Transcript Slide 1
On The Use Of MA47 Solution Procedure In The Finite Element Steady State Heat Analysis
Ana Žiberna and Dubravka Mijuca Faculty of Mathematics Department of Mechanics University of Belgrade Studentski trg 16 - 11000 Belgrade - P.O.Box 550 Serbia and Montenegro www.matf.bg.ac.yu/~dmijuca 1
Physical problem
The steady state heat analysis problem in solid mechanics Novel mixed finite element approach (saddle point problem) on the contrary to the frequently used primal approach (extremal principle) Simultaneous simulations of both field variables of interest : temperature
T
and heat flux
q
Any numerical procedure of analysis which threats all variable of interest as fundamental ones (in the present case temperature and heat flux) is more reliable and more convenient for real engineering application Additional number of unknowns raise the need for reliable and fast solution procedure 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 2
Present Scheme
The adjusted large linear system of equations solver MA47 is used The basic motive for the use of the MA47 method is found in the fact that it is primarily designed for solving system of equations with symmetric, quadratic, sparse, indefinite and large system matrix The method is based on the multifrontal approach (frontal methods have their origin in the solution of finite element problems in structural mechanics) Achieving better efficiency XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 3
Keywords
Sparse Matrices Indefinite Matrices Direct Methods Multifrontal Methods Solid Mechanics Steady State Heat Finite Element Large Scale Systems XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 4
Aim
Aim of this presentation is a preliminary validation of the new solution approach in the mixed finite element steady state heat analysis, its effectiveness and reliability 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 5
Heat Transfer Problem
Temperature
T
– primal variable Heat Flux
q
- dual variable
k
– Material thermal conductivity
f
– Heat source 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 6
Field Equations
Equation of Balance
div
q
f
0;
q
,
i i
f
0
Fourrier’s Law
28-Sep-04
q k
T
;
q i
(
ij k T
,
j
) XI Congress of Mathematics of Serbia and Montenegro 7
Boundary Conditions
Prescribed Temperature
T
T na
T
Prescribed Flux
q h
h na
q q c
c
T
0 )
na
c q r
r
4
T
0 4 )
na
r
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Symmetric weak mixed formulation
1
qk Q
d
L
2
Q
d
q
H
1
L
2
f d T
T
T
q
and
c q c
T
0 XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 9
Sub-spaces of the FE functions FOR TEMPERATURE, FLUX AND APPROPRIATE TEST FUNCTIONS
T h
T
H
1
Q h
h h
q Q
H
1
H
1
H
1
T
|
T
_
L T P L
( ),
i i C h
: : |
T
| |
q
h
,
M P M
( |
c
c
0,
Q
i c
Q
M V M
(
i C h
T
0 ),
q
i
i
q
L V L
( ),
i i C h C h
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System Matrix
after discretization of the starting problem, writing in componential form and separating by temperature and flux test functions we obtain a system of order:
n
n q
n T
A B vv vv B vv T
D vv
q v
A vp
B vp
B vp T D vp
q
F p
0
H p K p
A LpMr B LpM D LM
e
e
e
e
e
ce g a g a V r g L ab
(
b V P L h P P c L M d
e
ce V d M
e F M H M K M
e
e e
he
e
ce
e P hd M
he P h T d M c
0
ce
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Symmetric Sparse Indefinite Systems
A matrix is sparse if many of its coefficient are zero There is an advantage in exploiting its zeros A matrix is indefinite if there exists a vector x and vector y such that
x
T
A
x
0
y T
A
y
0 Both positive and negative eigenvalues XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 12
MA47 from HSL
The
Harwell Subroutine Library
(HSL) is an ISO Fortran Library packages for many areas in scientific computations. It is probably best known for its codes for the direct solution of sparse linear systems Written by
I. S. Duff and J. K. Reid
, represents a version of sparse Gaussian elimination, which is implemented using a multifrontal method Follows the sparsity structure of the matrix more closely in the case when some of the diagonal entries are zero Provide a stable factorization by using a combination of 1x1 and 2x2 pivots from the diagonal XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 13
Block pivots
oxo
pivot
tile
× ×
0 × × 0
× 0
0 ×
structured
pivot - either a tile or an oxo pivot
× ×
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Maintaining sparsity
crucial requirement (perhaps the most crucial) in the elimination process -
we want factors to be also sparse
process of factorization causes so called
fill-ins
(generation of new nonzero entries) no efficient general algorithms to solve this problem are known there are some algorithms used to reduce the number of fill-ins XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 15
Markowitz algorithm
most commonly used and quite successful we use the variant of the
Markowitz criterion
Markowitz measure of fill-ins in
k
-th stage of elimination process for a tile pivot for an oxo pivot (
r i
1)(
r i
j
3) (
r i
1)(
r j
1) XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 16
Numerical stability
all the pivots are tested numerically additional symmetric permutations for the sake of numerical stability
a ij
u
max
l a lj
(
a kk k
)
a k
(
k
) 1,
k a
(
k
) 1
a k
(
k
) 1,
k
1 1 max
l
max
l a lj
(
k
)
a l
(
k
) 1,
j
u
1
u
1
threshold parameter
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Principal Phases of code
ANALYSE -
the matrix structure is analysed to produce a suitable ordering, determine a good pivotal sequence and prepare data structures for efficient factorization
FACTORIZE –
numerical factorization is performed using the chosen pivotal sequence
SOLVE -
the stored factors are used to solve the system performing forward and backward substitution XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 18
Numerical example
Multi-material hollow sphere Performance has been examined on the PC configuration Pentium IV on 2.4 GHz, 2GB RAM, SCSI HDD 2x36GB 14000 12000 10000 8000 6000 4000 2000 0 200 1000 number of rows XI Congress of Mathematics of Serbia and Montenegro MA47 Gauss 10000 28-Sep-04 19
Relative errors in target points
MA47 GAUSS 1.6
1.2
0.8
0.4
0 400 800 1200 execution time (seconds) PointA PointB PointC 1600 1.6
1.2
0.8
0.4
0 2000 4000 6000 execution time (seconds) PointA PointB PointC 8000 10000 XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 20
Hollow cylinder
120 110 100 90 80 70 60 50 40 30 20 10 0 100 28-Sep-04 MA47 Gauss 1000 number of rows XI Congress of Mathematics of Serbia and Montenegro 10000 21
Future research
Perform matrix scaling to increase accuracy in solution when matrix has entries widely differing in magnitude 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 22
References
Duff, I. S., Erisman, A. M., and Reid, J. K. (1986). Direct methods for sparse matrices. Oxford University Press, London.
Duff, I. S. and Reid, J. K. (1983). The multifrontal solution of indefinite sparse symmetric linear systems. ACM Trans. Math. Softw. 9, 302-325.
Bunch, J. R. and Parlett, B. N. (1971). Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal. 8, 639-655. Duff, I. S., Gould, N. I. M., Reid, J. K., Scott, J. A. and Turner, K. (1991). The factorization of sparse symmetric indefinite matrices. IMA J. Numer. Anal. 11, 181-204.
Dubravka M. MIJUCA, Ana M. ŽIBERNA & Bojan I. MEDJO(2004). A New multifield finite element method in steady state heat analysis. Thermal Science, Vinca A.A. Cannarozzi, F. Ubertini (2001)
A mixed variational method for linear coupled thermoelastic analysis
, International Journal of Solids and Structures 38, 717-739 J. Jaric, (1988) Mehanika kontinuuma
,
Gradjevinska knjiga, Beograd XI Congress of Mathematics of Serbia and Montenegro 28-Sep-04 23