Linear smoothed finite element method for quasi-incompressible hyperelastic media

Abstract

This work presents a linear smoothing scheme over high-order triangular elements within the framework of the cell-based strain smoothed finite element method for two-dimensional nonlinear problems. The main idea behind the proposed linear smoothing scheme is that it unlike the classical SFEM, it does not require the subdivision of the finite element cells into smoothing sub-cell. The other features of the classical SFEM are retained, such as: it does not require an explicit form of the derivatives of the basis functions, all the computations are done in the physical space, and the results are less sensitive to mesh distortion. A series of benchmark tests are done to demonstrate the validity and the stability of the proposed scheme. The validity and accuracy are confirmed by comparing the obtained numerical results with the standard FEM using quadratic triangular element and the exact solutions.

Appendix

1.1 The smoothed strain–displacement matrix

The following discrete trial and test functions are given using Lagrangian shape functions $\varPsi _I$ at node I:

$$\begin{aligned} {\mathbf {u}}^{\mathrm {h}}\left( {\mathbf {x}}\right) =\sum _{I=1}^N{u_I\varPsi _I\left( {\mathbf {x}}\right) },\quad {\mathbf {v}}^{\mathrm {h}}\left( {\mathbf {x}}\right) =\sum _{I=1}^N{u_I\varPsi _I\left( {\mathbf {x}}\right) } \end{aligned}$$

(37)

Thus, from the smoothed strain of Eq. (4), the smoothed strain–displacement matrix for 2D can be evaluated as:

$$\begin{aligned} \bar{\mathbf {B}}_I\left( {\mathbf {x}}\right) =\dfrac{1}{A_k^s}\int _{\varGamma _K^s}{{\mathbf {n}}\left( {\mathbf {x}}\right) \varPsi _I{\mathrm {d}}\varGamma }= \left[ \begin{array}{cc} \bar{B}_{I1}&{}\quad 0\\ 0&{}\quad \bar{B}_{I2}\\ \bar{B}_{I2}&{}\quad \bar{B}_{I1} \end{array}\right] \end{aligned}$$

(38)

where $\bar{B}_{Ii}$ is given as:

$$\begin{aligned} \bar{B}_{Ii}=\dfrac{1}{A_k^s}\int _{\varGamma _k^s}{\psi _I\left( {\mathbf {x}}\right) n_i\left( {\mathbf {x}}\right) {\mathrm {d}}\varGamma } \end{aligned}$$

(39)

1.2 Consistency of the nodal derivatives

For linear smoothing approximation, as given in Eq. (16), the approximation consistency is given as [14]:

$$\begin{aligned} \sum _I{\varPsi _I\left( {\mathbf {x}}\right) }=1,\quad \sum _I{\varPsi _I\left( {\mathbf {x}}\right) X_I}=x,\quad \sum _I{\varPsi _I\left( {\mathbf {x}}\right) Y_I}=y \end{aligned}$$

(40)

where the sampling point ${\mathbf {x}}=\left[ x,y\right]$.

Equation (40) can be reproduced as:

$$\begin{aligned} f\left( {\mathbf {x}}\right) =\sum _I{\varPsi _I\left( {\mathbf {x}}\right) f\left( {\mathbf {x}}_I\right) } \end{aligned}$$

(41)

Therefore, Eq. (41) can be rewritten by taking spatial derivatives:

$$\begin{aligned} f_{,i}\left( {\mathbf {x}}\right) =\sum _I{f\left( {\mathbf {x}}_I\right) \varPsi _{I,a}\left( {\mathbf {x}}\right) } \end{aligned}$$

(42)

which can be extended as:

$$\begin{aligned} \sum _I\varPsi _{I,1}\left( {\mathbf {x}}\right) =0,\quad \sum _I\varPsi _{I,1}\left( {\mathbf {x}}\right) X_I=1,\quad \sum _I\varPsi _{I,1}\left( {\mathbf {x}}\right) Y_I=0 \end{aligned}$$

(43)

for $\varPsi _{I,1}$ and

$$\begin{aligned} \sum _I\varPsi _{I,2}\left( {\mathbf {x}}\right) =0,\quad \sum _I\varPsi _{I,2}\left( {\mathbf {x}}\right) X_I=0,\quad \sum _I\varPsi _{I,2}\left( {\mathbf {x}}\right) Y_I=1 \end{aligned}$$

(44)

for $\varPsi _{I,2}$. Note that, the shape functions $\varPsi _I\left( {\mathbf {x}}\right)$ need to meet a consistency condition with their derivatives $\varPsi _{I,i}\left( {\mathbf {x}}\right)$ in terms of the divergence theorem. Therefore, the divergence consistency is expressed as:

$$\begin{aligned} \int _{\varOmega _S}{\varPsi _{I,i}\left( {\mathbf {x}}\right) f\left( {\mathbf {x}}\right) {\mathrm {d}}\varOmega }= & {} \int _{\varGamma _S}{\varPsi _I\left( {\mathbf {x}}\right) f\left( {\mathbf {x}}\right) n_i{\mathrm {d}}\varGamma }\nonumber \\&- \int _{\varOmega _S}{\varPsi _I\left( {\mathbf {x}}\right) f_{,i}\left( {\mathbf {x}}\right) {\mathrm {d}}\varOmega } \end{aligned}$$

(45)

where $\varOmega _S$ and $\varGamma _S$ are sub-domain and the boundaries of sub-domain under the control of the shape functions $\varPsi _I$.