A restricted additive smoother for finite cell flow problems

TL;DR

The paper tackles ill-conditioning in finite cell flow problems caused by cut cells by developing an adaptive geometric multigrid solver for the Stokes equations. At its core is a restricted additive smoother, defined as $\mathbf{S}=\sum_{i=1}^{n_p} (\tilde{\mathbf{R}}_{i}^{T} \boldsymbol{\omega}_{i} \mathbf{L}_{i}^{-1} \mathbf{R}_{i})$, which enables exact parallel replication with minimal communication. Three cache policies—$\text{cache\_matrix}$, $\text{cache\_inverse}$, and $\text{cache\_none}$—balance memory footprint and on-the-fly computation, with $\text{cache\_inverse}$ delivering the best runtime in practice. The GMG solver maintains iteration counts that are bounded independently of problem size and grid depth, and demonstrates strong and weak scalability on large-scale problems up to over $6.65\times10^{8}$ DoF. This approach provides an efficient and scalable option for solving large finite cell flow problems on massively parallel HPC systems.

Abstract

In this work, we propose an adaptive geometric multigrid method for the solution of large-scale finite cell flow problems. The finite cell method seeks to circumvent the need for a boundary-conforming mesh through the embedding of the physical domain in a regular background mesh. As a result of the intersection between the physical domain and the background computational mesh, the resultant systems of equations are typically numerically ill-conditioned, rendering the appropriate treatment of cutcells a crucial aspect of the solver. To this end, we propose a smoother operator with favorable parallel properties and discuss its memory footprint and parallelization aspects. We propose three cache policies that offer a balance between cached and on-the-fly computation and discuss the optimization opportunities offered by the smoother operator. It is shown that the smoother operator, on account of its additive nature, can be replicated in parallel exactly with little communication overhead, which offers a major advantage in parallel settings as the geometric multigrid solver is consequently independent of the number of processes. The convergence and scalability of the geometric multigrid method is studied using numerical examples. It is shown that the iteration count of the solver remains bounded independent of the problem size and depth of the grid hierarchy. The solver is shown to obtain excellent weak and strong scaling using numerical benchmarks with more than 665 million degrees of freedom. The presented geometric multigrid solver is, therefore, an attractive option for the solution of large-scale finite cell problems in massively parallel high-performance computing environments.

Figures (3)

• Figure 1: The convergence of the geometric multigrid solver in the mesh study, where the grid hierarchy in Table \ref{['tab:grid']} is solved. $\Omega_{e,h}^{l}, l = 2 \ldots 7$ denotes the fine problem. All problems employ $\Omega_{e,h}^{1}$ as the coarse grid; therefore, finer problems use a deeper grid hierarchy. The reduction of the relative residual by $10^{9}$ is used as the convergence criterion
• Figure 2: The strong scaling of the geometric multigrid solver using different cache policies in the channel flow benchmark with $\Omega_{e,h}^{5}$ as the fine grid, see Table \ref{['tab:grid']} with $n_{\texttt{proc}} = 1, \ldots, 512$. $t_{\mathbf{S}}$ denotes the runtime of the smoother per iteration on the fine grid, and $t_{\text{sol}}$ denotes the total solver runtime including the setup time. We note that the required memory by the cache_matrix policy exceeds the available main memory of up to two compute nodes
• Figure 3: The weak scaling of the geometric multigrid solver in the channel flow benchmark using the grid hierarchy in Table \ref{['tab:grid']}, where the number of degrees of freedom per process is kept roughly constant. $t_{\mathbf{S}}$ denotes the runtime of the smoother per iteration on the fine grid, and $t_{\text{sol}}$ denotes the total solver runtime