Weak Boundary Conditions for Lagrangian Shock Hydrodynamics: A High-Order Finite Element Implementation on Curved Boundaries

TL;DR

This paper introduces a novel Nitsche-type weak boundary enforcement method for Lagrangian shock hydrodynamics using high-order finite elements on curved boundaries, ensuring energy conservation and simplifying traditional approaches.

Contribution

It presents a new variational formulation for weakly enforcing boundary conditions in Lagrangian shock hydrodynamics that preserves energy and simplifies implementation.

Findings

Method conserves total energy.

Mass matrices remain constant over time.

Validated robustness and accuracy on curved boundaries.

Abstract

We propose a new Nitsche-type approach for weak enforcement of normal velocity boundary conditions for a Lagrangian discretization of the compressible shock-hydrodynamics equations using high-order finite elements on curved boundaries. Specifically, the variational formulation is appropriately modified to enforce free-slip wall boundary conditions, without perturbing the structure of the function spaces used to represent the solution, with a considerable simplification with respect to traditional approaches. Total energy is conserved and the resulting mass matrices are constant in time. The robustness and accuracy of the proposed method are validated with an extensive set of tests involving nontrivial curved boundaries.