A portable coding strategy to exploit vectorization on combustion simulations

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A portable coding strategy to exploit vectorization on

combustion simulations

Fabio Banchellia, Guillermo Oyarzuna,∗

, Marta Garcia-Gasullaa, Filippo

Mantovania, Ambrus Botha, Guillaume Houzeauxa, Daniel Miraa

aBarcelona Supercomputing Center, Plaza Eusebi Guell, 1-3, 08034 Barcelona (Spain)

Abstract

The complexity of combustion simulations demands the latest high-performance

computing tools to accelerate its time-to-solution results. A current trend on

HPC systems is the utilization of CPUs with SIMD or vector extensions to ex-

ploit data parallelism. Our work proposes a strategy to improve the automatic

vectorization of ﬁnite-element-based scientiﬁc codes. The approach applies a

parametric conﬁguration to the data structures to help the compiler detect the

block of codes that can take advantage of vector computation while maintain-

ing the code portable. A detailed analysis of the computational impact of this

methodology on the diﬀerent stages of a CFD solver is studied on the PREC-

CINSTA burner simulation. Our parametric implementation has proven to help

the compiler generate more vector instructions in the assembly operation: this

results in a reduction of up to 9.39×of the total executed instruction maintain-

ing constant the Instructions Per Cycle and the CPU frequency. The proposed

strategy improves the performance of the CFD case under study up to 4.67×

on the MareNostrum 4 supercomputer.

Keywords: vectorization, high performance computing, combustion

simulations, performance analysis

∗Corresponding author

Email address: guillermo.oyarzun@bsc.es (Guillermo Oyarzun)

Preprint submitted to Computers & Fluids October 24, 2022

arXiv:2210.11917v1 [cs.DC] 21 Oct 2022

1. Introduction and related work

The decarbonization of the transportation sector is one of the ﬁelds with

high strategic importance for our society [1,2]. Implementing new greener fu-

els in real combustion systems demands advanced combustion simulations, as

their physical and chemical properties are expected to be signiﬁcantly diﬀerent

from those of conventional transportation fuels [3]. In such complex simula-

tions, the investigation of more accurate and eﬃcient numerical algorithms is of

key importance to increase the accuracy and reduce the time-to-solution. The

diﬃculty relies on the constant evolution of the High-Performance Computing

(HPC) systems. Consequently, scientiﬁc software requires periodic updates to

exploit the new features and run eﬃciently on those systems.

On modern CPUs, the use of vector or Single Instruction Multiple Data

(SIMD) extensions is becoming more and more relevant. Beside the AVX-512

SIMD extension by Intel, we detect appearing on the market the ﬁrst CPU im-

plementing the Arm SVE extension (Fujitsu A64FX, ranked ﬁrst in the Top500)

and the NEC SX-Aurora vector engine, a discrete accelerator leveraging vector

CPUs able to operate with registers of up to 256 double precision elements. On

top of this market movements, we can not ignore the RISC-V architecture which

recently ratiﬁed v1.0 of the V-extension, boosting vector computation from the

academic world and the open-source community.

The eﬃcient use of vector units within CPUs relies on auto-vectorization by

the compiler and often requires to adapt or rewrite classical algorithms to exploit

their full computing power [4]. Large-scale CFD codes are generally dominated

by two operations: the linear solver and the matrix assembly. The ﬁrst can be

considered a black-box component that receives a matrix and a right-hand-side

as an input and returns a solution vector [5]. The solver is composed of al-

gebraic operations that can exploit vectorization by using speciﬁc libraries [6].

This strategy allows to port a part of large scientiﬁc codes to vector accelerators

in a relatively smooth way [7]. Regarding the matrix assembly, the algorithm

for unstructured meshes depends on the discretization method, where ﬁnite vol-

ume (FV) or ﬁnite elements (FE) are the most common strategies. Obtaining

gains from vectorization in FV assembly requires introducing changes that have

proved not practical on large-scale combustion codes [8,9]. On the contrary, the

FE assembly is constituted by matrix-like structures with the potential applica-

tion of SIMD-friendly functions [10]. Our work is implemented on Alya [11], a

large-scale computational mechanics code (FE-based) that is one of the thirteen

Uniﬁed European Applications Benchmark Suite codes. We propose and ana-

lyze a parametric conﬁguration to its data structure, allowing the compiler to

enable auto-vectorization. We evaluate the proposed implementation on a state-

of-the-art supercomputer, MareNostrum 4, powered by Intel Skylake CPUs. We

show that Alya takes advantage of AVX-512 SIMD units present in the Skylake

CPUs while keeping the code portable. The strategy is extensible to any other

FE-based code.

The main contributions of this paper are: i) we propose a parametric conﬁg-

uration of the data structure for a complex ﬂuid-dynamic code; ii) we measure

and explain the impact of the proposed conﬁguration from a computational

point of view; iii) we quantify the overall performance gain on a state-of-the-art

HPC supercomputer.

The remaining part of the paper is structured as follows: Section 2sum-

marizes the computational ﬂuid-dynamics problem solved with Alya; Section 3

brieﬂy presents the technological context of the study performed in this paper,

including details of the hardware and software conﬁgurations. Section 4ana-

lyzes the optimizations applied to Alya in terms of execution time, instruction

mix and cache eﬀects to quantify the overall performance gain. Section 5closes

the paper with general remarks and conclusions.

2. Application context

2.1. Governing equations

The governing equations describing the reacting ﬂow ﬁeld in the turbu-

lent premixed ﬂame correspond to the low-Mach number approximation of the

Navier-Stokes equations with the energy equation represented by the total en-

thalpy. The combustion process is assumed to take place in the ﬂamelet regime

and the ﬂamelet database is based on the tabulation of a laminar premixed

ﬂamelet at constant equivalence ratio that uses the chemistry from the San

Diego mechanism [12]. A Favre-ﬁltered description of the governing equations

is followed to avoid modelling of terms including density ﬂuctuations [13]. The

governing equations are given by:

∂ρ

∂t +∇ · (ρe

u) = 0 (1)

∂(ρe

∂t +∇ · (ρe

u) = −∇p+∇ · ρ(ν+νt)2S−2

3(∇e

u)I (2)

∂ρe

h

∂t +∇ · ρe

h=∇ · ρD+νt

P rt∇e

h(3)

where ρ,t,eu,p,ν,e

hand Drepresent the density, time, velocity vector, pressure,

kinematic viscosity, total enthalpy and thermal diﬀusion coeﬃcient respectively.

Heating due to viscous forces is neglected in the enthalpy equation and the

unresolved heat ﬂux is modelled using a gradient diﬀusion approach [14]. The

formulation is closed by an appropriate expression for the subgrid-scale or eddy-

viscosity νtthat in this study is deﬁned by the closured proposed by Vreman

[15] with a model constant of cs= 0.1. The viscous stress tensor is deﬁned

based on Stokes’ assumption and the turbulence contribution is determined by

the use of the Boussinesq approximation [13], in which S=1

2h∇e

u+ (∇e

u)Ti

and Iare the strain and the identity tensor respectively. A unity Lewis number

assumption has been made to simplify the multicomponent transport in the

governing equations. Turbulent Schmidt and Prandtl numbers are both set

constant with value of 0.7.

For the present combustion model, a controlling variable based on a reactive

scalar is used to couple the chemical states with the ﬂuid ﬂow. This controlling

variable can be understood as a progress variable Ycthat is used to describe

the thermochemical state from an unreacted mixture to a fully reacted mixture.

For numerical reasons [16], a scaled progress variable cis deﬁned as:

c=Yc−Y0

Yeq

c−Y0

(4)

where Y0

cand Yeq

care the values of the progress variable of the unreacted

mixture and at equilibrium conditions respectively. Considering the application

of this ﬂamelet combustion model to premixed combustion in LES, the subscale

eﬀects need to be addressed. The tabulated properties ψare integrated with

a presumed-shape probability density function (PDF) that is constructed from

the ﬁltered progress variable ecand the subgrid variance f

c002=ecc −ececusing

aβ-function [16]. A closure for the subgrid scale variance is provided by the

solution of the transport of f

c002following Domingo et al. (2005) [16].

The chemical state of the perfectly premixed ﬂame in the LES framework is

ultimately described by the two controlling variables: ecand f

c002, so the governing

equations describing the chemical evolution of the ﬂame are given by:

∂(ρec)

∂t +∇ · (ρe

uec) = ∇ · ρD+νt

Sct∇ec+ ˙ωc(5)

∂ρf

c002

∂t +∇ · ρe

c002=∇ · ρe

D+νt

Sct∇f

c002(6)

+ 2ρe

D|∇ec|2(7)

+ 2 c˙ωc−ec˙ωc(8)

−ρeχc(9)

where eχcrepresents the scalar dissipation rate and ˙ωcis the ﬁltered source term

of the progress variable. The scalar dissipation rate is composed by the resolved

and unresolved parts, which are given by:

eχc= 2 e

D|∇ec|2+χsgs

c= 2 e

D|∇ec|2+Cd

τtf

c002

where τtis a turbulent time scale, which is obtained following Ventosa et al. [17].

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AportablecodingstrategytoexploitvectorizationoncombustionsimulationsFabioBanchellia,GuillermoOyarzuna,,MartaGarcia-Gasullaa,FilippoMantovania,AmbrusBotha,GuillaumeHouzeauxa,DanielMiraaaBarcelonaSupercomputingCenter,PlazaEusebiGuell,1-3,08034Barcelona(Spain)AbstractThecomplexityofcombustionsimulatio...

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A portable coding strategy to exploit vectorization on combustion simulations

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