Method

Abstract

Numerical simulations are the most powerful tool to investigate resultant shapes of asteroids produced through various collisional events. To conduct such numerical simulations, we need knowledge about not only hydrodynamics but also elastic dynamics for intact rocks, fracturing of rocky materials, and friction of completely damaged rocks. In this chapter, we introduce the detailed methods necessary for numerical simulations of asteroidal collisions, i.e., equations for elastic dynamics, discretized equations for smoothed particle hydrodynamics method, equation of states for impact simulations, a model for fracture of rocks, a model for friction of damaged rocks, parallelization of simulation codes, and a time development procedure.

Appendix: Linear Stability Analysis for Godunov SPH Method

Here, we introduce the details of the linear stability analysis of the equations for the Godunov SPH method. In this linear stability analysis, we use the following simplification. We ignore the terms of the deviatoric stress tensor and set $S_{i}^{\alpha \beta }=0$ because the deviatoric stress tensor generally prevents the clumping of SPH particles, which leads to prevent the tensile instability. We also ignore the Riemann solver and set $P_{ij}^{*}=(P_{i}+P_{j})/2$ because numerical viscosity terms including the Riemann solver generally prevent numerical instabilities. We consider the cases that SPH particles are put on cubic lattices, and the smoothing length h is set to be the side length of the cubic lattices $\Delta x$. We conduct the linear stability analysis for the longitudinal perturbations because the tensile instability is the instability of compressional waves. We utilize the equation of state of $P=C_{s}^{2}(\rho - \rho _{0})$. Unperturbed states have the uniform density and pressure, and all SPH particles have the same mass m.

Unperturbed positions of particles are expressed as,

$$\begin{aligned} \overline{\varvec{x}}_{i}=(\overline{x}_{i},\overline{y}_{i},\overline{z}_{i}). \end{aligned}$$

(2.125)

We add the perturbation of positions in the x-direction. Then the positions of particles are represented as

$$\begin{aligned}&\varvec{x}_{i}=(\overline{x}_{i}+\delta x_{i},\overline{y}_{i},\overline{z}_{i}), \nonumber \\&\delta x_{i}=\epsilon _{x} \exp [I(k\overline{x}_{i}-\omega t)], \end{aligned}$$

(2.126)

where $\epsilon _{x}$ is very small value, k and $\omega$ represent the wavenumber and frequency of the perturbation respectively, and I represents the imaginary unit. Hereafter, $\epsilon$ shows very small value, and we ignore second or higher order of $\epsilon$.

From Eq. (2.126) and $(d/dt)\varvec{x}_{i}=\varvec{v}_{i}$, the velocity of the ith particle is represented as

$$\begin{aligned} \varvec{v}_{i}=(-I\omega \delta x_{i},0,0). \end{aligned}$$

(2.127)

The density in unperturbed states is now defined as $\overline{\rho }$, and the density of the ith particle is written as

$$\begin{aligned}&\rho _{i}=\overline{\rho }+\delta \rho _{i},\nonumber \\&\delta \rho _{i}=\epsilon _{\rho }\exp [I(k\overline{x}_{i}-\omega t)]. \end{aligned}$$

(2.128)

From Eq. (2.19), we represent $\delta \rho _{i}$ using $\delta x_{i}$ as,

$$\begin{aligned}&\delta \rho _{i}=-I\overline{\rho }D\delta x_{i}, \nonumber \\&D\equiv \sum _{j}-\sin [k(\overline{x}_{i}-\overline{x}_{j})]\frac{\partial }{\partial \overline{x}_{i}}W(|\overline{\varvec{x}}_{i}-\overline{\varvec{x}}_{j}|,h)\frac{m}{\overline{\rho }}. \end{aligned}$$

(2.129)

Then we represent the pressure of the ith particle as

$$\begin{aligned} P_{i}=\overline{P}+\delta P_{i}=\overline{P}+C_{s}^{2}\delta \rho _{i}=\overline{P}-IC_{s}^{2}\overline{\rho }D\delta x_{i}, \end{aligned}$$

(2.130)

where $\overline{P}=C_{s}^{2}(\overline{\rho }-\rho _{0})$ represents the pressure in unperturbed states.

From Eq. (2.54), the specific volume is represented as

$$\begin{aligned} V_{i}=\frac{1}{\overline{\rho }}(1+ID\delta x_{i}). \end{aligned}$$

(2.131)

From Eq. (2.54), the x-component of the specific volume gradient is represented as

$$\begin{aligned} \frac{\partial V_{i}}{\partial x_{i}}&=-\frac{1}{\rho _{i}^{2}}\frac{\partial \rho _{i}}{\partial x_{i}} \nonumber \\&\approx -\frac{1}{\overline{\rho }^{2}}\sum _{j}m(\delta x_{i}-\delta x_{j})\frac{\partial ^{2}}{\partial \overline{x_{i}}^{2}}W(|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|,h) =-\frac{1}{\overline{\rho }}C_{\rho }\delta x_{i}, \nonumber \\ C_{\rho }&\equiv \sum _{j}(1-\cos [k(\overline{x_{i}}-\overline{x_{j}})])\frac{\partial ^{2}}{\partial \overline{x_{i}}^{2}}W(|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|,h)\frac{m}{\overline{\rho }}. \end{aligned}$$

(2.132)

The xx-component of the second derivative of $V_{i}$ is represented using Eq. (2.54) as

$$\begin{aligned} \frac{\partial ^{2}V_{i}}{\partial x_{i}^{2}}=\sum _{j}\frac{m}{\rho _{j}}\frac{\partial V_{j}}{\partial x_{j}}\frac{\partial }{\partial x_{i}}W(|\varvec{x}_{i}-\varvec{x}_{j}|,h)\approx -\frac{I}{\overline{\rho }}C_{\rho }D\delta x_{i}. \end{aligned}$$

(2.133)

We then substitute linearized quantities into Eq. (2.65), and finally we obtain the dispersion relations (DRs) for the three interpolation methods. DR for the linear interpolation is expressed as

$$\begin{aligned} \omega ^{2}_{\mathrm{linear}}=-C_{s}^{2}Da+\frac{\overline{P}}{\overline{\rho }}\Bigg [ 2Da + 2b \Bigg ], \end{aligned}$$

(2.134)

where

$$\begin{aligned}&a=\sum _{j\ne i}\sin [k(\overline{x_{i}}-\overline{x_{j}})]\frac{\partial }{\partial \overline{x_{i}}}W(|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|,\sqrt{2}h)\frac{m}{\overline{\rho }}, \nonumber \\&b=\sum _{j\ne i}(1-\cos [k(\overline{x_{i}}-\overline{x_{j}})])\frac{\partial ^{2}}{\partial \overline{x_{i}}^{2}}W(\overline{|\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|,\sqrt{2}h)\frac{m}{\overline{\rho }}. \end{aligned}$$

(2.135)

DR for the cubic spline interpolation is expressed as

$$\begin{aligned} \omega _{\mathrm{cubic}}^{2}=-C_{s}^{2}Da + \frac{\overline{P}}{\overline{\rho }}\Bigg [ 2Da + 2b - \frac{1}{2}h^{2}C_{\rho }c + \frac{1}{2}C_{\rho }d \Bigg ], \end{aligned}$$

(2.136)

where

$$\begin{aligned}&c=\sum _{j\ne i}\frac{\overline{x_{i}}-\overline{x_{j}}}{|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|^{2}}(1-\cos [k(\overline{x_{i}}-\overline{x_{j}})])\frac{\partial }{\partial \overline{x_{i}}}W(|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|,\sqrt{2}h)\frac{m}{\overline{\rho }}, \nonumber \\&d=\sum _{j\ne i}(\overline{x_{i}}-\overline{x_{j}})(1-\cos [k(\overline{x_{i}}-\overline{x_{j}})])\frac{\partial }{\partial \overline{x_{i}}}W(|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|,\sqrt{2}h)\frac{m}{\overline{\rho }}. \end{aligned}$$

(2.137)

DR for the quintic spline interpolation is expressed as

$$\begin{aligned} \omega _{\mathrm{quintic}}^{2} =&-C_{s}^{2}Da + \frac{\overline{P}}{\overline{\rho }}\Bigl [ 2Da + 2b + \frac{3}{8}h^{4}C_{\rho }B_{1,4} + \frac{3}{16}h^{4}C_{\rho }DA_{2,4} - \frac{3}{4}h^{2}C_{\rho }B_{1,2} \nonumber \\&- \frac{1}{8}h^{2}C_{\rho }DA_{2,2} + \frac{5}{8}C_{\rho }B_{1,0} + \frac{1}{16}C_{\rho }DA_{2,0} \Bigr ], \end{aligned}$$

(2.138)

where

$$\begin{aligned}&A_{n,m}=\sum _{j}\frac{(\overline{x_{i}}-\overline{x_{j}})^{n}}{|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|^{m}}\sin [k(\overline{x_{i}}-\overline{x_{j}})]\frac{\partial }{\partial \overline{x_{i}}}W(|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|,\sqrt{2}h)\frac{m}{\overline{\rho }}, \nonumber \\&B_{n,m}=\sum _{j}\frac{(\overline{x_{i}}-\overline{x_{j}})^{n}}{|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|^{m}}(1-\cos [k(\overline{x_{i}}-\overline{x_{j}})])\frac{\partial }{\partial \overline{x_{i}}}W(|\overline{\varvec{x}_{i}}-\overline{\varvec{x}_{j}}|,\sqrt{2}h)\frac{m}{\overline{\rho }}. \end{aligned}$$

(2.139)