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Machine-precision energy conservative reduced models for Lagrangian hydrodynamics by quadrature methods

Chris Vales, Siu Wun Cheung, Dylan M. Copeland, Youngsoo Choi

Abstract

We present an energy conservative, quadrature based model reduction framework for the compressible Euler equations of Lagrangian hydrodynamics. Building on a finite element discretization of the governing equations, we develop reduced models using data based reduced basis functions and the empirical quadrature procedure (EQP). We introduce a strongly energy conservative variant of EQP that enforces exact energy conservation in the reduction process. Numerical experiments for four benchmark problems -- Sedov blast, Gresho vortex, triple point and Taylor-Green vortex -- demonstrate that the numerical implementation of our proposed method conserves total energy to near machine precision, while maintaining accuracy comparable to the basic EQP formulation.

Machine-precision energy conservative reduced models for Lagrangian hydrodynamics by quadrature methods

Abstract

We present an energy conservative, quadrature based model reduction framework for the compressible Euler equations of Lagrangian hydrodynamics. Building on a finite element discretization of the governing equations, we develop reduced models using data based reduced basis functions and the empirical quadrature procedure (EQP). We introduce a strongly energy conservative variant of EQP that enforces exact energy conservation in the reduction process. Numerical experiments for four benchmark problems -- Sedov blast, Gresho vortex, triple point and Taylor-Green vortex -- demonstrate that the numerical implementation of our proposed method conserves total energy to near machine precision, while maintaining accuracy comparable to the basic EQP formulation.

Paper Structure

This paper contains 25 sections, 1 theorem, 63 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Projection based reduction
  3. Hyper-reduction methods
  4. Lagrangian hydrodynamics
  5. Main results
  6. Full model
  7. Spatial discretization
  8. Temporal discretization
  9. Reduced model
  10. Empirical quadrature procedure
  11. Basic EQP hyper-reduction
  12. Temporal discretization
  13. Time windowing
  14. Simulation stages
  15. Energy conservative EQP
  16. ...and 10 more sections

Key Result

Theorem 1

Assume that the following three conditions are satisfied. Then the discrete hyper-reduced model eq:hrom-ceqp-rk2avg conserves the discrete total energy; namely, for all $k\geq 0$.

Figures (1)

  • Figure 1: Final snapshots of the velocity field for each simulated problem case. The top row corresponds to the results obtained using the full model; the middle row using the model hyper-reduced by the basic EQP method; the bottom row using the model hyper-reduced by the energy conservative EQP method. From left to right, the first column corresponds to the Sedov problem; the second to Gresho; the third to triple point; the fourth to Taylor-Green.

Theorems & Definitions (2)

  • Theorem 1
  • proof