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

• Ana iberna 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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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
• Fourriers Law

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Boundary Conditions

• Prescribed Temperature
• Prescribed Flux

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Symmetric weak mixed formulation

• Find such that
  and
• for all such that
  

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Sub-spaces of the FE functionsFOR TEMPERATURE,
FLUX AND APPROPRIATE TEST FUNCTIONS
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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

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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

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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

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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

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Numerical stability

• all the pivots are tested numerically
• additional symmetric permutations for the sake of
  numerical stability
• where -
  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

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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

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Relative errors in target points
MA47
GAUSS
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Hollow cylinder
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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. IBERNA 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