Isogeometric analysis of first and second strain gradient elasticity

Abstract

The elastic energy in the conventional Cauchy continuum model depends on the gradient of the displacement field, which is not adequate to show the behavior of a system under point and line forces. Applying these kinds of boundary conditions to the continuum will lead to singularities. To overcome this problem, a generalization of the Cauchy continuum concept is a choice. In this paper, isogeometric analysis is used to simulate the behavior of second- and third-gradient materials. As expected, a significant improvement in the efficiency and accuracy of the attained results compared to the conventional finite element method is noticed.

Appendix: Global derivatives of shape functions

Here we will show the process to transform the local derivatives of the shape functions (derivatives with respect to the parametric coordinates) to their global (spatial) counterparts. We will use the same approach employed by R.L.Taylor’s team in nurbFEAP [52] for the implementation of two-dimensional global second-derivatives and will expand it to three-dimensions and global third-derivatives.

The calculations here are independent of the shape function nature and demonstrate merely the mapping from local to global coordinates. For the calculations related to deriving the NURBS shape functions, we refer the reader to Cottrell et al. [17]. We will do the calculations for the most general case, i.e. $\frac{\partial ^3 N_a}{\partial \xi \partial \eta \partial \zeta }$, all the other third derivatives are in fact parts of this one (e.g. replacing $\eta$ with $\xi$ results in $\frac{\partial ^3 N_a}{\partial \xi ^2 \partial \zeta }$).

First, we need to find the relation between the local and global first derivatives (i.e. $\frac{\partial N_a}{\partial \xi }$ in terms of $\frac{\partial N_a}{\partial x}$, $\frac{\partial N_a}{\partial y}$ and $\frac{\partial N_a}{\partial z}$). Using the chain rule,

$$\begin{aligned}&\frac{\partial N_a}{\partial \xi } = \frac{\partial x}{\partial \xi } \cdot \frac{\partial N_a}{\partial x} + \frac{\partial y}{\partial \xi } \cdot \frac{\partial N_a}{\partial y} + \frac{\partial z}{\partial \xi } \cdot \frac{\partial N_a}{\partial z} ,&\end{aligned}$$

(15)

where ($\frac{\partial x}{\partial \xi }, \cdots$) are components of the Jacobian matrix. Deriving the relation for all the parametric directions ($\xi$, $\eta$, and $\zeta$), solving the following system of equations gives us the global first derivatives:

$$\begin{aligned}&\begin{Bmatrix} \frac{\partial N_a}{\partial \xi }\\ \frac{\partial N_a}{\partial \eta }\\ \frac{\partial N_a}{\partial \zeta } \end{Bmatrix} = \begin{bmatrix} \frac{\partial x }{\partial \xi }&\frac{\partial y }{\partial \xi }&\frac{\partial z }{\partial \xi } \\ \frac{\partial x }{\partial \eta }&\frac{\partial y }{\partial \eta }&\frac{\partial z }{\partial \eta } \\ \frac{\partial x }{\partial \zeta }&\frac{\partial y }{\partial \zeta }&\frac{\partial z }{\partial \zeta } \end{bmatrix} \cdot \begin{Bmatrix} \frac{\partial N_a}{\partial x}\\ \frac{\partial N_a}{\partial y}\\ \frac{\partial N_a}{\partial z} \end{Bmatrix} \end{aligned}$$

(16)

$$\begin{aligned}&\begin{Bmatrix} \frac{\partial N_a}{\partial x}\\ \frac{\partial N_a}{\partial y}\\ \frac{\partial N_a}{\partial z} \end{Bmatrix} = \begin{bmatrix} \frac{\partial x }{\partial \xi }&\frac{\partial y }{\partial \xi }&\frac{\partial z }{\partial \xi } \\ \frac{\partial x }{\partial \eta }&\frac{\partial y }{\partial \eta }&\frac{\partial z }{\partial \eta } \\ \frac{\partial x }{\partial \zeta }&\frac{\partial y }{\partial \zeta }&\frac{\partial z }{\partial \zeta } \end{bmatrix}^{-1} \cdot \begin{Bmatrix} \frac{\partial N_a}{\partial \xi }\\ \frac{\partial N_a}{\partial \eta }\\ \frac{\partial N_a}{\partial \zeta } \end{Bmatrix}.&\end{aligned}$$

(17)

The next step is to find the relation between second order derivatives.

$$\begin{aligned} \frac{\partial ^2 N_a}{\partial \xi \partial \eta }&= \frac{\partial }{\partial \eta } \bigg ( \frac{\partial N_a}{\partial \xi } \bigg )&\nonumber \\&= \frac{\partial }{\partial \eta } \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial N_a}{\partial x} + \frac{\partial y}{\partial \xi } \cdot \frac{\partial N_a}{\partial y} + \frac{\partial z}{\partial \xi } \cdot \frac{\partial N_a}{\partial z} \bigg )&\end{aligned}$$

(18)

Applying the product rule, Eq. (18) becomes

$$\begin{aligned}&\frac{\partial ^2 N_a}{\partial \xi \partial \eta } = \frac{\partial ^2 x}{\partial \xi \partial \eta } \cdot \frac{\partial N_a}{\partial x} + \frac{\partial x}{\partial \xi } \cdot \frac{\partial }{\partial \eta } \bigg ( \frac{\partial N_a}{\partial x} \bigg )&\nonumber \\&\quad \qquad \qquad +\,\frac{\partial ^2 y}{\partial \xi \partial \eta } \cdot \frac{\partial N_a}{\partial y} + \frac{\partial y}{\partial \xi } \cdot \frac{\partial }{\partial \eta } \bigg ( \frac{\partial N_a}{\partial y} \bigg )&\nonumber \\&\qquad \quad \qquad +\,\frac{\partial ^2 z}{\partial \xi \partial \eta } \cdot \frac{\partial N_a}{\partial z} + \frac{\partial z}{\partial \xi } \cdot \frac{\partial }{\partial \eta } \bigg ( \frac{\partial N_a}{\partial z} \bigg ) .&\end{aligned}$$

(19)

In Eq. (19), there are four new kinds of components. First, $\frac{\partial ^2 N_a}{\partial \xi \partial \eta }$ which is the local derivative and is known. The second type includes the elements like $\frac{\partial ^2 x}{\partial \xi \partial \eta }$ which can be calculated using the same procedure as the Jacobian matrix (only this time with second order derivatives). Components like $\frac{\partial N_a}{\partial x}$ represent the global first derivatives. Finally there are $\frac{\partial }{\partial \eta } \bigg (\frac{\partial N_a}{\partial x}\bigg )$ kind of components, which need special treatment. Using the chain rule yields

$$\begin{aligned}&\frac{\partial }{\partial \eta } \bigg (\frac{\partial N_a}{\partial x}\bigg )= \frac{\partial x}{\partial \eta } \cdot \frac{\partial ^2 N_a}{\partial x^2} + \frac{\partial y}{\partial \eta } \cdot \frac{\partial ^2 N_a}{\partial y^2} + \frac{\partial z}{\partial \eta } \cdot \frac{\partial ^2 N_a}{\partial z^2} .&\end{aligned}$$

(20)

Doing the same procedure for all the similar elements and replacing them in Eq. (19), after re-ordering, Eq. (19) becomes

$$\begin{aligned}&\frac{\partial ^2 N_a}{\partial \xi \partial \eta } = \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial x}{\partial \eta } \bigg ) \frac{\partial ^2 N_a}{\partial x^2} + \bigg ( \frac{\partial y}{\partial \xi } \cdot \frac{\partial y}{\partial \eta } \bigg ) \frac{\partial ^2 N_a}{\partial y^2}&\nonumber \\&\quad + \bigg ( \frac{\partial z}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } \bigg ) \frac{\partial ^2 N_a}{\partial z^2} + \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial y}{\partial \eta } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial y}{\partial \xi } \bigg ) \frac{\partial ^2 N_a}{\partial x \partial y}&\nonumber \\&\quad + \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial z}{\partial \xi } \bigg ) \frac{\partial ^2 N_a}{\partial x \partial z}&\nonumber \\&\quad + \bigg ( \frac{\partial y}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial y}{\partial \eta } \cdot \frac{\partial z}{\partial \xi } \bigg ) \frac{\partial ^2 N_a}{\partial y \partial z}&\nonumber \\&\quad + \bigg ( \frac{\partial ^2 x}{\partial \xi \partial \eta } \bigg ) \frac{\partial N_a}{\partial x } + \bigg ( \frac{\partial ^2 y}{\partial \xi \partial \eta } \bigg ) \frac{\partial N_a}{\partial y } + \bigg ( \frac{\partial ^2 z}{\partial \xi \partial \eta } \bigg ) \frac{\partial N_a}{\partial z }.&\end{aligned}$$

(21)

The same applies for other second-derivatives. Now, to calculate the unknowns, the following system of equations must be solved (see Eq. (22)).

$$\begin{aligned}&\underbrace{ \begin{Bmatrix} \frac{\partial ^2 N_a}{\partial \xi ^2} \\ \frac{\partial ^2 N_a}{\partial \eta ^2}\\ \frac{\partial ^2 N_a}{\partial \zeta ^2} \\ \frac{\partial ^2 N_a}{\partial \xi \partial \eta }\\ \frac{\partial ^2 N_a}{\partial \xi \partial \zeta }\\ \frac{\partial ^2 N_a}{\partial \eta \partial \zeta } \end{Bmatrix} }_{\text {known}} = \underbrace{ \begin{bmatrix} \left( \frac{\partial x}{\partial \xi } \right) ^2&\left( \frac{\partial y}{\partial \xi } \right) ^2&\left( \frac{\partial z}{\partial \xi } \right) ^2&2 \ \frac{\partial x}{\partial \xi } \ \frac{\partial y}{\partial \xi }&2 \ \frac{\partial x}{\partial \xi } \ \frac{\partial z}{\partial \xi }&2 \ \frac{\partial y}{\partial \xi } \ \frac{\partial z}{\partial \xi } \\ \left( \frac{\partial x}{\partial \eta } \right) ^2&\left( \frac{\partial y}{\partial \eta } \right) ^2&\left( \frac{\partial z}{\partial \eta } \right) ^2&2 \ \frac{\partial x}{\partial \eta } \ \frac{\partial y}{\partial \eta }&2 \ \frac{\partial x}{\partial \eta } \ \frac{\partial z}{\partial \eta }&2 \ \frac{\partial y}{\partial \eta } \ \frac{\partial z}{\partial \eta } \\ \left( \frac{\partial x}{\partial \zeta } \right) ^2&\left( \frac{\partial y}{\partial \zeta } \right) ^2&\left( \frac{\partial z}{\partial \zeta } \right) ^2&2 \ \frac{\partial x}{\partial \zeta } \ \frac{\partial y}{\partial \zeta }&2 \ \frac{\partial x}{\partial \zeta } \ \frac{\partial z}{\partial \zeta }&2 \ \frac{\partial y}{\partial \zeta } \ \frac{\partial z}{\partial \zeta } \\ \frac{\partial x}{\partial \xi } \frac{\partial x}{\partial \eta }&\frac{\partial y}{\partial \xi } \frac{\partial y}{\partial \eta }&\frac{\partial z}{\partial \xi } \frac{\partial z}{\partial \eta }&\left( \frac{\partial x}{\partial \xi } \frac{\partial y}{\partial \eta } + \frac{\partial x}{\partial \eta } \frac{\partial y}{\partial \xi } \right)&\left( \frac{\partial x}{\partial \xi } \frac{\partial z}{\partial \eta } + \frac{\partial x}{\partial \eta } \frac{\partial z}{\partial \xi } \right)&\left( \frac{\partial y}{\partial \xi } \frac{\partial z}{\partial \eta } + \frac{\partial y}{\partial \eta } \frac{\partial z}{\partial \xi } \right) \\ \frac{\partial x}{\partial \xi } \frac{\partial x}{\partial \zeta }&\frac{\partial y}{\partial \xi } \frac{\partial y}{\partial \zeta }&\frac{\partial z}{\partial \xi } \frac{\partial z}{\partial \zeta }&\left( \frac{\partial x}{\partial \xi } \frac{\partial y}{\partial \zeta } + \frac{\partial x}{\partial \zeta } \frac{\partial y}{\partial \xi } \right)&\left( \frac{\partial x}{\partial \xi } \frac{\partial z}{\partial \zeta } + \frac{\partial x}{\partial \zeta } \frac{\partial z}{\partial \xi } \right)&\left( \frac{\partial y}{\partial \xi } \frac{\partial z}{\partial \zeta } + \frac{\partial y}{\partial \zeta } \frac{\partial z}{\partial \xi } \right) \\ \frac{\partial x}{\partial \eta } \frac{\partial x}{\partial \zeta }&\frac{\partial y}{\partial \eta } \frac{\partial y}{\partial \zeta }&\frac{\partial z}{\partial \eta } \frac{\partial z}{\partial \zeta }&\left( \frac{\partial x}{\partial \eta } \frac{\partial y}{\partial \zeta } + \frac{\partial x}{\partial \zeta } \frac{\partial y}{\partial \eta } \right)&\left( \frac{\partial x}{\partial \eta } \frac{\partial z}{\partial \zeta } + \frac{\partial x}{\partial \zeta } \frac{\partial z}{\partial \eta } \right)&\left( \frac{\partial y}{\partial \eta } \frac{\partial z}{\partial \zeta } + \frac{\partial y}{\partial \zeta } \frac{\partial z}{\partial \eta } \right) \end{bmatrix} }_{\text {known}} \cdot \underbrace{ \begin{Bmatrix} \frac{\partial ^2 N_a}{\partial x^2} \\ \frac{\partial ^2 N_a}{\partial y^2} \\ \frac{\partial ^2 N_a}{\partial z^2} \\ \frac{\partial ^2 N_a}{\partial x \partial y}\\ \frac{\partial ^2 N_a}{\partial x \partial z}\\ \frac{\partial ^2 N_a}{\partial y \partial z} \end{Bmatrix} }_{\text {unknown}}&\nonumber \\&\qquad \qquad \qquad + \underbrace{ \begin{bmatrix} \frac{\partial ^2 x}{\partial \xi ^2}&\frac{\partial ^2 y}{\partial \xi ^2}&\frac{\partial ^2 z}{\partial \xi ^2} \\ \frac{\partial ^2 x}{\partial \eta ^2}&\frac{\partial ^2 y}{\partial \eta ^2}&\frac{\partial ^2 z}{\partial \eta ^2} \\ \frac{\partial ^2 x}{\partial \zeta ^2}&\frac{\partial ^2 y}{\partial \zeta ^2}&\frac{\partial ^2 z}{\partial \zeta ^2} \\ \frac{\partial ^2 x}{\partial \xi \partial \eta }&\frac{\partial ^2 y}{\partial \xi \partial \eta }&\frac{\partial ^2 z}{\partial \xi \partial \eta } \\ \frac{\partial ^2 x}{\partial \xi \partial \zeta }&\frac{\partial ^2 y}{\partial \xi \partial \zeta }&\frac{\partial ^2 z}{\partial \xi \partial \zeta } \\ \frac{\partial ^2 x}{\partial \eta \partial \zeta }&\frac{\partial ^2 y}{\partial \eta \partial \zeta }&\frac{\partial ^2 z}{\partial \eta \partial \zeta } \end{bmatrix} }_{\text {known}} \cdot \underbrace{ \begin{Bmatrix} \frac{\partial N_a}{\partial x} \\ \frac{\partial N_a}{\partial y} \\ \frac{\partial N_a}{\partial z} \end{Bmatrix} }_{\text {known from Eq. (17)}}&\end{aligned}$$

(22)

Moving the second part of Eq. (22) RHS to left,

$$\begin{aligned}&\begin{Bmatrix} \frac{\partial ^2 \bar{N}_a}{\partial \xi ^2} \\ \frac{\partial ^2 \bar{N}_a}{\partial \eta ^2} \\ \frac{\partial ^2 \bar{N}_a}{\partial \zeta ^2} \\ \frac{\partial ^2 \bar{N}_a}{\partial \xi \partial \eta }\\ \frac{\partial ^2 \bar{N}_a}{\partial \xi \partial \zeta }\\ \frac{\partial ^2 \bar{N}_a}{\partial \eta \partial \zeta } \end{Bmatrix} = \begin{Bmatrix} \frac{\partial ^2 N_a}{\partial \xi ^2} \\ \frac{\partial ^2 N_a}{\partial \eta ^2} \\ \frac{\partial ^2 N_a}{\partial \zeta ^2} \\ \frac{\partial ^2 N_a}{\partial \xi \partial \eta }\\ \frac{\partial ^2 N_a}{\partial \xi \partial \zeta }\\ \frac{\partial ^2 N_a}{\partial \eta \partial \zeta } \end{Bmatrix} - \begin{bmatrix} \frac{\partial ^2 x}{\partial \xi ^2}&\frac{\partial ^2 y}{\partial \xi ^2}&\frac{\partial ^2 z}{\partial \xi ^2} \\ \frac{\partial ^2 x}{\partial \eta ^2}&\frac{\partial ^2 y}{\partial \eta ^2}&\frac{\partial ^2 z}{\partial \eta ^2} \\ \frac{\partial ^2 x}{\partial \zeta ^2}&\frac{\partial ^2 y}{\partial \zeta ^2}&\frac{\partial ^2 z}{\partial \zeta ^2} \\ \frac{\partial ^2 x}{\partial \xi \partial \eta }&\frac{\partial ^2 y}{\partial \xi \partial \eta }&\frac{\partial ^2 z}{\partial \xi \partial \eta } \\ \frac{\partial ^2 x}{\partial \xi \partial \zeta }&\frac{\partial ^2 y}{\partial \xi \partial \zeta }&\frac{\partial ^2 z}{\partial \xi \partial \zeta } \\ \frac{\partial ^2 x}{\partial \eta \partial \zeta }&\frac{\partial ^2 y}{\partial \eta \partial \zeta }&\frac{\partial ^2 z}{\partial \eta \partial \zeta } \end{bmatrix} \begin{Bmatrix} \frac{\partial N_a}{\partial x} \\ \frac{\partial N_a}{\partial y} \\ \frac{\partial N_a}{\partial z} \end{Bmatrix} .&\end{aligned}$$

(23)

Finally, the global derivatives can be calculated using Eq. (24).

$$\begin{aligned}&\begin{Bmatrix} \frac{\partial ^2 N_a}{\partial x^2} \\ \frac{\partial ^2 N_a}{\partial y^2} \\ \frac{\partial ^2 N_a}{\partial z^2} \\ \frac{\partial ^2 N_a}{\partial x \partial y}\\ \frac{\partial ^2 N_a}{\partial x \partial z}\\ \frac{\partial ^2 N_a}{\partial y \partial z} \end{Bmatrix} = \begin{bmatrix} \left( \frac{\partial x}{\partial \xi } \right) ^2&\left( \frac{\partial y}{\partial \xi } \right) ^2&\left( \frac{\partial z}{\partial \xi } \right) ^2&2 \ \frac{\partial x}{\partial \xi } \ \frac{\partial y}{\partial \xi }&2 \ \frac{\partial x}{\partial \xi } \ \frac{\partial z}{\partial \xi }&2 \ \frac{\partial y}{\partial \xi } \ \frac{\partial z}{\partial \xi } \\ \left( \frac{\partial x}{\partial \eta } \right) ^2&\left( \frac{\partial y}{\partial \eta } \right) ^2&\left( \frac{\partial z}{\partial \eta } \right) ^2&2 \ \frac{\partial x}{\partial \eta } \ \frac{\partial y}{\partial \eta }&2 \ \frac{\partial x}{\partial \eta } \ \frac{\partial z}{\partial \eta }&2 \ \frac{\partial y}{\partial \eta } \ \frac{\partial z}{\partial \eta } \\ \left( \frac{\partial x}{\partial \zeta } \right) ^2&\left( \frac{\partial y}{\partial \zeta } \right) ^2&\left( \frac{\partial z}{\partial \zeta } \right) ^2&2 \ \frac{\partial x}{\partial \zeta } \ \frac{\partial y}{\partial \zeta }&2 \ \frac{\partial x}{\partial \zeta } \ \frac{\partial z}{\partial \zeta }&2 \ \frac{\partial y}{\partial \zeta } \ \frac{\partial z}{\partial \zeta } \\ \frac{\partial x}{\partial \xi } \frac{\partial x}{\partial \eta }&\frac{\partial y}{\partial \xi } \frac{\partial y}{\partial \eta }&\frac{\partial z}{\partial \xi } \frac{\partial z}{\partial \eta }&\left( \frac{\partial x}{\partial \xi } \frac{\partial y}{\partial \eta } + \frac{\partial x}{\partial \eta } \frac{\partial y}{\partial \xi } \right)&\left( \frac{\partial x}{\partial \xi } \frac{\partial z}{\partial \eta } + \frac{\partial x}{\partial \eta } \frac{\partial z}{\partial \xi } \right)&\left( \frac{\partial y}{\partial \xi } \frac{\partial z}{\partial \eta } + \frac{\partial y}{\partial \eta } \frac{\partial z}{\partial \xi } \right) \\ \frac{\partial x}{\partial \xi } \frac{\partial x}{\partial \zeta }&\frac{\partial y}{\partial \xi } \frac{\partial y}{\partial \zeta }&\frac{\partial z}{\partial \xi } \frac{\partial z}{\partial \zeta }&\left( \frac{\partial x}{\partial \xi } \frac{\partial y}{\partial \zeta } + \frac{\partial x}{\partial \zeta } \frac{\partial y}{\partial \xi } \right)&\left( \frac{\partial x}{\partial \xi } \frac{\partial z}{\partial \zeta } + \frac{\partial x}{\partial \zeta } \frac{\partial z}{\partial \xi } \right)&\left( \frac{\partial y}{\partial \xi } \frac{\partial z}{\partial \zeta } + \frac{\partial y}{\partial \zeta } \frac{\partial z}{\partial \xi } \right) \\ \frac{\partial x}{\partial \eta } \frac{\partial x}{\partial \zeta }&\frac{\partial y}{\partial \eta } \frac{\partial y}{\partial \zeta }&\frac{\partial z}{\partial \eta } \frac{\partial z}{\partial \zeta }&\left( \frac{\partial x}{\partial \eta } \frac{\partial y}{\partial \zeta } + \frac{\partial x}{\partial \zeta } \frac{\partial y}{\partial \eta } \right)&\left( \frac{\partial x}{\partial \eta } \frac{\partial z}{\partial \zeta } + \frac{\partial x}{\partial \zeta } \frac{\partial z}{\partial \eta } \right)&\left( \frac{\partial y}{\partial \eta } \frac{\partial z}{\partial \zeta } + \frac{\partial y}{\partial \zeta } \frac{\partial z}{\partial \eta } \right) \end{bmatrix} ^{-1} \cdot \begin{Bmatrix} \frac{\partial ^2 \bar{N}_a}{\partial \xi ^2} \\ \frac{\partial ^2 \bar{N}_a}{\partial \eta ^2} \\ \frac{\partial ^2 \bar{N}_a}{\partial \zeta ^2} \\ \frac{\partial ^2 \bar{N}_a}{\partial \xi \partial \eta }\\ \frac{\partial ^2 \bar{N}_a}{\partial \xi \partial \zeta }\\ \frac{\partial ^2 \bar{N}_a}{\partial \eta \partial \zeta } \end{Bmatrix}.&\end{aligned}$$

(24)

The same procedure applies for the third-derivatives. Here, to avoid tremendous matrix representations, a summary of the necessary steps to derive the unknowns is presented. The Eq. (21) counterpart for the third-derivatives is

$$\begin{aligned}&\frac{\partial ^3 N_a}{\partial \xi \partial \eta \partial \zeta } = \frac{\partial x}{\partial \xi } \cdot \frac{\partial x}{\partial \eta } \cdot \frac{\partial x}{\partial \zeta } \cdot \frac{\partial ^3 N_a}{\partial x^3} + \frac{\partial y}{\partial \xi } \cdot \frac{\partial y}{\partial \eta } \cdot \frac{\partial y}{\partial \zeta } \cdot \frac{\partial ^3 N_a}{\partial y^3}&\\&\quad + \frac{\partial z}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } \cdot \frac{\partial z}{\partial \zeta } \cdot \frac{\partial ^3 N_a}{\partial z^3}&\\&\quad + \bigg [ \frac{\partial x}{\partial \xi } \cdot \frac{\partial x}{\partial \eta } \cdot \frac{\partial y}{\partial \zeta } + \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial y}{\partial \eta } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial y}{\partial \xi } \bigg ) \frac{\partial x}{\partial \zeta } \bigg ] \cdot \frac{\partial ^3 N_a}{\partial x^2 \partial y}&\\&\quad + \bigg [ \frac{\partial x}{\partial \xi } \cdot \frac{\partial x}{\partial \eta } \cdot \frac{\partial z}{\partial \zeta } + \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial z}{\partial \xi } \bigg ) \frac{\partial x}{\partial \zeta } \bigg ] \cdot \frac{\partial ^3 N_a}{\partial x^2 \partial z}&\\&\quad + \bigg [ \frac{\partial y}{\partial \xi } \cdot \frac{\partial y}{\partial \eta } \cdot \frac{\partial x}{\partial \zeta } + \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial y}{\partial \eta } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial y}{\partial \xi } \bigg ) \frac{\partial y}{\partial \zeta } \bigg ] \cdot \frac{\partial ^3 N_a}{\partial x \partial y^2}&\\&\quad + \bigg [ \frac{\partial y}{\partial \xi } \cdot \frac{\partial y}{\partial \eta } \cdot \frac{\partial z}{\partial \zeta } + \bigg ( \frac{\partial y}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial y}{\partial \eta } \cdot \frac{\partial z}{\partial \xi } \bigg ) \frac{\partial y}{\partial \zeta } \bigg ] \cdot \frac{\partial ^3 N_a}{\partial y^2 \partial z}&\\&\quad + \bigg [ \frac{\partial z}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } \cdot \frac{\partial x}{\partial \zeta } + \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial z}{\partial \xi } \bigg ) \frac{\partial z}{\partial \zeta } \bigg ] \cdot \frac{\partial ^3 N_a}{\partial x \partial z^2}&\\&\quad + \bigg [ \frac{\partial z}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } \cdot \frac{\partial y}{\partial \zeta } + \bigg ( \frac{\partial y}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial y}{\partial \eta } \cdot \frac{\partial z}{\partial \xi } \bigg ) \frac{\partial z}{\partial \zeta } \bigg ] \cdot \frac{\partial ^3 N_a}{\partial y \partial z^2}&\\&\quad + \bigg [ \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial y}{\partial \eta } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial y}{\partial \xi } \bigg ) \frac{\partial z}{\partial \zeta } + \bigg ( \frac{\partial x}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial z}{\partial \xi } \bigg ) \frac{\partial y}{\partial \zeta }&\\&\quad + \bigg ( \frac{\partial y}{\partial \xi } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial y}{\partial \eta } \cdot \frac{\partial z}{\partial \xi } \bigg ) \frac{\partial x}{\partial \zeta } \bigg ] \cdot \frac{\partial ^3 N_a}{\partial x \partial y \partial z}&\\&\quad + \bigg ( \frac{\partial ^2 x}{\partial \xi \partial \zeta } \cdot \frac{\partial x}{\partial \eta } + \frac{\partial x}{\partial \xi } \cdot \frac{\partial ^2 x}{\partial \eta \partial \zeta } + \frac{\partial ^2 x}{\partial \xi \partial \eta } \cdot \frac{\partial x}{\partial \zeta } \bigg ) \cdot \frac{\partial ^2 N_a}{\partial x^2}&\\ \end{aligned}$$

$$\begin{aligned}&\quad + \bigg ( \frac{\partial ^2 y}{\partial \xi \partial \zeta } \cdot \frac{\partial y}{\partial \eta } + \frac{\partial y}{\partial \xi } \cdot \frac{\partial ^2 y}{\partial \eta \partial \zeta } + \frac{\partial ^2 y}{\partial \xi \partial \eta } \cdot \frac{\partial y}{\partial \zeta } \bigg ) \cdot \frac{\partial ^2 N_a}{\partial y^2}&\nonumber \\&\quad + \bigg ( \frac{\partial ^2 z}{\partial \xi \partial \zeta } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial z}{\partial \xi } \cdot \frac{\partial ^2 z}{\partial \eta \partial \zeta } + \frac{\partial ^2 z}{\partial \xi \partial \eta } \cdot \frac{\partial z}{\partial \zeta } \bigg ) \cdot \frac{\partial ^2 N_a}{\partial z^2}&\nonumber \\&\quad + \bigg ( \frac{\partial ^2 x}{\partial \xi \partial \zeta } \cdot \frac{\partial y}{\partial \eta } + \frac{\partial x}{\partial \xi } \cdot \frac{\partial ^2 y}{\partial \eta \partial \zeta } + \frac{\partial ^2 x}{\partial \eta \partial \zeta } \cdot \frac{\partial y}{\partial \xi } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial ^2 y}{\partial \xi \partial \zeta }&\nonumber \\&\quad + \frac{\partial ^2 x}{\partial \xi \partial \eta } \cdot \frac{\partial y}{\partial \zeta } + \frac{\partial ^2 y}{\partial \xi \partial \eta } \cdot \frac{\partial x}{\partial \zeta } \bigg ) \cdot \frac{\partial ^2 N_a}{\partial x \partial y}&\nonumber \\&\quad + \bigg ( \frac{\partial ^2 x}{\partial \xi \partial \zeta } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial x}{\partial \xi } \cdot \frac{\partial ^2 z}{\partial \eta \partial \zeta } + \frac{\partial ^2 x}{\partial \eta \partial \zeta } \cdot \frac{\partial z}{\partial \xi } + \frac{\partial x}{\partial \eta } \cdot \frac{\partial ^2 z}{\partial \xi \partial \zeta }&\nonumber \\&\quad + \frac{\partial ^2 x}{\partial \xi \partial \eta } \cdot \frac{\partial z}{\partial \zeta } + \frac{\partial ^2 z}{\partial \xi \partial \eta } \cdot \frac{\partial x}{\partial \zeta } \bigg ) \cdot \frac{\partial ^2 N_a}{\partial x \partial z}&\nonumber \\&\quad + \bigg ( \frac{\partial ^2 y}{\partial \xi \partial \zeta } \cdot \frac{\partial z}{\partial \eta } + \frac{\partial y}{\partial \xi } \cdot \frac{\partial ^2 z}{\partial \eta \partial \zeta } + \frac{\partial ^2 y}{\partial \eta \partial \zeta } \cdot \frac{\partial z}{\partial \xi } + \frac{\partial y}{\partial \eta } \cdot \frac{\partial ^2 z}{\partial \xi \partial \zeta }&\nonumber \\&\quad + \frac{\partial ^2 y}{\partial \xi \partial \eta } \cdot \frac{\partial z}{\partial \zeta } + \frac{\partial ^2 z}{\partial \xi \partial \eta } \cdot \frac{\partial y}{\partial \zeta } \bigg ) \cdot \frac{\partial ^2 N_a}{\partial y \partial z}&\nonumber \\&\quad + \frac{\partial ^3 x}{\partial \xi \partial \eta \partial \zeta } \cdot \frac{\partial N_a}{\partial x} + \frac{\partial ^3 y}{\partial \xi \partial \eta \partial \zeta } \cdot \frac{\partial N_a}{\partial y} + \frac{\partial ^3 z}{\partial \xi \partial \eta \partial \zeta } \cdot \frac{\partial N_a}{\partial z}. \end{aligned}$$

(25)

After doing the same calculations for all the third-derivatives and re-ordering, this results to a system of equations which looks similar to Eq. (22),

$$\begin{aligned}&\mathbf {a}_{10\times 1} = \mathbf {B}_{10\times 10}\cdot \mathbf {c}_{10\times 1} \ + \ \mathbf {D}_{10\times 6}\cdot \mathbf {e}_{6\times 1} \ + \ \mathbf {F}_{10\times 3}\cdot \mathbf {g}_{3\times 1},&\end{aligned}$$

(26)

where the subscripts denote the dimensions. In Eq. (26), $\mathbf {a}$ is the vector of local third-derivatives (known), $\mathbf {c}$ represents the unknown vector of global third-derivatives, $\mathbf {e}$ is the vector of global second-derivatives (known from Eq. (24)), $\mathbf {g}$ denotes the vector of global first-derivatives (known from Eq. (17)), and $\mathbf {B}$, $\mathbf {D}$ and $\mathbf {F}$ (all known) are coefficient matrices. At last, defining

$$\begin{aligned}&\mathbf {h}_{10\times 1} = \mathbf {a}_{10\times 1} \ - \ \mathbf {D}_{10\times 6}\cdot \mathbf {e}_{6\times 1} \ - \ \mathbf {F}_{10\times 3}\cdot \mathbf {g}_{3\times 1},&\end{aligned}$$

(27)

the global third-derivatives vector, $\mathbf {c}$, can be calculated from,

$$\begin{aligned}&\mathbf {c}_{10\times 1} = \mathbf {B}^{-1}_{10\times 10}\cdot \mathbf {h}_{10\times 1} .&\end{aligned}$$

(28)

The complete matrix multiplication is provided as a supplementary file for the online-version.