A three-dimensional finite element arbitrary Lagrangian–Eulerian method for shock hydrodynamics on unstructured grids

• We present a 3D arbitrary Lagrangian–Euerlian method for shock hydrodynamics on unstructured grids.
• Our method combines an unsplit formulation with mesh motion based on the fluid velocity.
• Shock-dominated flows show global convergence rates of 0.8–1.0.
• Smooth problems show global convergence rates of 1.9.
• The method satisfies the geometric conservation law to truncation error.

We present a three-dimensional (3D) finite element (FE) arbitrary Lagrangian–Eulerian (ALE) method for shock hydrodynamics on unstructured grids. The method is based on an FE Eulerian Godunov scheme for linear tetrahedra that has been extended to include mesh motion in an unsplit, flux-conservative formulation. The proposed method eliminates the splitting errors present in traditional Lagrange-plus-remap methods that occur during the remap phase. Unlike typical unsplit approaches, the mesh velocity is not determined by boundary motion but is instead based on the local fluid velocity. Smoothing operations are then applied to the mesh velocity to avoid mesh tangling. This approach allows the mesh to follow the fluid motion in a robust manner and leverage one of the primary advantages of Lagrangian schemes for shock hydrodynamics, namely that the resolution follows the flow. An approximate Riemann solver is used to calculate fluxes in the co-moving frame of the mesh. Results for a number of standard test problems are presented for 3D meshes of up to 107107 tetrahedra. Global convergence rates of 0.80.8–1.01.0 are observed for shock dominated flows and 1.91.9 for smooth flows. We also demonstrate that the method satisfies the discrete geometric conservation law to truncation error, conserves total energy to machine precision, and preserves symmetry.