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A Fast Multifrontal Solver for Non-Linear Multi-Physics Problems
[A. Bertoldo, M. Bianco, G. Pucci] {cyberto, bianco1, geppo} @dei.unipd.it
ACGADVANCED COMPUTING GROUP
DEI - UNIVERSITY OF PADOVA
The multifrontal approachUnifrontal
Sequential assemblystrategy
Recursive assembly strategyin a bottom-up fashion
Multifrontal
June 6-9, 2004Krakow, Poland
Regions
Mesh region E
An elementaryregion
A compositeregion
At each node:1) Assemble the region2) Eliminate fully-summed variables
The assembly tree
Leaves: The assembly phase gets elements from the FEM formulation
At the root we havefully decomposed the linear system into LU factors
Strip phaseIII.Strip factors away for
subsequent use in the finalbackward and forward substitution
Elimination phaseII.Eliminate FS variables
using a blocked UL decomposition
BLASLevel 3
0.0
59
0.6
27
9
1.1
54
7.1
36
0.0
6
0.9
9
2.0
72
23
.26
4
0.1
14
0.6
80
2
1.3
57
12
.59
9
0.0
53
9
1.0
62 4.5
84
33
.19
4
0
5
10
15
20
25
30
35
100 400 625 2500
Number of elements
Ex
ec
uti
on
tim
es
(s
)
PMMS
Unifrontal
MUMPS
SuperLU
31
2
49
4
51
8
66
6
32
5
58
4 64
2
84
6
45
0
55
3 60
6
86
0
74
28
4
31
7 35
4
0
100
200
300
400
500
600
700
800
900
1000
100 400 625 2500
Number of elements
Flo
ps
ra
te (
MF
LO
PS
)
PMMS
Unifrontal
MUMPS
SuperLU
Execution time Flops
PMMS is faster than Unifrontal solver,but they have the same solving kernel
The multifrontal assembly scheme is more efficient
PMMS is faster than both SuperLU andMUMPS for all significant problem sizes
The super-assembly phase and theuse of BLAS boosts the computation
The larger is the problem size, the faster is PMMS with respect to the other solvers
For larger test cases we expect abigger performance improvement
MUMPS and Unifrontal solver exhibitlarger flop rates than PMMS does
They are better tuned but theiralgorithm has higher complexity
Performance ResultsAlgorithm of super-assembly phase
MIUR Center for Science and Applicationof Advanced Computing Paradigms at the University of Padova, Italy
Consorzio Roma RicercheRoma, ItalyInternational Centre
for Mechanical Sciences,Udine, Italy
Ente per le Nuove tecnologie,l’Energia e l’Ambiente, Roma, Italy
Department of InformationEngineering University of Padova, Italy
? IBM Power3@375MHz with 4 GB mem? HPM Toolkit for performance measurements? FE square meshes with 100, 400, 625, and 2500 square
8-node elements
? PMMS is our multifrontal solver? SuperLU version 3.0? MUMPS version 4.3
Assembly phaseIa.
Merge the two reduced components into a new
composite region
Swap phaseIb.
Pack FS rows and columnsat the bottom-right cornerof the non-reduced region
Copy phaseIc.
Copy FS blocks intotemporary buffers
+ +
= Super-assembly phase
Computation properties
Symbolicanalysis
Symbolic dataThese data are computed once at the beginning of
the computation, but used at each iteration to performthe super-assembly phase.
Assembly treetopology
Finite elementmesh data
Simulation of porous mediaunder high temperature
Area of interest
Non-linear coupledmulti-physics problem
Large non-linearsystems of PDEs
LARGE LINEAR SYSTEMS
FEM
Physical model
Mathematical model
Linearization
Our Goal
Main features of our solver
Other specific features:F Multifrontal assembly/elimination strategyF Implicit minimum degree pivotingF Symbolic preprocessing phaseF Super-assembly phaseF Blocked LU decomposition for elimination
+Frontal solvers are direct methods since they first transform the system using Gaussian elimination or LU decomposition, then get the final solution using forward and backward substitution.+They do not operate on the completely
assembled linear system, but rather interleave assembly phases with elimination phases.+They require low memory space and can exploit
efficient dense linear algebra kernels.
Regions can be both elementary and composite. The former are obtained from the finite element formulation. The latter are unions of two component regions from an assembly phase.
The “spalling” phenomenon in a concrete-made pillar
after a simulated fire.
The FE simulation of the “spalling”phenomenon in a (section of)
concrete-made pillar in case of fire
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