1. 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

2. 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

3. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 3 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

4. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 4 Keywords Sparse Matrices Indefinite Matrices Direct Methods Multifrontal Methods Solid Mechanics Steady State Heat Finite Element Large Scale Systems

5. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 5 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

6. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 6 Heat Transfer Problem Temperature T – primal variable Heat Flux q - dual variable k – Material thermal conductivity f – Heat source

7. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 7 Field Equations Equation of Balance Fourrier’s Law

8. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 8 Boundary Conditions Prescribed Temperature Prescribed Flux

9. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 9 Symmetric weak mixed formulation Find such that and for all such that

10. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 10 Sub-spaces of the FE functions FOR TEMPERATURE, FLUX AND APPROPRIATE TEST FUNCTIONS

11. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 11 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:

12. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 12 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

13. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 13 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

14. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 14 Block pivots oxo pivot tile pivot or structured pivot - either a tile or an oxo pivot

15. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 15 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

16. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 16 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

17. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 17 Numerical stability all the pivots are tested numerically additional symmetric permutations for the sake of numerical stability where - threshold parameter

18. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 18 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

19. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 19 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

20. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 20 Relative errors in target points MA47 GAUSS

21. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 21 Hollow cylinder

22. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 22 Future research Perform matrix scaling to increase accuracy in solution when matrix has entries widely differing in magnitude

23. 28-Sep-04 XI Congress of Mathematics of Serbia and Montenegro 23 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