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On The Use Of MA47 Solution Procedure In The Finite Element Steady State Heat Analysis

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.

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On The Use Of MA47 Solution Procedure In The Finite Element Steady State Heat Analysis

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  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. 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 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 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 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 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 XI Congress of Mathematics of Serbia and Montenegro

  7. Field Equations • Equation of Balance • Fourrier’s Law XI Congress of Mathematics of Serbia and Montenegro

  8. Boundary Conditions • Prescribed Temperature • Prescribed Flux XI Congress of Mathematics of Serbia and Montenegro

  9. Symmetric weak mixed formulation Find such thatand for all such that XI Congress of Mathematics of Serbia and Montenegro

  10. Sub-spaces of the FE functionsFOR TEMPERATURE, FLUX AND APPROPRIATE TEST FUNCTIONS 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: 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 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 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 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 offactorization 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

  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 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 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 performingforward and backward substitution 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 XI Congress of Mathematics of Serbia and Montenegro

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

  21. Hollow cylinder 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 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 XI Congress of Mathematics of Serbia and Montenegro

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