A discontinuous Galerkin method based on a hierarchical orthogonal basis for Lagrangian hydrodynamics on curvilinear grids

TL;DR

This paper introduces a high-order Lagrangian discontinuous Galerkin method using a hierarchical orthogonal basis for simulating material dynamics on curvilinear grids, achieving up to fourth-order accuracy.

Contribution

It develops a novel hierarchical orthogonal basis for DG methods that decouples solution moments and employs a unique vertex-based quadrature approach for surface integrals.

Findings

Achieves up to fourth-order accuracy in simulations.

Demonstrates robustness on complex curved grids.

Effectively handles diverse gas and solid dynamics problems.

Abstract

We present a new high-order accurate Lagrangian discontinuous Galerkin (DG) hydrodynamic method to simulate material dynamics (for e.g., gasses, fluids, and solids) with up to fourth-order accuracy on cubic meshes. The variables, such as specific volume, velocity, specific total energy, and deformation gradient fields within a cell, are represented with a polynomial constructed from a novel hierarchical orthogonal basis about the center of mass, which decouples the moments of the solution because the mass matrix is diagonal. The discontinuity in the polynomials at the cell boundary is addressed by solving a multi-directional Riemann problem at the vertices of the cell and a 1D Riemann problem at additional non-vertex quadrature points along the edges so that the surface integral is exact for the polynomial order. The uniqueness lies in that the vertices of the curvilinear grid work as the quadrature points for the surface integral of DG methods. To ensure robust mesh motion, the pressure for the Riemann problem accounts for the difference between the density variation over the cell and a density field from subcell mesh stabilization (SMS). The accuracy and robustness of the new high-order accurate Lagrangian DG hydrodynamic method is demonstrated by simulating a diverse suite of challenging test problems covering gas and solid dynamic problems on curved grids.