Eigendecomposition-based Poisson Solvers for Fluid Flow Simulations in Non-uniform Grids
D. Seixas Carlos Palancha (TU Delft - Mechanical Engineering)
P. Simões Costa – Mentor (TU Delft - Mechanical Engineering)
B. Font – Graduation committee member (TU Delft - Mechanical Engineering)
M.J.B.M. Pourquie – Graduation committee member (TU Delft - Mechanical Engineering)
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Abstract
Fundamental research on fluid dynamics allows us to progress in our technological endeavours, ensuring safety in our daily lives. Understanding the underlying mechanics of turbulence is a challenge due to its unsteady, chaotic, three-dimensional structure. Mathematically, this is portrayed in the continuity and Navier-Stokes equations.
In incompressible flows, the velocity field must satisfy a global, instantaneous divergence-free constraint. The fractional-step method imposes this by solving a Poisson equation for the pressure field. The present work aims at formulating, implementing and evaluating an eigendecomposition-based framework for solving the discretized pressure Poisson equation in direct numerical simulation for non-uniform meshes.
The eigendecomposition-based approach was implemented into CaNS, a CPU- and GPU-based code which solves the discretized pressure Poisson equation in a massively-parallel fashion through an FFT-based approach on a finite-difference scheme. It was validated against lid-driven cavity and turbulent square duct flows, showing good agreement with the data of Gavrilakis.
The CPU-based computational performance of the framework was assessed through strong and weak scaling tests, and compared against results of SNaC, an implementation of an iterative solver for the pressure Poisson equation through geometric multigrid methods. It is shown that performance becomes communication-bound at 4096 cores, with wall-clock times higher than the FFT-based approach and lower than the multigrid approach. This bound has been hypothesized to be a consequence of the small computational domain tested (1024³) and of possible memory fragmentation. In every test performed, regardless of scale, the eigendecomposition-based implementation is shown to have lower wall-clock times than the geometric multigrid implementation, despite its higher operation count.
Finally, the environmental and financial cost of developing the eigendecomposition-based approach was assessed. 525 thousand core hours were spent, representing ≈ 3000 kWh of energy, close to 1 ton of CO2 emitted, ≈ 400€ spent and over 160 hours of elapsed compute time. Deploying this approach in a large-scale environment for solving a turbulent square duct would require 300 to 400 sand core hours.
All in all, the eigendecomposition-based approach here developed and implemented is deemed suitable for performing large-scale simulations on a many-CPU framework, with lower wall-clock times than comparable geometric multigrid methods. Unfortunately, in cases where a uniform mesh can discretize the problem, the FFT-based approach still demonstrates comparatively lower wall-clock times, meaning it should be the preferred approach where suitable. The present framework will be made available in a separate release of CaNS at github.com/CaNS-World.
https://github.com/CaNS-World