\ &\quad - \frac{1}{2}\lambda_{n + 1} {\mathbf{t}}_{n + 1/2} \otimes {\hat{\mathbf{t}}}_{n + 1} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \\ & \quad + \frac{1}{2}\lambda_{n + 1} {\varvec{\Phi}}_{n + 1/2} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} {\hat{\mathbf{t}}}_{n + 1} \\ &\quad - \frac{1}{2}\lambda_{n + 1} {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ \end{aligned} $$

$$ {\bar{\mathbf{H}}}_{\alpha 4}^{o} = {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} $$

$$ \begin{aligned} {\bar{\mathbf{H}}}_{3\alpha }^{o} & = - \frac{1}{2}\lambda_{n + 1} {\hat{\mathbf{t}}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \otimes {\mathbf{t}}_{n + 1/2} \\ &\quad - \frac{1}{2}\lambda_{n + 1} {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\hat{\mathbf{t}}}_{n + 1/2} \\ & \quad + \frac{1}{4}\lambda_{n + 1} \left( {{\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} + {\varvec{\Lambda}}_{n} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n}^{T} } \right){\varvec{\Psi}}_{1(n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ &\quad - \frac{1}{2}\lambda_{n + 1} {\hat{\mathbf{t}}}_{n + 1/2} {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ \end{aligned} $$

$$ \begin{aligned} {\bar{\mathbf{H}}}_{33}^{o} & = \frac{1}{4}\lambda_{n + 1} \left( {{\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} + {\varvec{\Lambda}}_{n} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n}^{T} } \right){\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} {\hat{\mathbf{t}}}_{n + 1} \\ &\quad + \frac{1}{2}\lambda_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{n + 1/2} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ & \quad - \frac{1}{2}\lambda_{n + 1} \left( {\mathbf{t}}_{n + 1} \otimes {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \right. \\ &\quad \left. - {\mathbf{t}}_{n + 1}^{T} {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} \right) \\ & \quad - \frac{1}{2}\lambda_{n + 1} {\hat{\mathbf{t}}}_{n + 1/2} \left( {\mathbf{r}},_{\alpha (n + 1/2)} \otimes {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \right. \\ &\quad \left. + {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \otimes {\mathbf{r}},_{\alpha (n + 1/2)} \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + \frac{1}{2}\lambda_{n + 1} {\hat{\mathbf{t}}}_{n + 1/2} {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} {\hat{\mathbf{t}}}_{n + 1} \\ &\quad - \frac{1}{2}\lambda_{n + 1} {\hat{\mathbf{t}}}_{n + 1/2} {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ \end{aligned} $$

$$ {\bar{\mathbf{H}}}_{34}^{o} = \left( {{\hat{\mathbf{t}}}_{n + 1/2} {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} - {\boldsymbol{\hat{\bar{\tau }}}}_{n + 1/2} } \right){\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} $$

$$ {\bar{\mathbf{H}}}_{4\alpha }^{o} = {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1}^{T} {\varvec{\Psi}}_{\alpha (n + 1)} {\varvec{\Phi}}_{n + 1} $$

$$ {\bar{\mathbf{H}}}_{43}^{o} = {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1}^{T} {\varvec{\Psi}}_{\alpha (n + 1)} {\varvec{\Upsilon}}_{\alpha (n + 1)}^{T} {\hat{\mathbf{t}}}_{n + 1} + {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1}^{T} {\boldsymbol{\hat{\bar{\tau }}}}_{n + 1} $$

$$ {\bar{\mathbf{H}}}_{\alpha \beta }^{o} = {\mathbf{0}}_{3 \times 3} ,\quad {\bar{\mathbf{H}}}_{44}^{o} = 0 $$

for case (ii) in Table 1

$$ \begin{aligned} {\mathbf{H}}_{\alpha \alpha }^{a} & = \frac{{\lambda_{n + 1} \text{sgn} (\left\| {{\mathbf{r}}_{n + 1} ,_{\alpha } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\alpha } } \right\|^{2} )}}{{\sum\limits_{\gamma = 1}^{2} {\left| {\left\| {{\mathbf{r}}_{n + 1} ,_{\gamma } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\gamma } } \right\|^{2} } \right|} }}\left[ \Delta {\mathbf{I}}_{3 \times 3} + {\mathbf{r}}_{n + 1/2} ,_{\alpha }\right. \\ &\quad \left. \otimes \left( {{\varvec{\Phi}}_{n + 1} {\varvec{\Psi}}_{\alpha (n + 1)} {\varvec{\Lambda}}_{n + 1} - {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} } \right){\bar{\boldsymbol{\tau }}}_{0} \right. \\ & \quad - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right){\varvec{\Phi}}_{n + 1/2} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ &\quad + {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ & \quad + {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\mathbf{t}}_{n + 1/2} \\ &\quad + {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ & \quad \left. { - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} {\varvec{\Psi}}_{1(n + 1/2)} {\varvec{\Phi}}_{n + 1/2} } \right] \\ &\quad - 4\lambda_{n + 1} \Delta {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\mathbf{r}}_{n + 1} ,_{\alpha } / \\ &\quad \left( {\sum\limits_{\gamma = 1}^{2} {\left| {\left\| {{\mathbf{r}}_{n + 1} ,_{\gamma } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\gamma } } \right\|^{2} } \right|} } \right)^{2} \\ \end{aligned} $$

$$ \begin{aligned} {\mathbf{H}}_{\alpha \beta }^{a} & = \frac{{\lambda_{n + 1} \text{sgn} (\left\| {{\mathbf{r}}_{n + 1} ,_{\alpha } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\alpha } } \right\|^{2} )}}{{\sum\limits_{\gamma = 1}^{2} {\left| {\left\| {{\mathbf{r}}_{n + 1} ,_{\gamma } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\gamma } } \right\|^{2} } \right|} }} \\ &\quad \left[ {{\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\varvec{\Phi}}_{n + 1} {\varvec{\Psi}}_{\beta (n + 1)} {\varvec{\Lambda}}_{n + 1} - {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} } \right){\bar{\boldsymbol{\tau }}}_{0} } \right. \\ & \quad - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right){\varvec{\Phi}}_{n + 1/2} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ &\quad + {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ & \quad + {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1/2}^{T} {\varvec{\Psi}}_{2(n + 1/2)} \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\mathbf{t}}_{n + 1/2} \\ &\quad {\mathbf{ + r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ & \quad \left. { - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} } \right] \\ & \quad + 4\lambda_{n + 1} \text{sgn} \left( {(\left\| {{\mathbf{r}}_{n + 1} ,_{\alpha } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\alpha } } \right\|^{2} )(\left\| {{\mathbf{r}}_{n + 1} ,_{\beta } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\beta } } \right\|^{2} )} \right)\\ &\quad Z_{n + 1/2} {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\mathbf{r}}_{n + 1} ,_{\beta } /\sum\limits_{\gamma = 1}^{2} {\left| {\left\| {{\mathbf{r}}_{n + 1} ,_{\gamma } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\gamma } } \right\|^{2} } \right|} \\ \end{aligned} $$

(A.15)

$$ \begin{aligned} {\mathbf{H}}_{\alpha 3}^{a} & = \frac{{\lambda_{n + 1} \text{sgn} (\left\| {{\mathbf{r}}_{n + 1} ,_{\alpha } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\alpha } } \right\|^{2} )}}{{\sum\limits_{\gamma = 1}^{2} {\left| {\left\| {{\mathbf{r}}_{n + 1} ,_{\gamma } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\gamma } } \right\|^{2} } \right|} }}\\ &\quad \left\{ {{\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes } \right.\left[ {{\hat{\mathbf{t}}}_{n + 1} \left( {{\varvec{\Upsilon}}_{\alpha (n + 1)} {\varvec{\Psi}}_{\alpha (n + 1)} + {\varvec{\Upsilon}}_{\beta (n + 1)} {\varvec{\Psi}}_{\beta (n + 1)} } \right) - {\boldsymbol{\hat{\bar{\tau }}}}_{n + 1} } \right]{\varvec{\Lambda}}_{n + 1} {\bar{\boldsymbol{\tau }}}_{0} \\ & \quad - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right) \\ & \quad \left( {{\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} + {\mathbf{t}}_{n + 1/2} \otimes {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} } \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right){\varvec{\Phi}}_{n + 1/2} \\ &\quad \left( {{\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\mathbf{ - \varOmega }}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\hat{\mathbf{t}}}_{n + 1} } \right) \\ & \quad + {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right)\\ &\quad \left( {{\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} + {\mathbf{t}}_{n + 1/2} \otimes {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} } \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right){\varvec{\Phi}}_{n + 1/2} \\ &\quad \left( {{\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\mathbf{ - \varOmega }}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\hat{\mathbf{t}}}_{n + 1} } \right) \\ & \quad - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ &\quad \left( {\mathbf{r}}_{n + 1/2} ,_{\beta } \otimes {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} + {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \right. \\ &\quad \left. \otimes {\mathbf{r}}_{n + 1/2} ,_{\beta } - {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)}^{T} \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ &\quad \left( {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \right. \\ &\quad \left. + {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \otimes {\mathbf{r}}_{n + 1/2} ,_{\alpha } - {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ &\quad \left( {{\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} + {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Psi}}_{\beta (n + 1/2)} } \right){\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ &\quad - 2{\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\bar{\boldsymbol{\tau }}}_{n + 1/2} \\ & \quad - {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} \\ &\quad \left( {{\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} + {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)}^{T} } \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + 2{\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\varvec{\Lambda}}_{n + 1} {\hat{\mathbf{t}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ & \quad \left. \left( {{\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Psi}}_{\beta (n + 1/2)} + {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} } \right)\varLambda_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \right\} \\ \end{aligned} $$

$$ {\mathbf{H}}_{\alpha 4}^{a} = \frac{{2\lambda_{n + 1} \text{sgn} (\left\| {{\mathbf{r}}_{n + 1} ,_{\alpha } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\alpha } } \right\|^{2} )\Delta {\mathbf{r}}_{n + 1/2} ,_{\alpha } }}{{\sum\limits_{\gamma = 1}^{2} {\left| {\left\| {{\mathbf{r}}_{n + 1} ,_{\gamma } } \right\|^{2} - \left\| {{\mathbf{r}}_{n} ,_{\gamma } } \right\|^{2} } \right|} }} $$

$$ \begin{aligned} {\mathbf{H}}_{3\alpha }^{a} & = {\mathbf{H}}_{33}^{a} = {\mathbf{0}}_{3 \times 3} ; \\ {\mathbf{H}}_{4\alpha }^{a} & = {\mathbf{H}}_{43}^{a} = {\mathbf{0}}_{1 \times 3} ; \\ {\mathbf{H}}_{34}^{a} & = {\mathbf{0}}_{3 \times 1} ; \\ {\mathbf{H}}_{44}^{a} & = 0; \\ \end{aligned} $$

for case (iii) in Table 1

$$ \begin{aligned} {\mathbf{H}}_{\beta \beta }^{a} & = \frac{{\lambda_{n + 1} }}{{2\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)}} \\ &\quad \left[ {2{\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\varvec{\Phi}}_{n + 1} {\varvec{\Psi}}_{\beta (n + 1)} {\varvec{\Lambda}}_{n + 1} - {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} } \right){\bar{\boldsymbol{\tau }}}_{0} } \right. \\ & \quad - {\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Phi}}_{n + 1/2} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \left( {{\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right) \\ &\quad + {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ & \quad + {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\beta (n + 1/2)} \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\mathbf{e}}_{j}^{n + 1/2} \otimes {\mathbf{t}}_{n + 1/2} \\ &\quad + {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ & \quad \left. { - {\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\beta (n + 1/2)} \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} } \right] \\ & \quad - \lambda_{n + 1} \Delta {\mathbf{e}}_{j}^{n + 1/2} \otimes {\mathbf{e}}_{i}^{n + 1} /\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)^{2} \\ \end{aligned} $$

$$ \begin{aligned} {\mathbf{H}}_{\beta \alpha }^{a} & = \frac{{\lambda_{n + 1} }}{{2\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)}} \\ &\quad \left[ {2{\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\varvec{\Phi}}_{n + 1} {\varvec{\Psi}}_{\alpha (n + 1)} {\varvec{\Lambda}}_{n + 1} - {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} } \right){\bar{\boldsymbol{\tau }}}_{0} } \right. \\ & \quad - {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right){\varvec{\Phi}}_{n + 1/2} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ &\quad + {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ & \quad + {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\mathbf{e}}_{j}^{n + 1/2} \otimes {\mathbf{t}}_{n + 1/2} \\ &\quad + {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ & \quad \left. { - {\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} } \right] \\ \end{aligned} $$

$$ \begin{aligned} {\mathbf{H}}_{\beta 3}^{a} & = \frac{{\lambda_{n + 1} }}{{2\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)}}\left\{ { - \Delta {\hat{\mathbf{e}}}_{j}^{n + 1} - 2{\mathbf{e}}_{j}^{n + 1/2} \otimes } \right. \\ &\quad \left[ {{\hat{\mathbf{t}}}_{n + 1} \left( {{\varvec{\Upsilon}}_{\alpha (n + 1)} {\varvec{\Psi}}_{\alpha (n + 1)} + {\varvec{\Upsilon}}_{\beta (n + 1)} {\varvec{\Psi}}_{\beta (n + 1)} } \right) - {\boldsymbol{\hat{\bar{\tau }}}}_{n + 1} } \right]{\varvec{\Lambda}}_{n + 1} {\bar{\boldsymbol{\tau }}}_{0} \\ & \quad - {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{(n + 1)} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right)\left( {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} \right. \\ &\quad \left. + {\mathbf{t}}_{n + 1/2} \otimes {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{(n + 1)} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right){\varvec{\Phi}}_{n + 1/2} \\ &\quad \left( {{\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\mathbf{ - \varOmega }}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)}^{T} {\hat{\mathbf{t}}}_{n + 1} } \right) \\ & \quad - {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{(n + 1)} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right) \\&\quad \left( {{\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} + {\mathbf{t}}_{n + 1/2} \otimes {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} } \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{(n + 1)} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right){\varvec{\Phi}}_{n + 1/2} \\ &\quad \left( {{\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\mathbf{ - \varOmega }}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} {\hat{\mathbf{t}}}_{n + 1} } \right) \\ & \quad - {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} \right) \left( {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \right. \\ &\quad \left. + {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \otimes {\mathbf{r}}_{n + 1/2} ,_{\alpha } - {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad - {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} \right)\left( {\mathbf{r}}_{(n + 1/2)} ,_{\beta } \otimes {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \right.\\ &\quad \left. + {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \otimes {\mathbf{r}}_{n + 1/2} ,_{\beta } - {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)}^{T} \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} \right)\left( {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} \right. \\ &\quad \left. + {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Psi}}_{\beta (n + 1/2)} \right){\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ & \quad - {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} \\ &\quad \left( {{\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)}^{T} - {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} } \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad - 2{\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Lambda}}_{n + 1} {\hat{\mathbf{t}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ &\quad \left( {{\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} + {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Psi}}_{\beta (n + 1/2)} } \right)\varLambda_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \\ & \quad \left. { - 2{\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\bar{\boldsymbol{\tau }}}_{n + 1/2} } \right\} - \lambda_{n + 1} \Delta {\mathbf{e}}_{j}^{n + 1/2} \\ &\quad \otimes \left( {{\mathbf{e}}_{j}^{n + 1} \times {\mathbf{r}}_{n + 1} ,_{\beta } } \right)/\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)^{2} \\ \end{aligned} $$

(A.16)

$$ \begin{aligned} {\mathbf{H}}_{3\beta }^{a} & = \frac{{\lambda_{n + 1} }}{{2\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)}}\\ &\quad \left[ {{\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\mathbf{ - \varPhi }}_{n + 1} {\varvec{\Psi}}_{\beta (n + 1)} {\varvec{\Lambda}}_{n + 1} } \right){\bar{\boldsymbol{\tau }}}_{0} } \right. \\ & \quad + \frac{\Delta }{2}\left( {{\varvec{\Lambda}}_{n + 1} {\hat{\mathbf{e}}}_{j} {\varvec{\Lambda}}_{n + 1}^{T} + {\varvec{\Lambda}}_{n} {\hat{\mathbf{e}}}_{j} {\varvec{\Lambda}}_{n}^{T} } \right) + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \\ &\quad \otimes {\varvec{\Phi}}_{n + 1/2} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \left( {{\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right) \\ & \quad - {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \\ &\quad \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) - {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1/2}^{T} {\varvec{\Psi}}_{1(n + 1/2)} \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\mathbf{t}}_{n + 1/2} \\ & \quad - {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ &\quad \left. { + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\beta (n + 1/2)} \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} } \right] \\ & \quad + \lambda_{n + 1} \Delta {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\mathbf{e}}_{j}^{n + 1} /\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)^{2} \\ \end{aligned} $$

$$ \begin{aligned} {\mathbf{H}}_{3\alpha }^{a} & = \frac{{\lambda_{n + 1} }}{{2\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)}} \\ &\quad \left[ {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\varvec{\Lambda}}_{n + 1/2} {\mathbf{e}}_{j} \otimes \left( {{\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} - {\varvec{\Phi}}_{n + 1} {\varvec{\Psi}}_{\alpha (n + 1)} {\varvec{\Lambda}}_{n + 1} } \right){\bar{\boldsymbol{\tau }}}_{0} \right. \\ & \quad + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Phi}}_{n + 1/2} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \left( {{\mathbf{r}}_{n + 1} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right) \\ &\quad - {\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ & \quad - {\bar{\boldsymbol{\tau }}}_{0}^{T} {\varvec{\Lambda}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\mathbf{t}}_{n + 1/2} \\ &\quad - {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right){\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Phi}}_{n + 1/2} \\ & \quad \left. {{\mathbf{ + \hat{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Phi}}_{n + 1/2} {\varvec{\Psi}}_{\alpha (n + 1/2)} \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} } \right] \\ \end{aligned} $$

$$ \begin{aligned} {\mathbf{H}}_{33}^{a} & = \frac{{\lambda_{n + 1} }}{{2\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)}} \\ &\quad \left\{ {\Delta {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\varvec{\Lambda}}_{n + 1} {\hat{\mathbf{e}}}_{j} {\varvec{\Lambda}}_{n}^{T} + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} } \right. \\ &\quad \left. {\otimes \left[ {{\hat{\mathbf{t}}}_{n + 1} \left( {{\varvec{\Upsilon}}_{\alpha (n + 1)} {\varvec{\Psi}}_{\alpha (n + 1)} {\mathbf{ + \varUpsilon }}_{\beta (n + 1)} {\varvec{\Upsilon}}_{\beta (n + 1)} } \right) - {\boldsymbol{\hat{\bar{\tau }}}}_{n + 1} } \right]{\varvec{\Lambda}}_{n + 1} {\bar{\boldsymbol{\tau }}}_{0} } \right. \\ & \quad + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right) \\ &\quad \left( {{\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} + {\mathbf{t}}_{n + 1/2} \otimes {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} } \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{r}}_{n} ,_{\beta } } \right){\varvec{\Phi}}_{n + 1/2} \\ &\quad \left( {{\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\mathbf{ - \varOmega }}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)}^{T} {\hat{\mathbf{t}}}_{n + 1} } \right) \\ & \quad + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{n + 1} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right) \\ &\quad \left( {{\mathbf{t}}_{n + 1/2}^{T} {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} + {\mathbf{t}}_{n + 1/2} \otimes {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} } \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{r}}_{(n + 1)} ,_{\alpha } - {\mathbf{r}}_{n} ,_{\alpha } } \right){\varvec{\Phi}}_{n + 1/2} \\ &\quad \left( {{\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\mathbf{ - \varOmega }}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} {\hat{\mathbf{t}}}_{n + 1} } \right) \\ & \quad + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ &\quad \left( {\mathbf{r}}_{n + 1/2} ,_{\alpha } \otimes {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} + {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \otimes {\mathbf{r}}_{n + 1/2} ,_{\alpha } \right. \\ &\quad \left. - {\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Omega}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ &\quad \left( {\mathbf{r}}_{n + 1/2} ,_{\beta } \otimes {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} + {\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Lambda}}_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \otimes {\mathbf{r}}_{n + 1/2} ,_{\beta } \right. \\ &\quad \left. - {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Omega}}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)}^{T} \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad + {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\mathbf{t}}_{n + 1} - {\mathbf{t}}_{n} } \right) \\ &\quad \left( {{\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} + {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Psi}}_{\beta (n + 1/2)} } \right){\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ & \quad - {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes \left( {{\varvec{\Lambda}}_{n + 1} - {\varvec{\Lambda}}_{n} } \right){\bar{\boldsymbol{\tau }}}_{0} \\ &\quad \left( {{\varvec{\Psi}}_{\beta (n + 1/2)} {\varvec{\Upsilon}}_{\beta (n + 1/2)}^{T} + {\varvec{\Psi}}_{\alpha (n + 1/2)} {\varvec{\Upsilon}}_{\alpha (n + 1/2)}^{T} } \right){\hat{\mathbf{t}}}_{n + 1} \\ & \quad - 2{\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Lambda}}_{n + 1} {\hat{\mathbf{t}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} \\ &\quad \left( {{\varvec{\Upsilon}}_{\alpha (n + 1/2)} {\varvec{\Psi}}_{\alpha (n + 1/2)} + {\varvec{\Upsilon}}_{\beta (n + 1/2)} {\varvec{\Psi}}_{\beta (n + 1/2)} } \right)\varLambda_{n + 1/2} {\bar{\boldsymbol{\tau }}}_{0} \\ & \quad \left. { + 2{\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\varvec{\Lambda}}_{n + 1} {\boldsymbol{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}_{n + 1}^{T} {\bar{\boldsymbol{\tau }}}_{n + 1/2} } \right\} \\ &\quad + \lambda_{n + 1} \Delta {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} \otimes {\hat{\mathbf{e}}}_{j}^{n + 1} {\mathbf{r}}_{n + 1} ,_{\beta } / \\ &\quad \left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)^{2} \\ \end{aligned} $$

$$ {\mathbf{H}}_{\beta 4}^{a} = \frac{{\Delta {\mathbf{e}}_{j}^{n + 1/2} }}{{\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)}} $$

$$ {\mathbf{H}}_{34}^{a} = - \frac{{\Delta {\hat{\mathbf{r}}}_{n + 1/2} ,_{\beta } {\mathbf{e}}_{j}^{n + 1/2} }}{{\left( {{\mathbf{e}}_{j}^{n + 1} \cdot {\mathbf{r}}_{n + 1} ,_{\beta } - {\mathbf{e}}_{j}^{n} \cdot {\mathbf{r}}_{n} ,_{\beta } } \right)}} $$

$$ {\mathbf{H}}_{21}^{a} = {\mathbf{H}}_{22}^{a} = {\mathbf{H}}_{23}^{a} = {\mathbf{0}}_{3 \times 3} ;\quad {\mathbf{H}}_{24}^{a} = {\mathbf{0}}_{3 \times 1} $$

$$ {\mathbf{H}}_{41}^{a} = {\mathbf{H}}_{42}^{a} = {\mathbf{H}}_{43}^{a} = {\mathbf{0}}_{1 \times 3} ;\quad {\mathbf{H}}_{44}^{a} = 0 $$