A comparative study of 2D numerical simulations using vector and potential methods for extending 3D modeling of planetary evolution

Abstract

Numerical simulations provide an improved understanding of the evolution and core formation processes of terrestrial bodies. The differentiation of silicate and iron metal has been simulated by solving the two-dimensional Stokes equation using either the vector or potential method. To describe the realistic geometry of planetesimals and planets, the development of a 3D model is necessary. Here, we developed a vector method model implemented in FEniCS project via scripting the weak forms of governing equations. Subsequently, we compared the solutions of the vector method models with those of the benchmark potential method implemented in Python. Three cases were modeled using the two methods and confirmed for consistency to verify the feasibility of developing a 3D model for core formation. Case 1 corresponds to planetary evolution triggered by impact heating in early terrestrial bodies with a homogeneous metal fraction. The vector method developed in the current study showed the consistency with the potential methods. In Case 2, the model mimicked a scenario depicting post-evolutionary impact heating by assuming heterogeneous metal fractions. The results of Case 2 simultaneously represent the behavior of the solid mixture based on the density contrast and multiphase flow of the solid matrix and metallic pores. The checkerboard test (Case 3) used to evaluate the resolution of the numerical model as a function of the heat source size also indicated identical spatial resolutions for both methods. In all cases, two methods simulated identical physical behaviors, indicating that a three-dimensional model can be developed using the vector method. The 2D FEM vector method developed in this study was effectively utilized to simulate the advection scheme, showing the fast descent of metal phases and relatively slower silicate phases. We extended the 2D FEM vector method to calculate the velocity fields of silicate, Darcy, and iron metal in a 3D model with an impact heating case (similar to Case 1). The 3D results show that the velocity of fluid metal in 3D model is faster than that in 2D model, suggesting the difference in the resistance to viscous flow depending the dimension of model. Our 3D vector method, implemented in FEniCS, demonstrates the numerical prospect of the methods necessary to model the multiphase fluid dynamics of a 3D planetary evolution.