11:30 AM - 11:45 AM
[MIS21-04] High accuracy particle method for simulating planetary interiors and its numerical stability analysis
Keywords:computational fluid dynamics, SPH method, numerical instability, particle method, planetary interiors
To clarify the full picture of the planetary evolution, it is extremely important to understand global dynamics processes continuously. The strong dependence of the physicochemical evolution of the planetary interiors on the impact conditions and the starting compositions of planetesimals [1] makes it impossible to consider the planetary formation and interior evolution separately. However, no computational fluid dynamics method that can accurately represent multiphase flows with large deformations and simultaneously treat different spatial and temporal scales has been established. This lack of numerical tools is a factor that hinders solving the planetary evolution.
In order to treat the formation and interior evolution continuously, we have developed a framework based on the Smoothed Particle Hydrodynamics method (SPH method [2,3]) since it is widely used in the planetary accretion because of its benefits in treating multiple phases and large deformations, and have applied it to the planetary core formation and mantle convection [4,5]. However, recent studies [6] have revealed that the accuracy of the SPH method deteriorates dramatically when the particle configuration is disturbed, because the standard SPH equations are derived assuming a uniform particle configuration. Therefore, we developed a high accuracy SPH method that does not assume a symmetric arrangement of particles, extending previous studies [7, 8]. The figure below shows the numerical convergence of the conventional SPH method (ST0, CSPH) and our method (LSSPH). The horizontal axis represents the particle spacing, the vertical axis represents the root mean square error (RMSE) from the analytical solution, and the epsilon in the title of each figure represents the perturbation from the uniform configuration. These results indicate that our newly developed method is highly accurate, regardless of the particle configuration.
It is well known that the conventional SPH method involves particle clustering as the simulation develops [9,10]. This clumping is due to non-physical instability (i.e. numerical instability) and not only reduces the effective resolution, but also calculates the incorrect strain (or stress), which may lead to numerical divergence. This instability is considered to be caused by the sound wave instability, which is investigated as “paring instability” by Dehnen and Aly [9] in the astronomical field, but as “tensile instability” by Swegle et al. [10] in the computational engineering field. These conventional numerical stability analyses are limited to low accuracy SPH methods, and there is no discussion for high accuracy SPH methods. Therefore, we applied the linear theory to our method and investigated how the instability region changes. In this presentation, we are going to present the key concepts of our high accuracy SPH method and the results of the numerical stability analysis.
Reference
[1] D. C. Rubie, et al., Earth Planet. Sci. Lett., 205, 2003
[2] R.A. Gingold, J. J., Monaghan, Mon. Notices Royal Astron. Soc., 181, 1977
[3] L. B.Lucy, Astron. J., 82, 1977
[4] K. Shobuzako, et al., JpGU2022, Chiba, Japan, May 2022
[5] K. Shobuzako, et al., the Japanese Society for Planetary Sciences, Hiroshima, Japan, October 2023
[6] M. Asai, Comput. Methods Appl. Mech. Eng., 415, 2023
[7] T. Tamai, S. Koshizuka, Computational Particle Mechanics, 1, 2014
[8] S. Yamamoto, J. Makino, Publ. Astron. Soc. Jpn., 69, 2017
[9] W. Dehnen, H. Aly, Mon. Notices Royal Astron. Soc., 425, 2012
[10] J. W. Swegle, et al., J. Comput. Phys., 116, 1995
In order to treat the formation and interior evolution continuously, we have developed a framework based on the Smoothed Particle Hydrodynamics method (SPH method [2,3]) since it is widely used in the planetary accretion because of its benefits in treating multiple phases and large deformations, and have applied it to the planetary core formation and mantle convection [4,5]. However, recent studies [6] have revealed that the accuracy of the SPH method deteriorates dramatically when the particle configuration is disturbed, because the standard SPH equations are derived assuming a uniform particle configuration. Therefore, we developed a high accuracy SPH method that does not assume a symmetric arrangement of particles, extending previous studies [7, 8]. The figure below shows the numerical convergence of the conventional SPH method (ST0, CSPH) and our method (LSSPH). The horizontal axis represents the particle spacing, the vertical axis represents the root mean square error (RMSE) from the analytical solution, and the epsilon in the title of each figure represents the perturbation from the uniform configuration. These results indicate that our newly developed method is highly accurate, regardless of the particle configuration.
It is well known that the conventional SPH method involves particle clustering as the simulation develops [9,10]. This clumping is due to non-physical instability (i.e. numerical instability) and not only reduces the effective resolution, but also calculates the incorrect strain (or stress), which may lead to numerical divergence. This instability is considered to be caused by the sound wave instability, which is investigated as “paring instability” by Dehnen and Aly [9] in the astronomical field, but as “tensile instability” by Swegle et al. [10] in the computational engineering field. These conventional numerical stability analyses are limited to low accuracy SPH methods, and there is no discussion for high accuracy SPH methods. Therefore, we applied the linear theory to our method and investigated how the instability region changes. In this presentation, we are going to present the key concepts of our high accuracy SPH method and the results of the numerical stability analysis.
Reference
[1] D. C. Rubie, et al., Earth Planet. Sci. Lett., 205, 2003
[2] R.A. Gingold, J. J., Monaghan, Mon. Notices Royal Astron. Soc., 181, 1977
[3] L. B.Lucy, Astron. J., 82, 1977
[4] K. Shobuzako, et al., JpGU2022, Chiba, Japan, May 2022
[5] K. Shobuzako, et al., the Japanese Society for Planetary Sciences, Hiroshima, Japan, October 2023
[6] M. Asai, Comput. Methods Appl. Mech. Eng., 415, 2023
[7] T. Tamai, S. Koshizuka, Computational Particle Mechanics, 1, 2014
[8] S. Yamamoto, J. Makino, Publ. Astron. Soc. Jpn., 69, 2017
[9] W. Dehnen, H. Aly, Mon. Notices Royal Astron. Soc., 425, 2012
[10] J. W. Swegle, et al., J. Comput. Phys., 116, 1995