A kinematic comparison of meshfree and mesh-based Lagrangian approximations using manufactured extreme deformation fields

Abstract

Meshfree methods for solid mechanics have been in development since the early 1990’s. Initial motivations included alleviation of the burden of mesh creation and the desire to overcome the limitations of traditional mesh-based discretizations for extreme deformation applications. Here, the accuracy and robustness of both meshfree and mesh-based Lagrangian discretizations are compared using manufactured extreme deformation fields. For the meshfree discretizations, both moving least squares and maximum entropy are considered. Quantitative error and convergence results are presented for the best approximation in the $H^1$ norm.

Appendices

Appendix A: Deformation field $\mathscr {D}_2$

The following stream function is a classical example from fluid mechanics representing the incompressible inviscid flow around a cylinder [1]. By choosing an appropriate initial material body, this deformation resembles the impact of projectile onto a ductile plate. Actual penetration is not possible, however, due to continuity of the deformation. The stream function, $\Psi$, is defined using polar coordinates $(r,\theta )$, as

$$\begin{aligned} \Psi (r, \theta ) = V_\infty \,r \sin \theta \left( 1 - \frac{r_{\mathrm {o}}^2}{r^2} \right) , \end{aligned}$$

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where $r_{\mathrm {o}}$ is the cylinder radius, $V_\infty$ is the upstream velocity. The velocity components are then given by

$$\begin{aligned} \begin{aligned} v_r&= \frac{1}{r} \frac{\partial \Psi }{\partial \theta } = V_\infty \cos \theta \left( 1 - \frac{r_{\mathrm {o}}^2}{r^2} \right) , \\ v_\theta&= - \frac{\partial \Psi }{\partial r} = -V_\infty \sin \theta \left( 1 + \frac{r_{\mathrm {o}}^2}{r^2} \right) . \end{aligned} \end{aligned}$$

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In rectangular coordinates, the velocity components are

$$\begin{aligned} \begin{aligned} v_1&= \frac{1}{r}\left( x_1 v_r - x_2 v_\theta \right) \\&= \frac{V_\infty }{r^2} \left[ x_1^2\left( 1-\frac{r_{\mathrm {o}}^2}{r^2} \right) + x_2^2\left( 1+\frac{r_{\mathrm {o}}^2}{r^2} \right) \right] , \\ v_2&= \frac{1}{r}\left( x_2 v_r + x_1 v_\theta \right) =- V_\infty \, x\,y \, \frac{r_{\mathrm {o}}^2}{r^4} \end{aligned} \end{aligned}$$

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with $x_1 = r \cos \theta$, $x_2 = r \sin \theta$, $\tan \theta = x_2/x_1$, and $r^2 = x_1^2 + x_2^2$.

The components of the velocity gradient are then

$$\begin{aligned} \begin{aligned} L_{11}&= V_\infty \left\{ \frac{1}{r^2} \left[ \frac{2r_{\mathrm {o}}^2 x_1^3}{r^4} + 2x_1\left( 1-\frac{r_{\mathrm {o}}^2}{r^2}\right) - \frac{2r_{\mathrm {o}}^2x_1x_2^2}{r^4} \right] \right. \\&\quad \left. {} - \frac{2x_1}{r^4} \left[ x_1^2\left( 1-\frac{r_{\mathrm {o}}^2}{r^2}\right) +x_2^2\left( 1+\frac{r_{\mathrm {o}}^2}{r^2}\right) \right] \right\} , \\ L_{12}&= V_\infty \left\{ \frac{1}{r^2} \left[ \frac{2r_{\mathrm {o}}^2 x_1^2x_2}{r^4} + 2x_2\left( 1+\frac{r_{\mathrm {o}}^2}{r^2}\right) - \frac{2r_{\mathrm {o}}^2x_2^3}{r^4} \right] \right. \\&\quad \left. {} - \frac{2x_2}{r^4} \left[ x_1^2\left( 1-\frac{r_{\mathrm {o}}^2}{r^2}\right) +x_2^2\left( 1+\frac{r_{\mathrm {o}}^2}{r^2}\right) \right] \right\} , \\ L_{21}&= -2V_\infty \,r_{\mathrm {o}}^2\, x_2 \frac{r^2 - 4 x_1^2}{r^6} ,\\ L_{22}&= -2V_\infty \,r_{\mathrm {o}}^2\, x_1 \frac{r^2 - 4 x_2^2}{r^6} . \end{aligned} \end{aligned}$$

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Fig. 19

Deformation $\mathscr {D}_2$ at times a$t = 0$, b$t = 0.4$, c$t=0.8$

Normalizing, let $V_\infty =1$ and $r_{\mathrm {o}}=1$, and if the material body $\mathscr {B}$ is chosen to be the rectangle body $[-2,-1]\times [-3,3]$, the deformation shown in Fig. 19 is obtained.

Additional two-dimensional deformation fields can be defined similarly using the stream function approach including axisymmetric flows. For example, the stream function for axisymmetric flow around an ellipsoid is given in [11].

Appendix B: Convergence data

For completeness, the convergence results presented in Figs. 14–17 are also given in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. Several of the finite element results were not computable (NC) for the highly distorted meshes ($s=0.4$ and $s=0.5$). All results are normalized by the respective norm of the exact solution.

Table 1 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{L_2} / || \mathbf {u}||_{L_2}$, for the finite element approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of mesh refinement and mesh distortion s

Table 2 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{H^1} / || \mathbf {u}||_{H^1}$, for the finite element approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of mesh refinement and mesh distortion s

Table 3 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{L_2} / || \mathbf {u}||_{L_2}$, for the moving least squares approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of node refinement and node distortion s

Table 4 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{H^1} / || \mathbf {u}||_{H^1}$, for the moving least squares approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of node refinement and node distortion s

Table 5 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{L_2} / || \mathbf {u}||_{L_2}$, for the moving least squares approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of node refinement and node distortion s

Table 6 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{H^1} / || \mathbf {u}||_{H^1}$, for the moving least squares approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of node refinement and node distortion s

Table 7 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{L_2} / || \mathbf {u}||_{L_2}$, for the maximum entropy approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of node refinement and node distortion s

Table 8 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{H^1} / || \mathbf {u}||_{H^1}$, for the maximum entropy approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of node refinement and node distortion s

Table 9 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{L_2} / || \mathbf {u}||_{L_2}$, for the maximum entropy approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of node refinement and node distortion s

Table 10 Normalized error, $|| \mathbf {u}- \mathbf {u}^h_{H^1} ||_{H^1} / || \mathbf {u}||_{H^1}$, for the maximum entropy approximation (best approximation in the $H^1$ norm) of deformation $\mathscr {D}_1$ at time $t=0.1$ as a function of node refinement and node distortion s