Adaptive mesh refinement (AMR) provides an attractive means of significantly reducing computational costs while simultaneously maintaining a high degree of fidelity in regions of the domain requiring it. In the present work, an analysis of the performance of AMR supported by simulations is undertaken for liquid injection and spray formation problems. These problems are particularly challenging from a computational cost perspective since the associated interfacial area typically grows by orders of magnitude, leading to similar growth in the number of highly refined cells. While this increase in cell numbers directly contributes to a declining performance for AMR, a second less obvious factor is the decaying trend for the cell-based speedup, $\Theta$. A theoretical analysis is presented, leading to a closed-form estimate for this cell-based speedup, namely ${\Theta }_E=\sqrt{{\kappa }_{F,SM}}/\sqrt{{\kappa }_{F,AMR}},$ where ${\kappa }_F$ is the Frobenius condition number, and SM corresponds to a static mesh case. It is shown that for spray formation problems, the typical growth in ${\kappa }_{F,AMR}$ is more pronounced than ${\kappa }_{F,SM}$ causing a decline in $\Theta$ and consequently diminishing the AMR performance. Additional contributing sources are also examined, which include the role of load balancing and the choice of linear solvers for the Poisson system.

自适应网格细化（AMR）提供了一种有吸引力的方法，可以显着降低计算成本，同时在需要它的域的区域中保持高度的保真度。在目前的工作中，针对液体注入和喷雾形成问题，对由模拟支持的AMR性能进行了分析。从计算成本的角度来看，这些问题尤其具有挑战性，因为相关的界面区域通常会增长几个数量级，从而导致高度精炼单元的数量出现相似的增长。虽然细胞数量的增加直接导致AMR的性能下降，但第二个不太明显的因素是基于细胞的加速的下降趋势，$\Theta$。进行了理论分析，得出了这种基于单元格的加速的闭式估计，即${\Theta }_E=\sqrt{{\kappa }_{F，\mathrm{小号}\mathrm{中号}}}/\sqrt{{\kappa }_{F，\mathrm{一种}\mathrm{中号}\mathrm{[R}}}，$ 在哪里 ${\kappa }_F$是Frobenius条件编号，SM对应于静态网格物体。结果表明，对于喷雾形成问题，典型的增长是${\kappa }_{F，\mathrm{一种}\mathrm{中号}\mathrm{[R}}$ 比 ${\kappa }_{F，\mathrm{小号}\mathrm{中号}}$ 导致下降 $\Theta$因此会降低AMR的性能。还检查了其他贡献来源，包括负载平衡的作用和Poisson系统的线性求解器的选择。