Presentacin de PowerPoint

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Numerical simulation of viscoelastic problems
with contact related to aluminum casting
Peregrina Quintela Estévez, Patricia Barral
Rodiño María Teresa Sánchez. Department of
Applied Mathematics. University of Santiago de
Compostela Spain
Workshop in numerical simulations for aluminum
industry Sept. 30, 2004 Oct. 2, 2004 EPFL
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Numerical simulation of viscoelastic problems
with contact related to aluminum casting
The physical problem the casting process. A
mechanical model for simulation of
thermomechanical deformations. A weak
formulation of the problem. Space
discretization. Contact condition Time
integration. Viscoplastic-law. Numerical
procedures to improve the efficiency of the
algorithm. Numerical results.
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The casting process
Electromagnetic casting
Direct chill casting
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The whole casting process
P. Barral A. BermúdezM.C. MuñizV. OteroP.
QuintelaP. Salgado. Numerical simulation of some
problems related to aluminium casting. J. of
Materials and Processing Tech. Vol. 142/2, pp.
383-399. 2003.
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The mechanical simulation
Butt curl
The start phase
Lateral faces contraction
The stationary stage
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The mechanical problem
Unknowns
Boundary conditions
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Boundary conditions
Unknowns
Boundary conditions
Metallostatic pressure on the solidification
front.
Signorini contact condition.
Symmetry conditions
Free of forces
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The mechanical problem
Equilibrium equations
Constitutive law
Hookes law
Arrheniuss law
Norton-Hoffs law
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Mathematical model (Problem P)
Equilibrium equations
Boundary conditions
Drezet, Plata Mariaux, Rappaz
Constitutive law
Initial conditions
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Weak formulation
Weak formulation (WP) .- Find
and

verifying the inequality a.e.
and

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Perturbed problem
Elastic law
Constitutive law
Initial conditions
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Space discretization

• The weak problem is discretized in space using a
  finite element method.
• The approximation of displacements is performed
  using four nodes tetrahedral elements.
• The stresses are assumed to be constant for each
  element.

Perturbed problem discretized in space
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Contact condition
Penalized problem
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Update the multiplier
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Time discretization
Discretization of the constitutive law
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Updated viscoplasticity multiplier
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Flow chart to the viscoplasticity law
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Numerical procedures to improve algorithm
efficiency

• Problems with large spacial stress gradients
  adimensionalization technique to improve a major
  approximation and convergence.
• New unknowns
• New coefficients

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Numerical procedures to improve algorithm
efficiency

• Adimensionalization Dt 1.e-5

• Problems with large gradients with respect to
  time small and variable time step
• Armijo rule on the viscoplastic algorithm.
• Optimization of the time step.

Dt 0.125
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Real casting process
Start stage
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Numerical results
Start stage
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Numerical results
Stationary stage

• Computational difficulties
• The domain grows with time
• Remesh at each time step

The butt curl doesn't influence the contraction
of the lateral faces
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References

• P. Barral P. Quintela. A numerical method for
  simulation of thermal stresses during casting of
  aluminum slabs. Computer Methods in Applied
  Mechanics and Engineering. Vol. 178, pp.69-88.
  1999.
• P. Barral P. Quintela. A numerical algorithm
  for prediction of deformation during the casting
  of aluminum alloy ingots. Finite elements in
  analysis and design Mechanical Engineering. Vol.
  34, pp. 125-143. 2000.
• P. Barral P. Quintela. Numerical analysis of a
  viscoplastic problem with contact condition
  taking place in an aluminium casting. Journal of
  Computational and Applied Mathematics. Vol. 115,
  pp. 63-86. 2000
• P. Barral P. Quintela. Asymptotic justification
  of the treatment of a metallostatic pressure type
  boundary condition in an aluminium casting. Math.
  Mod. Meth. Appl. S. Vol. 11, Nº 6 , pp.
  951-977, 2001.

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References

• P. Barral - P. Quintela. Existence of solution
  for a Signorini contact problem for
  Maxwell-Norton materials. IMA J. of Appl. Math.
  Vol. 67, pp. 525-549. 2002.
• P. Barral P. Quintela. Asymptotic analysis of a
  metallostatic pressure type boundary condition
  modelled by a fictitious domain method in an
  aluminium casting. Asymptotic Analysis. Vol.
  30, pp. 93-116, 2002.
• P. Barral A. BermúdezM.C. MuñizV. OteroP.
  QuintelaP. Salgado. Numerical simulation of some
  problems related to aluminium casting. J. of
  Materials and Processing Tech. Vol. 142/2, pp.
  383-399. 2003.
• P. Barral C. MorenoP. QuintelaM.T. Sánchez. A
  numerical algorithm for a Signorini problem
  associated with Maxwell-Norton materials by using
  Newton's methods. Submitted to Comp. Meth. in
  Appl. Mech. and Eng. 2004.