Analytic Solution to the Piecewise Linear Interface Construction Problem and Its Application in Curvature Calculation for Volume-of-Fluid Simulation Codes

Abstract

The plane–cube intersection problem has been discussed in the literature since 1984 and iterative solutions to it have been used as part of piecewise linear interface construction (PLIC) in computational fluid dynamics simulation codes ever since. In many cases, PLIC is the bottleneck of these simulations regarding computing time, so a faster analytic solution to the plane–cube intersection would greatly reduce the computing time for such simulations. We derive an analytic solution for all intersection cases and compare it to the previous solution from Scardovelli and Zaleski (Scardovelli, R.; Zaleski, S. Analytical relations connecting linear interfaces and volume fractions in rectangular grids. J. Comput. Phys. 2000, 164, 228–237), which we further improve to include edge cases and micro-optimize to reduce arithmetic operations and branching. We then extend our comparison regarding computing time and accuracy to include two different iterative solutions as well. We find that the best choice depends on the employed hardware platform: on the CPU, Newton–Raphson is fastest with compiler optimization enabled, while analytic solutions perform better than iterative solutions without. On the GPU, the fastest method is our optimized version of the analytic SZ solution. We finally provide details on one of the applications of PLIC—curvature calculation for the Volume-of-Fluid model used for free surface fluid simulations in combination with the lattice Boltzmann method.

Accuracy and performance of the lattice Boltzmann method with 64-bit, 32-bit, and customized 16-bit number formats

“…This offers a large benefit, most prominently on FP16 accuracy, by substantially reducing numerical loss of significance at no additional computational cost. Since it is also beneficial for regular FP32 accuracy, it is already widely used in LBM codes such as our FluidX3D [6][7][8][9][10][11][12], OpenLB [68][69][70][71], ESPResSo [24][25][26], Palabos [72][73][74][75][76], and some versions of waLBerla [53]. In Appendix A 2, we provide the entire algorithm without and with DDF-shifting for comparison and in Appendix A 3 we clarify our notation.…”

“…While most LBM implementations are limited to one particular hardware platform-either CPUs [67][68][69][70][71][72][73][74][75][76][77][78][79], Nvidia FP64/FP64 q 16 q 17 q FP64/FP32 q 8 q 9 q FP32/FP32 q 8 q 9 q FP32/16-bit q 4 q 5 q GPUs , CPUs and Nvidia GPUs [18][19][20][21][22][23][24][25][26][27][28][29] or mobile SoCs [122,123]-only few use OpenCL [5][6][7][8][9][10][11][12][13][14][15][16][17]. With FluidX3D also being implemented in OpenCL, we are able to benchmark our code across a large variety of hardware, from the world's fastest data-center GPUs over gaming GPUs and CPUs to even the GPUs of mobile phone ARM SoCs.…”

“…While in the most of this paper, we used the FluidX3D code [6][7][8][9][10][11][12], we also confirmed selected results with the LB3D lattice Boltzmann simulation package [79]. For this, we ported the FP64/FP64 routines to FP32/FP32 also in LB3D.…”

“…This approach requires minimal code interventions and can be easily combined with any velocity set, collision operator, swap algorithm, or LBM extension. In addition to using custombuilt floating-point formats, we show that shifting the DDFs by subtracting the lattice weights and computing the equilibrium DDFs in a specific order of operations as originally proposed by Skordos [66] and further investigated by He and Luo [94] and Gray and Boek [60]-an optimization beneficial across all floating-point formats and already widely used [6][7][8][9][10][11][12][24][25][26]31,35,53,60,66,[68][69][70][71][72][73][74][75][76]88,94]-turns out absolutely crucial for the 16-bit compression.…”

“…Regarding LBM accuracy, we study Poiseuille flow through a cylinder [95], Taylor-Green vortex energy dissipation [66,96], Karman vortices [97] from flow around a cylinder, lid-driven cavity [30,33,37,39,49,52,[98][99][100][101][102][103], deformation of a microcapsule in shear flow [104][105][106] with the immersed-boundary method (IBM) extension, and microplastic particle transport during a raindrop impact [10] with the volume-of-fluid and IBM extensions. Regarding performance, we exploit the capability of our FluidX3D LBM implementation written in OpenCL [6][7][8][9][10][11][12] to provide benchmarks for all floating-point variants on a large variety of hardware, from the world's fastest datacenter GPU over various consumer GPUs and CPUs from different vendors to even a mobile phone ARM system-on-a-chip (SoC), and show roofline analysis [64,87,107] for one hardware example.…”