11th Symposium

学際大規模情報基盤共同利用・共同研究拠点公募型共同研究 平成31年度採択課題

jh190022-NAJ鈴木 厚 (大阪大学 サイバーメディアセンター)

High performance simulations using FreeFem++ on mixed distributedplus shared-memory architecture
abstract

member of the project

We are focused on the finite element method to discretize partial differentialequations (PDE). FreeFem++ developed by F. Hecht in LJLL, UPMC is a powerfultool to describe weak formulation of the PDE and it can connect to modern domain
decomposition solver, “HPDDM” (by P. Jolivet) for distributed parallelism. The aimof this project is to combine efficient direct solver “Dissection” (by A. Suzuki), inside
one node through shared-memory parallelism and HPDDM with “GenEO”
preconditioner for numerical robustness of mixed direct- and iterative solver.

Atsushi Suzuki : Cybermedia Center, Osaka University
Pierre Jolivet :CNRS, ENSEEIHT, France
Daisuke Furihata : Cybermedia Center, Osaka University

target systemoctopus (Intel Xeon Gold 6126), NEC SX-ACE@ Cybermedia Center, Osaka University

Numerical simulation by FreeFem++target problem
Domain specific language FreeFem++is designed to solve nonlinear partial
differential equations for industrial
problem, e.g., fluid-structure interaction,
fluid-thermal coupling, semi-conductor,
electro-magnetic problems.

Rayleigh-Benard thermal convection

computing resource, and strategy for linear solver
Time dependent nonlinear system needs to be solved with internal linear solver
for large sparse matrix with high condition number.
Linear system consists of 10 to 100 million DOF, which is arising from adaptive
mesh refinement technique to 1003-based mesh for 5 or 7 physical components.

definition of a stiffness matrix by weak formulation in
Newton iteration for Rayleigh-Benard thermal convection
equation (stationary Navier-Stokes eqs. + heat equation)

iterative substructuring-type domain decomposition method
Neumann-Neumann preconditioner
local stiffness matrix
with Neumann b.c.
local Schurcomplement
inverse on

when subdomain
is floating

Balancing Neumann-Neumann preconditioner
kernel of the local Schur complement
consists of zero energy mode (e.g. rigid
body modes for elasticity problem)coarse space is enriched by spectral information
with eigenvalue and egienvectorof a generalized eigenvalue problem

coarse space is used to accelerate
convergence of the interface probem

global Schurcomplement
interface problem

GenEO preconditioner

ARPACK is used to compute the eigenvalue
problem with dense local Schur compelemt.
local Schur complement needs to beformed explicitly for eigenvalue analysis

no necessary to form Schur complement nor
BNN preconditioner

zero eigenvalue corresponds tozero energy mode

improvement of sparse direct solver for subdomain problem
local Schur complement matrix is explicitly computed with two
forward substitutions of sparse multiple right-hand sides

GenEO with enriched coarse space converges faster than standard
BDD with coarse space of zero energy mode, but construction of
Schur complement consumes more time than solution phase.

entries of sparse RHS locate
on the interface of the local
subdomain.

forward substitution contributes
only on half of the bisection tree.

学際大規模情報基盤共同利用・共同研究拠点 第11回シンポジウム
2019年 7月 11日，12日

THE GRAND HALL （品川）