Three-Dimensional Finite-Element Analysis of Magnetotelluric Data Using Coulomb-Gauged Potentials in General Anisotropic Media

Abstract

The magnetotelluric (MT) method has been widely used in geophysical electromagnetic (EM) exploration. However, complex geometries and anisotropic structures still pose challenges for the simulations of 3D MT problems. This study presents a nodal finite-element (FE) solution to simulate 3D MT responses in 3D conductivity structures with general anisotropy. The method is based on the ${\mathbf{A}} - \psi$ decomposition of the electric field. The computational domain is discretized into hexahedral elements. The linear system equations that result from the FE discretization are solved by iterative solvers. We designed three examples to test the performance of the algorithm in this study. For the first example, we compare our results with those of other scholars to validate the effectiveness of the procedure. For the second example, the convergence behaviors of different iterative solvers with different preconditioners are tested. For the third example, a complex model is designed to demonstrate the robustness and effectiveness of the proposed code. Numerical experiments show that the convergence rate of the iterative solver of the ${\mathbf{A}} - \psi$ method is very fast, especially at low frequencies.

Appendix 1

In element e, Eqs. (25)–(28) can be written as

$$\begin{gathered} \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}xj}} \int_{e} {\left( { - \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} - \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial y} - \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + \sqrt { - 1} \omega \mu_{0} \sigma_{xx} N_{i} N_{j} } \right)} } \hfill \\ + \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}yj}} } \int_{e} {\left( {\sqrt { - 1} \omega \mu_{0} \sigma_{xy} N_{i} N_{j} } \right)} + \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}zj}} } \int_{e} {\left( {\sqrt { - 1} \omega \mu_{0} \sigma_{xz} N_{i} N_{j} } \right)} \hfill \\ + \sum\limits_{j = 1}^{8} {\psi_{j} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{xx} N_{i} \frac{{\partial N_{j} }}{\partial x} + \sigma_{xy} N_{i} \frac{{\partial N_{j} }}{\partial y} + \sigma_{xz} N_{i} \frac{{\partial N_{j} }}{\partial z}} \right)} \hfill \\ = - \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}xj}} } \int_{e} {\left( {N_{i} \delta_{xx} N_{j} } \right) - } \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}yj}} } \int_{e} {\left( {N_{i} \delta_{xy} N_{j} } \right)} - \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}zj}} } \int_{e} {\left( {N_{i} \delta_{xz} N_{j} } \right)} , \hfill \\ \end{gathered}$$

(42)

$$\begin{gathered} \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}yj}} \int_{e} {\left( { - \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} - \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial y} - \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + \sqrt { - 1} \omega \mu_{0} \sigma_{yy} N_{i} N_{j} } \right)} } \hfill \\ + \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}xj}} } \int_{e} {\left( {\sqrt { - 1} \omega \mu_{0} \sigma_{yx} N_{i} N_{j} } \right)} + \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}zj}} } \int_{e} {\left( {\sqrt { - 1} \omega \mu_{0} \sigma_{yz} N_{i} N_{j} } \right)} \hfill \\ + \sum\limits_{j = 1}^{8} {\psi_{j} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{yx} N_{i} \frac{{\partial N_{j} }}{\partial x} + \sigma_{yy} N_{i} \frac{{\partial N_{j} }}{\partial y} + \sigma_{yz} N_{i} \frac{{\partial N_{j} }}{\partial z}} \right)} \hfill \\ = - \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}xj}} } \int_{e} {\left( {N_{i} \delta_{yx} N_{j} } \right) - } \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}yj}} } \int_{e} {\left( {N_{i} \delta_{yy} N_{j} } \right)} - \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}zj}} } \int_{e} {\left( {N_{i} \delta_{yz} N_{j} } \right)} \hfill \\ \end{gathered}$$

(43)

$$\begin{gathered} \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}zj}} \int_{e} {\left( { - \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} - \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial y} - \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + \sqrt { - 1} \omega \mu_{0} \sigma_{zz} N_{i} N_{j} } \right)} } \hfill \\ + \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}xj}} } \int_{e} {\left( {\sqrt { - 1} \omega \mu_{0} \sigma_{zx} N_{i} N_{j} } \right)} + \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}yj}} } \int_{e} {\left( {\sqrt { - 1} \omega \mu_{0} \sigma_{zy} N_{i} N_{j} } \right)} \hfill \\ + \sum\limits_{j = 1}^{8} {\psi_{j} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{zx} N_{i} \frac{{\partial N_{j} }}{\partial x} + \sigma_{zy} N_{i} \frac{{\partial N_{j} }}{\partial y} + \sigma_{zz} N_{i} \frac{{\partial N_{j} }}{\partial z}} \right)} \hfill \\ = - \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}xj}} } \int_{e} {\left( {N_{i} \delta_{zx} N_{j} } \right) - } \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}yj}} } \int_{e} {\left( {N_{i} \delta_{zy} N_{j} } \right)} - \mu_{0} \sum\limits_{j = 1}^{8} {{\text{E}}_{{{\text{p}}zj}} } \int_{e} {\left( {N_{i} \delta_{zz} N_{j} } \right)} , \hfill \\ \end{gathered}$$

(44)

$$\begin{gathered} \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}xj}} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{xx} \frac{{\partial N_{i} }}{\partial x}N_{j} + \sigma_{yx} \frac{{\partial N_{i} }}{\partial y}N_{j} + \sigma_{zx} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right)} \hfill \\ + \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}yj}} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{xy} \frac{{\partial N_{i} }}{\partial x}N_{j} + \sigma_{yy} \frac{{\partial N_{i} }}{\partial y}N_{j} + \sigma_{zy} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right)} \hfill \\ + \sum\limits_{j = 1}^{8} {{\text{A}}_{{{\text{s}}zj}} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{xz} \frac{{\partial N_{i} }}{\partial x}N_{j} + \sigma_{yz} \frac{{\partial N_{i} }}{\partial y}N_{j} + \sigma_{zz} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right)} \hfill \\ + \sum\limits_{j = 1}^{8} {\psi_{j} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{xx} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + \sigma_{xy} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial y} + \sigma_{xz} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial z}} \right)} \hfill \\ + \sum\limits_{j = 1}^{8} {\psi_{j} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{yx} \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial x} + \sigma_{yy} \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial y} + \sigma_{yz} \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial z}} \right)} \hfill \\ + \sum\limits_{j = 1}^{8} {\psi_{j} } \int_{e} {\sqrt { - 1} \omega \mu_{0} \left( {\sigma_{zx} \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial x} + \sigma_{zy} \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial y} + \sigma_{zz} \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z}} \right)} \hfill \\ = - \mu_{0} \sum\limits_{j = 1}^{e} {{\text{E}}_{{{\text{p}}xj}} } \int_{e} {\left( {\delta_{xx} \frac{{\partial N_{i} }}{\partial x}N_{j} + \delta_{yx} \frac{{\partial N_{i} }}{\partial y}N_{j} + \delta_{zx} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right)} \hfill \\ - \mu_{0} \sum\limits_{j = 1}^{e} {{\text{E}}_{{{\text{p}}yj}} } \int_{e} {\left( {\delta_{xy} \frac{{\partial N_{i} }}{\partial x}N_{j} + \delta_{yy} \frac{{\partial N_{i} }}{\partial y}N_{j} + \delta_{zy} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right)} \hfill \\ - \mu_{0} \sum\limits_{j = 1}^{e} {{\text{E}}_{{{\text{p}}zj}} } \int_{e} {\left( {\delta_{xz} \frac{{\partial N_{i} }}{\partial x}N_{j} + \delta_{yz} \frac{{\partial N_{i} }}{\partial y}N_{j} + \delta_{zz} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right)} , \hfill \\ \end{gathered}$$

(45)

Equations (42)–(46) can be written into matrix form

$${\mathbf{k}}^{e} {\mathbf{x}} = {\mathbf{b}}^{e} ,$$

(46)

In this expression, ${\mathbf{k}}^{e}$ is

$${\mathbf{k}}^{e} = \left( {\begin{array}{*{20}c} {{\mathbf{k}}^{11} } & {{\mathbf{k}}^{12} } & {{\mathbf{k}}^{13} } & {{\mathbf{k}}^{14} } \\ {{\mathbf{k}}^{21} } & {{\mathbf{k}}^{22} } & {{\mathbf{k}}^{23} } & {{\mathbf{k}}^{24} } \\ {{\mathbf{k}}^{31} } & {{\mathbf{k}}^{32} } & {{\mathbf{k}}^{33} } & {{\mathbf{k}}^{34} } \\ {{\mathbf{k}}^{41} } & {{\mathbf{k}}^{42} } & {{\mathbf{k}}^{43} } & {{\mathbf{k}}^{44} } \\ \end{array} } \right),$$

(47)

where component ${\mathbf{k}}^{ij} \in {\mathbb{C}}^{8 \times 8}$ and

$$\left( {{\mathbf{k}}^{11} } \right)_{ij} = \int_{e} {\left( { - \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} - \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial y} - \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + i\omega \mu_{0} \sigma_{xx} N_{i} N_{j} } \right)} {\text{d}}v,$$

(48)

$$\left( {{\mathbf{k}}^{12} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \sigma_{xy} \int_{e} {N_{i} N_{j} } {\text{d}}v,$$

(49)

$$\left( {{\mathbf{k}}^{13} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \sigma_{xz} \int_{e} {N_{i} N_{j} } {\text{d}}v,$$

(50)

$$\left( {{\mathbf{k}}^{14} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \int_{e} {\left( {\sigma_{xx} N_{i} \frac{{\partial N_{j} }}{\partial x} + \sigma_{xy} N_{i} \frac{{\partial N_{j} }}{\partial y} + \sigma_{xz} N_{i} \frac{{\partial N_{j} }}{\partial z}} \right)} {\text{d}}v,$$

(51)

$$\left( {{\mathbf{k}}^{21} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \sigma_{yx} \int_{e} {N_{i} N_{j} } {\text{d}}v,$$

(52)

$$\left( {{\mathbf{k}}^{22} } \right)_{ij} = \int_{e} {\left( { - \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} - \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial y} - \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + \sqrt { - 1} \omega \mu_{0} \sigma_{yy} N_{i} N_{j} } \right)} {\text{d}}v,$$

(53)

$$\left( {{\mathbf{k}}^{23} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \sigma_{yz} \int_{e} {N_{i} N_{j} } {\text{d}}v,$$

(54)

$$\left( {{\mathbf{k}}^{24} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \int_{e} {\left( {\sigma_{yx} N_{i} \frac{{\partial N_{j} }}{\partial x} + \sigma_{yy} N_{i} \frac{{\partial N_{j} }}{\partial y} + \sigma_{yz} N_{i} \frac{{\partial N_{j} }}{\partial z}} \right)} {\text{d}}v,$$

(55)

$$\left( {{\mathbf{k}}^{31} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \sigma_{zx} \int_{e} {N_{i} N_{j} } {\text{d}}v,$$

(56)

$$\left( {{\mathbf{k}}^{32} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \sigma_{zy} \int_{e} {N_{i} N_{j} } {\text{d}}v,$$

(57)

$$\left( {{\mathbf{k}}^{33} } \right)_{ij} = \int_{e} {\left( { - \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} - \frac{{\partial N_{i} }}{\partial y}\frac{{\partial N_{j} }}{\partial y} - \frac{{\partial N_{i} }}{\partial z}\frac{{\partial N_{j} }}{\partial z} + \sqrt { - 1} \omega \mu_{0} \sigma_{zz} N_{i} N_{j} } \right)} {\text{d}}v,$$

(58)

$$\left( {{\mathbf{k}}^{34} } \right)_{ij} = \int_{e} {\left( {\sigma_{zx} N_{i} \frac{{\partial N_{j} }}{\partial x} + \sigma_{zy} N_{i} \frac{{\partial N_{j} }}{\partial y} + \sigma_{zz} N_{i} \frac{{\partial N_{j} }}{\partial z}} \right)} {\text{d}}v,$$

(59)

$$\left( {{\mathbf{k}}^{41} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \int_{e} {\left( {\sigma_{xx} \frac{{\partial N_{i} }}{\partial x}N_{j} + \sigma_{yx} \frac{{\partial N_{i} }}{\partial y}N_{j} + \sigma_{zx} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right)} {\text{d}}v,$$

(60)

$$\left( {{\mathbf{k}}^{42} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \int_{e} {\left( {\sigma_{xx} \frac{{\partial N_{i} }}{\partial x}N_{j} + \sigma_{yx} \frac{{\partial N_{i} }}{\partial y}N_{j} + \sigma_{zx} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right){\text{d}}v} ,$$

(61)

$$\left( {{\mathbf{k}}^{43} } \right)_{ij} = \sqrt { - 1} \omega \mu_{0} \int_{e} {\left( {\sigma_{xz} \frac{{\partial N_{i} }}{\partial x}N_{j} + \sigma_{yz} \frac{{\partial N_{i} }}{\partial y}N_{j} + \sigma_{zz} \frac{{\partial N_{i} }}{\partial z}N_{j} } \right){\text{d}}v} ,$$

(62)

$$\begin{aligned} \left( {{\mathbf{k}}^{44} } \right)_{ij} = & \sqrt { - 1} \omega \mu_{0} \int_{e} {\left( {\sigma_{xx} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + \sigma_{xy} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial y} + \sigma_{xz} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial z}} \right)} {\text{d}}v \\ & \quad + \sqrt { - 1} \omega \mu_{0} \int_{e} {\left( {\sigma_{yx} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + \sigma_{yy} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial y} + \sigma_{yz} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial z}} \right)} {\text{d}}v \\ & \quad + \sqrt { - 1} \omega \mu_{0} \int_{e} {\left( {\sigma_{zx} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial x} + \sigma_{zy} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial y} + \sigma_{zz} \frac{{\partial N_{i} }}{\partial x}\frac{{\partial N_{j} }}{\partial z}} \right){\text{d}}v,} \\ \end{aligned}$$

(63)

and

$${\mathbf{b}}^{e} = {\mathbf{sp}},$$

(64)

where

$${\mathbf{s}} = \left( {\begin{array}{*{20}c} {{\mathbf{s}}^{11} } & {{\mathbf{s}}^{12} } & {{\mathbf{s}}^{13} } \\ {{\mathbf{s}}^{21} } & {{\mathbf{s}}^{22} } & {{\mathbf{s}}^{23} } \\ {{\mathbf{s}}^{31} } & {{\mathbf{s}}^{32} } & {{\mathbf{s}}^{33} } \\ {{\mathbf{s}}^{41} } & {{\mathbf{s}}^{42} } & {{\mathbf{s}}^{43} } \\ \end{array} } \right),\;{\mathbf{p}} = \left( {\begin{array}{*{20}c} {{\mathbf{p}}^{1} } \\ {{\mathbf{p}}^{2} } \\ {{\mathbf{p}}^{3} } \\ \end{array} } \right).$$

(65)

In this expression, an arbitrary component of matrix S is an 8 × 8 complex matrix, and

$${\mathbf{p}}^{1} = \left( {E_{x1} ,E_{x2} , \ldots ,E_{x8} } \right)^{{\text{T}}} ,{\mathbf{p}}^{2} = \left( {E_{y1} ,E_{y2} , \ldots ,E_{y8} } \right)^{{\text{T}}} ,{\mathbf{p}}^{3} = \left( {E_{z1} ,E_{z2} , \ldots ,E_{z8} } \right)^{{\text{T}}} ,$$

(66)

where $E_{xi}$, $E_{yi}$, $E_{zi}$ are the primary electric fields in x-, y-, and z-directions of node i. We have

$${\mathbf{s}}^{11} = - \mu \int_{e} {\delta_{xx} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(67)

$${\mathbf{s}}^{12} = - \mu \int_{e} {\delta_{xy} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(68)

$${\mathbf{s}}^{13} = - \mu \int_{e} {\delta_{xz} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(69)

$${\mathbf{s}}^{21} = - \mu \int_{e} {\delta_{yx} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(70)

$${\mathbf{s}}^{22} = - \mu \int_{e} {\delta_{yy} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(71)

$${\mathbf{s}}^{23} = - \mu \int_{e} {\delta_{yz} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(72)

$${\mathbf{s}}^{31} = - \mu \int_{e} {\delta_{zx} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(73)

$${\mathbf{s}}^{32} = - \mu \int_{e} {\delta_{zy} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(74)

$${\mathbf{s}}^{33} = - \mu \int_{e} {\delta_{zz} {\mathbf{NN}}^{{\text{T}}} {\text{d}}v,}$$

(75)

$${\mathbf{s}}^{41} = - \mu \int_{e} {\left( {\delta_{xx} \frac{{\partial {\mathbf{N}}}}{\partial x}{\mathbf{N}}^{{\text{T}}} + \delta_{yx} \frac{{\partial {\mathbf{N}}}}{\partial y}{\mathbf{N}}^{{\text{T}}} + \delta_{zx} \frac{{\partial {\mathbf{N}}}}{\partial z}{\mathbf{N}}^{{\text{T}}} } \right)} {\text{d}}v,$$

(76)

$${\mathbf{s}}^{42} = - \mu \int_{e} {\left( {\delta_{xy} \frac{{\partial {\mathbf{N}}}}{\partial x}{\mathbf{N}}^{{\text{T}}} + \delta_{yy} \frac{{\partial {\mathbf{N}}}}{\partial y}{\mathbf{N}}^{{\text{T}}} + \delta_{zy} \frac{{\partial {\mathbf{N}}}}{\partial z}{\mathbf{N}}^{{\text{T}}} } \right)} {\text{d}}v,$$

(77)

$${\mathbf{s}}^{43} = - \mu \int_{e} {\left( {\delta_{xz} \frac{{\partial {\mathbf{N}}}}{\partial x}{\mathbf{N}}^{{\text{T}}} + \delta_{yz} \frac{{\partial {\mathbf{N}}}}{\partial y}{\mathbf{N}}^{{\text{T}}} + \delta_{zz} \frac{{\partial {\mathbf{N}}}}{\partial z}{\mathbf{N}}^{{\text{T}}} } \right)} {\text{d}}v.$$

(78)

In Eqs. (67)–(78),${\mathbf{N}} = \left( {N_{1} ,N_{2} , \ldots ,N_{8} } \right)$.