Multidimensional staggered grid residual distribution scheme for Lagrangian hydrodynamics

TL;DR

This paper introduces a second-order multidimensional residual distribution scheme for Lagrangian hydrodynamics that improves computational efficiency and conservation properties using staggered grids, Bernstein polynomials, and deferred correction time-stepping.

Contribution

It presents a novel residual distribution scheme with diagonal mass matrices, ensuring exact conservation and applicability to multi-material models in Lagrangian hydrodynamics.

Findings

Achieves second-order accuracy in multidimensional Lagrangian hydrodynamics.

Ensures exact conservation of mass, momentum, and energy.

Demonstrates effectiveness on challenging test problems.

Abstract

We present the second-order multidimensional Staggered Grid Hydrodynamics Residual Distribution (SGH RD) scheme for Lagrangian hydrodynamics. The SGH RD scheme is based on the staggered finite element discretizations as in [Dobrev et al., SISC, 2012]. However, the advantage of the residual formulation over classical FEM approaches consists in the natural mass matrix diagonalization which allows one to avoid the solution of the linear system with the global sparse mass matrix while retaining the desired order of accuracy. This is achieved by using Bernstein polynomials as finite element shape functions and coupling the space discretization with the deferred correction type timestepping method. Moreover, it can be shown that for the Lagrangian formulation written in non-conservative form, our residual distribution scheme ensures the exact conservation of the mass, momentum and total energy. In this paper we also discuss construction of numerical viscosity approximations for the SGH RD scheme allowing to reduce the dissipation of the numerical solution. Thanks to the generic formulation of the staggered grid residual distribution scheme, it can be directly applied to both single- and multimaterial and multiphase models. Finally, we demonstrate computational results obtained with the proposed residual distribution scheme for several challenging test problems.