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

1

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

28-Sep-04 XI Congress of Mathematics

of Serbia and Montenegro 2

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



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



Keywords

Sparse Matrices Indefinite Matrices Direct Methods Multifrontal Methods Solid Mechanics Steady State Heat Finite Element Large Scale Systems



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



Heat Transfer Problem

Temperature T – primal variable Heat Flux q - dual variable k – Material thermal conductivity f – Heat source



Field Equations

Equation of Balance

Fourrier’s Law

,0; 0iidiv f q f q

,; ( )i ijjT q k T q k



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



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



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



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



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



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



Block pivots

oxo pivot

tile pivot or

structured pivot - either a tile or an oxo pivot

0

0

××

0× ××

0 ×× ×



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



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



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



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



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



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



Hollow cylinder

100 1000 10000

0102030405060708090

100110120

MA47 Gauss

exec

utio

n tim

e (s

econ

ds)

number of rows



Future research

Perform matrix scaling to increase accuracy in solution when matrix has entries widely differing in magnitude



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

Documents

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