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A space-time LATIN-PGD strategy for solving Newtonian compressible flows

Élise Foulatier, Pierre-Alain Boucard, François Louf, David Néron, Philipp Junker

TL;DR

The paper tackles the computational challenge of simulating Newtonian compressible laminar flows with reduced cost and enhanced representation by introducing a space-time LATIN-PGD framework. It decouples pressure and velocity through a two-stage LATIN algorithm and embeds Proper Generalised Decomposition to obtain independent space-time bases for velocity and density, enabling efficient solution of the coupled PDE system. The approach is validated against an analytical Poiseuille flow, reproduces a 2D cylinder benchmark, and demonstrates extension to 3D geometries, with small PGD bases achieving high accuracy. This method offers a pathway to handle more complex material laws and parametric variations in fluid flows with potential computational speedups and modular multi-physics extensions.

Abstract

Simulating flow problems is at the core of many engineering applications but often requires high computational effort, especially when dealing with complex models. This work presents a novel approach for resolving flow problems using the LATIN-PGD solver. In this contribution, we place ourselves within the framework of Newtonian compressible and laminar flows. This specific and relatively simple case enables focusing on flows for which a state equation provides a direct relation between pressure and density. It is then possible to use the LATIN solver to set up a pressure-velocity decoupling algorithm. Moreover, Proper Generalised Decomposition (PGD) is natively included in the solver and yields two independent space-time decompositions for the velocity and the pressure fields. As a first step, the solver is validated on a problem for which an analytical solution is available. It is then applied to slightly more complex problems. The results show good agreement with the literature, and we expect that the solver could be used to compute more complicated material laws in the future.

A space-time LATIN-PGD strategy for solving Newtonian compressible flows

TL;DR

The paper tackles the computational challenge of simulating Newtonian compressible laminar flows with reduced cost and enhanced representation by introducing a space-time LATIN-PGD framework. It decouples pressure and velocity through a two-stage LATIN algorithm and embeds Proper Generalised Decomposition to obtain independent space-time bases for velocity and density, enabling efficient solution of the coupled PDE system. The approach is validated against an analytical Poiseuille flow, reproduces a 2D cylinder benchmark, and demonstrates extension to 3D geometries, with small PGD bases achieving high accuracy. This method offers a pathway to handle more complex material laws and parametric variations in fluid flows with potential computational speedups and modular multi-physics extensions.

Abstract

Simulating flow problems is at the core of many engineering applications but often requires high computational effort, especially when dealing with complex models. This work presents a novel approach for resolving flow problems using the LATIN-PGD solver. In this contribution, we place ourselves within the framework of Newtonian compressible and laminar flows. This specific and relatively simple case enables focusing on flows for which a state equation provides a direct relation between pressure and density. It is then possible to use the LATIN solver to set up a pressure-velocity decoupling algorithm. Moreover, Proper Generalised Decomposition (PGD) is natively included in the solver and yields two independent space-time decompositions for the velocity and the pressure fields. As a first step, the solver is validated on a problem for which an analytical solution is available. It is then applied to slightly more complex problems. The results show good agreement with the literature, and we expect that the solver could be used to compute more complicated material laws in the future.
Paper Structure (20 sections, 37 equations, 14 figures, 1 table)

This paper contains 20 sections, 37 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reference flow problem
  3. Numerical discretisation
  4. The LATIN-PGD solver for compressible flows
  5. Reformulation of the problem
  6. Iterative algorithm with two alternated search directions
  7. Practical developments
  8. Local stage
  9. Global stage
  10. Proper Generalised Decomposition within the LATIN framework
  11. Convergence indicator
  12. Summary of the algorithm
  13. Results
  14. Laminar flow in a channel
  15. Presentation of the problem
  16. ...and 5 more sections

Figures (14)

  • Figure 1: Flowchart of the LATIN-PGD algorithm for compressible flows
  • Figure 2: Scheme of the studied problem
  • Figure 3: Fields at steady state
  • Figure 4: Velocity and pressure results at steady state
  • Figure 5: Relative error between numerical and analytical solution
  • ...and 9 more figures