1 on the use of ma47 solution procedure in the finite element steady state heat analysis ana...
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On The Use Of MA47 Solution Procedure In The Finite Element Steady State Heat AnalysisAna Žiberna and Dubravka MijucaFaculty of MathematicsDepartment of MechanicsUniversity of BelgradeStudentski trg 16 - 11000 Belgrade - P.O.Box 550Serbia and Montenegrowww.matf.bg.ac.yu/~dmijuca
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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
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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
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Keywords
Sparse Matrices Indefinite Matrices Direct Methods Multifrontal Methods Solid Mechanics Steady State Heat Finite Element Large Scale Systems
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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
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Heat Transfer Problem
Temperature T – primal variable Heat Flux q - dual variable k – Material thermal conductivity f – Heat source
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Field Equations
Equation of Balance
Fourrier’s Law
,0; 0iidiv f q f q
,; ( )i ijjT q k T q k
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Boundary Conditions Prescribed Temperature
Prescribed Flux
TT T na
h qq h na q n
0( )c c cq h T T na q n
4 40( )r r rq h T T na q n
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Symmetric weak mixed formulationFind such that and
for all such that
12, ( ) ( )T H L q
TT T
1
q c
cd T d d f d hd q d
qk Q Q q
12, ( ) ( )H L Q 0
T
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Sub-spaces of the FE functionsFOR TEMPERATURE, FLUX AND APPROPRIATE TEST FUNCTIONS
_1
1
1 ( )0
1 ( )
( ) : | , ( ),
( ) : | 0, ( ),
( ) : | , | ( ), ( ),
( ) : | 0, ( ),
T
T
q c
q c
Lh L i i h
Mh M i i h
n Lh c L i i h
n Mh M i i h
T T H T T T T P C
H P C
Q H h h T T V C
H V C
q q n q n q q
Q Q n Q Q
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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:
q Tn n n
0
TTpv vp vpvv vv
p p p pv vp vpvv vv
qq A BA BT F H KT B DB D
0
e
he
ce
M M ee
M M hee
M M c cee
F P f d
H P hd
K P h T d
( ) ( )
( ) ,
e
e
ce
a bLpMr L p L ab M r M e
e
aLpM L p L M a e
e
LM c L M cee
A g V r g V d
B g V P d
D h P P
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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
Both positive and negative eigenvalues
, 0 0T Tx y x x y y A A
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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
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Block pivots
oxo pivot
tile pivot or
structured pivot - either a tile or an oxo pivot
0
0
××
0× ××
0 ×× ×
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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
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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
( 1)( 3)i i jr r r
( 1)( 1)i jr r
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Numerical stability
all the pivots are tested numerically
additional symmetric permutations for the sake of numerical stability
where - threshold parameter
( ) ( )maxk kij lj
la u a
1 ( )( ) ( ) 1, 1
( ) ( ) 1( )1, 1, 1 1,
max
max
kk kljkk k k l
k k kk k k k l j
l
aa a u
a a ua
0 1u
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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
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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
200 1000 10000
0
2000
4000
6000
8000
10000
12000
14000
MA47 Gauss
exec
utio
n tim
e (s
econ
ds)
number of rows
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Relative errors in target points
0 400 800 1200 1600
0.4
0.8
1.2
1.6
PointA PointB PointC
rela
tive
erro
r (%
)
execution time (seconds)0 2000 4000 6000 8000 10000
0.4
0.8
1.2
1.6
PointA PointB PointC
rela
tive
erro
r (%
)
execution time (seconds)
MA47 GAUSS
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Hollow cylinder
100 1000 10000
0102030405060708090
100110120
MA47 Gauss
exec
utio
n tim
e (s
econ
ds)
number of rows
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Future research
Perform matrix scaling to increase accuracy in solution when matrix has entries widely differing in magnitude
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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