Youssef Belhamadia et Andr Fortin

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Modélisation Numérique des problèmes de
Cryochirurgie INRIA 2003, Paris

• Youssef Belhamadia et André Fortin
• GIREF

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Liver cryosurgery
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Prostate cryosurgery
J.C. Bischof et Al, Cryobiology (1997)
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Cryosurgery
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Plan

• Introduction
• Stefans problem
• Enthalpy and semi-phase-field formulations
• Weak formulation
• Adaptive strategy
• Numerical results 2D
• Analytical solution
• Formation of a cusp
• Oscillating source problem
• Numerical results 3D
• Oscillating sphere
• Formation of a cusp
• Oscillating source problem
• Applications to cryosurgery
• Conclusions

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Introduction

• Motivation of this work part of a large project
  on cryosurgery (SKALPEL-ITC)
• Idea insert cryoprobes in a tumor. The
  freezing-thawing (phase change) will destroy
  cancerous cells.
• Our goal numerical study of phase change
  problems with an accurate prediction of the
  liquid-solid interface.

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• Stefans problem

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Enthalpy vs temperature
Enthalpy
Temperature
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Enthalpy formulation

• Introducing the enthalpy H as a function of T
• H-T formulation

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Vertical interface
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Vertical interface
Enthalpy at t1
Temperature at t1
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Enthalpy and semi-phase-field
where
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Semi-phase-field formulation

• formulation
• Initial and (time-dependent) boundary conditions

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Weak formulation

• Implicit Euler
• Find and
  such that

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Vertical interface
Regular mesh
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Mesh adaptation

• Difficult to capture the freezing interface on a
  uniform mesh
• In 3D, the number of elements necessary is huge
• The position of the interface evolves with time
• Time-dependent mesh adaptation

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Mesh adaptation

• 2D
• Hierarchical error estimator
• From a numerical solution of degree k, find a
  correction of degree k1
• 3D
• Definition of a solution dependent metric
• Valid only for linear solution
• Error related to the Hessian matrix

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Time dependent mesh adaptation

• Starting from on mesh
• Solve the system on mesh to obtain a
  first approximation
• Adapt the mesh on and
• to get mesh
• Reinterpolate on mesh
• Solve the system on mesh to get the
  solution

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Vertical interface
Adapted meshes
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Oscillating cylinder (2D)
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Oscillating cylinder
Initial mesh
Adapted meshes
8192 elements
around 3000 elements
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Analytical and numerical solutions
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Formation of a cusp
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Formation of a cusp
Adaptation on
Adaptation on and
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Formation of a cusp
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Oscillating source problem
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Oscillating source problem
Adaptation en
Adaptation en et
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Oscillating source problem
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Oscillating sphere
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Oscillating sphere (3D)
Exact (red) and numerical (blue) solutions
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Formation of a cusp (3D)
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Cusp regular meshes
279936 elements
105456 elements
584016 elements
2239488 elements
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Cusp adapted meshes
Adaptation on only
28 945 elements
Adaptation on
92 973 elements
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Source problem
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Rectal protection during prostate cryosurgery
J.C. Bischof et Al, Cryobiology (1997)
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Rectal protection during prostate cryosurgery
J.C. Bischof et Al, Cryobiology (1997)
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Liver cryosurgery
J.C. Bischof et Al, Cryobiology (1997)
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Liver cryosurgery
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Liver cryosurgery
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Conclusions

• The semi-phase-field formulation gives good
  results for phase change problems in 1D, 2D and
  3D problems
• Even better results when combined with mesh
  adaptation
• Mesh adaptation is necessary to obtain very
  accurate results both for 2D and 3D problems
• Our adaptive strategy is completely general and
  not specific to phase change problems

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Vertical interface
Mixed formulation
One-equation formulation
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Vertical interface
Mixed formulation
One-equation formulation
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• Solution analytique oscillation du cercle

• Nouvelle séquence dadaptation
• Raffinement des arêtes,
• Déraffinement des arêtes,
• Retournement des arêtes,
• Déplacement des sommets.
• Une seule adaptation pour chaque pas de temps.
• On adapte le maillage suivant